There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`
 - Each person has a unique birthday month: `april`, `sept`, `jan`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - People have unique heights: `average`, `short`, `very short`
 - They all have a unique favorite flower: `lilies`, `carnations`, `daffodils`

## Clues:
1. The person who is very short is the person who prefers city breaks.
2. The person whose birthday is in January is Peter.
3. The person whose birthday is in January is directly left of the person who loves a carnations arrangement.
4. The person who loves beach vacations is somewhere to the right of the person who loves a bouquet of daffodils.
5. Eric is the person who loves basketball.
6. The person who is short is the person whose birthday is in April.
7. The person who enjoys mountain retreats is the person who loves a carnations arrangement.
8. Eric is the person who is short.
9. The person who loves tennis is the person who loves a carnations arrangement.
 What is the value of attribute FavoriteSport for the person whose attribute Height is very short? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Alice`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `dog`, `cat`, `fish`, `bird`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `huawei p50`, `google pixel 6`, `samsung galaxy s21`

## Clues:
1. Bob is not in the fourth house.
2. The person who uses an iPhone 13 is somewhere to the left of the person who uses a OnePlus 9.
3. The person who uses a Huawei P50 is somewhere to the left of Alice.
4. Arnold is the dog owner.
5. Peter is somewhere to the left of the cat lover.
6. Arnold is not in the fourth house.
7. Peter is somewhere to the left of the fish enthusiast.
8. Peter is the person who uses a Huawei P50.
9. The person who uses a OnePlus 9 is directly left of the person who uses a Google Pixel 6.
10. The dog owner is somewhere to the right of the person who keeps horses.
11. The person who keeps horses is not in the first house.
12. There is one house between Bob and the person who uses an iPhone 13.
13. Eric is the person who uses a Google Pixel 6.
14. The person who uses a Google Pixel 6 is somewhere to the right of the fish enthusiast.
 What is the value of attribute House for the person whose attribute Animal is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Peter`, `Eric`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `roses`, `lilies`, `iris`, `tulips`, `daffodils`, `carnations`

## Clues:
1. Bob is somewhere to the right of the person who loves a carnations arrangement.
2. The person who loves the boquet of iris is in the fifth house.
3. Peter is directly left of the person who loves the boquet of lilies.
4. There is one house between the person who loves the rose bouquet and Bob.
5. Alice is in the fourth house.
6. The person who loves the vase of tulips is Peter.
7. Arnold is in the second house.
8. Alice is directly left of Carol.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$.  A polynomial $P$ of degree $2007$ has the property that
\[P\left(r_j+\dfrac{1}{r_j}\right)=0\]for $j=1,\ldots,2007$.  Determine the value of $\frac{P(1)}{P(-1)}$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 548


Given a triangle $ABC$ with angles $\angle A = 60^{\circ}$, $\angle B = 75^{\circ}$, $\angle C = 45^{\circ}$, let $H$ be its orthocenter, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\frac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.

Answer: 1132


Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.

Answer: 43


You have two blackboards, $A$ and $B$. You need to write some integers on them, where each integer is greater than or equal to $2$ and less than or equal to $20$. The condition is that each number on blackboard $A$ must be co-prime with each number on blackboard $B$. Determine the maximum possible value of the product of the number of integers written on $A$ and the number of integers written on $B$.

Answer: 65


Three students write on the blackboard three two-digit square numbers next to each other. In the end, they observe that the resulting 6-digit number is also a perfect square. Find this number!

Answer: 166464


The following figure shows a walk of length 6:

```asy
unitsize(20);
for (int x = -5; x <= 5; ++x)
  for (int y = 0; y <= 5; ++y)
    dot((x, y));
label("$O$", (0, 0), S);
draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3));
```

This walk has three interesting properties:

- It starts at the origin, labeled $O$.
- Each step is 1 unit north, east, or west. There are no south steps.
- The walk never comes back to a point it has been to.

Let's call a walk with these three properties a northern walk. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

Answer: 239


Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged, it is possible to travel between any two cities in Graphistan using only GA flights, provided that the absolute value of the difference between the number of flights originating from and terminating at any city is not more than $k$.

Answer: 1005


Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\frac{1}{\sqrt[4]{3}}$. What is the value of $900p$, rounded to the nearest integer?

Answer: 628


We wish to find the sum of $40$ given numbers using $40$ processors. Initially, each processor displays the number $0$ on its screen. Each processor can add the number on its screen with a number entered directly (only the given numbers can be entered directly into the processors) or transferred from another processor in a unit time. Whenever a number is transferred from one processor to another, the former processor resets to zero. Determine the minimum time required to compute the desired sum.

Answer: 7


Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$. For each $i$, where $1 \leq i \leq 200$, calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. Determine the sum of the distinct values of $g_i$.

Answer: 25


Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$, there exist at least $k$ consecutive composite integers in the set \{$n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$\}.

Answer: 7


In trapezoid $ABCD$, $AB < CD$, $AB \perp BC$, $AB \parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD \perp DC$. If
\[\frac{QP}{AP} + \frac{AP}{QP} = \left( \frac{51}{14} \right)^4 - 2,\]
then $\frac{BP}{AP} - \frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$.

Answer: 61


Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Answer: 95


Let $\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Assume that $B$ and $C$ lie on the same side of $l$, and that the distance from $B$ to $l$ is twice the distance from $C$ to $l$. Suppose also that the length $BA$ is twice that of $CA$. If $\triangle ABC$ has integer side lengths and is as small as possible, what is $AB^2+BC^2+CA^2$?

Answer: 61


Let $a$ be a positive number. Find the number of integer solutions $x$ of the inequality $2 < xa < 3$, given that the inequality $1 < xa < 2$ has exactly 3 integer solutions. Consider all possible cases.

Answer: 3


Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds.
[center]![Image](http://i.imgur.com/WNp59XW.png)[/center]

Answer: \frac{1}{3}


Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the  $\emph{last}$  forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?

Answer: 21


Let  $R$  be the set of points  $(x, y)$  such that  $x$  and  $y$  are positive,  $x + y$  is at most 2013, and
\[
  \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set  $R$ .  Recall that  $\lfloor a \rfloor$  is the greatest integer that is less than or equal to  $a$ , and  $\lceil a \rceil$  is the least integer that is greater than or equal to  $a$ .

Answer: 1006.5


There is a regular  $17$ -gon  $\mathcal{P}$  and its circumcircle  $\mathcal{Y}$  on the plane.
The vertices of  $\mathcal{P}$  are coloured in such a way that  $A,B \in \mathcal{P}$  are of different colour, if the shorter arc connecting  $A$  and  $B$  on  $\mathcal{Y}$  has  $2^k+1$  vertices, for some  $k \in \mathbb{N},$  including  $A$  and  $B.$
What is the least number of colours which suffices?

Answer: 4


On a board, there are  $n$  equations in the form  $*x^2+*x+*$ . Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After  $3n$  moves, there will be  $n$  quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?

Answer: 0


Find the smallest positive integer  $k$  such that  $ \underbrace{11\cdots 11}_{k\text{ 1's}}$  is divisible by  $9999$ .

Let  $T = TNYWR$ . Circles  $\omega_1$  and  $\omega_2$  intersect at  $P$  and  $Q$ .  The common external tangent  $\ell$  to the two circles closer to  $Q$  touches  $\omega_1$  and  $\omega_2$  at  $A$  and  $B$  respectively.  Line  $AQ$  intersects  $\omega_2$  at  $X$  while  $BQ$  intersects  $\omega_1$  again at  $Y$ .  Let  $M$  and  $N$  denote the midpoints of  $\overline{AY}$  and  $\overline{BX}$ , also respectively.  If  $AQ=\sqrt{T}$ ,  $BQ=7$ , and  $AB=8$ , then find the length of  $MN$ .

Answer: \frac{128}{21}


In a class with  $23$  students, each pair of students have watched a movie together. Let the set of movies watched by a student be his *movie collection*. If every student has watched every movie at most once, at least how many different movie collections can these students have?

Answer: 23


You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with  $3$  buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially  $3$  doors are closed and  $3$  mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?

Answer: 9


Each person stands on a whole number on the number line from  $0$  to  $2022$   . In each turn, two people are selected by a distance of at least  $2$ . These go towards each other by  $1$ . When no more such moves are possible, the process ends.
Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.)

*(Birgit Vera Schmidt)*

original wordingBei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person.
In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang.
Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)

Answer: 1011


In square  $ABCD$  with  $AB = 10$ , point  $P, Q$  are chosen on side  $CD$  and  $AD$  respectively such that  $BQ \perp AP,$  and  $R$  lies on  $CD$  such that  $RQ \parallel PA.$   $BC$  and  $AP$  intersect at  $X,$  and  $XQ$  intersects the circumcircle of  $PQD$  at  $Y$ . Given that  $\angle PYR = 105^{\circ},$   $AQ$  can be expressed in simplest radical form as  $b\sqrt{c}-a$  where  $a, b, c$  are positive integers. Find  $a+b+c.$

Answer: 23


The coordinate of  $ P$  at time  $ t$ , moving on a plane, is expressed by  $ x = f(t) = \cos 2t + t\sin 2t,\ y = g(t) = \sin 2t - t\cos 2t$ .

(1) Find the acceleration vector  $ \overrightarrow{\alpha}$  of  $ P$  at time  $ t$  .

(2) Let  $ L$  denote the line passing through the point  $ P$  for the time  $ t%Error. "neqo" is a bad command.
$ , which is parallel to the acceleration vector  $ \overrightarrow{\alpha}$  at the time. Prove that  $ L$  always touches to the unit circle with center the origin, then find the point of tangency  $ Q$ .

(3) Prove that  $ f(t)$  decreases in the interval  $ 0\leq t \leqq \frac {\pi}{2}$ .

(4) When  $ t$  varies in the range  $ \frac {\pi}{4}\leq t\leq \frac {\pi}{2}$ , find the area  $ S$  of the figure formed by moving the line segment  $ PQ$ .

Answer: \frac{7\pi^3}{192}


Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and subtracts 1, writing this second result on the same piece of paper. Malcolm then doubles the second integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm writes down are prime, determine all possible values for the first integer.

Answer: 23


Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.

Answer: 13


Last year, the U.S. House of Representatives passed a bill which would make Washington, D.C. into the $51$st state. Naturally, the mathematicians are upset that Congress won’t prioritize mathematical interest of flag design in choosing how many U.S. states there should be. Suppose the U.S. flag must contain, as it does now, stars arranged in rows alternating between $n$ and $n - 1$ stars, starting and ending with rows of n stars, where $n \ge 2$ is some integer and the flag has more than one row. What is the minimum number of states that the U.S. would need to contain so that there are at least three different ways, excluding rotations, to arrange the stars on the flag?

Answer: 53


Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

Answer: 289


Consider a regular $2020$-gon circumscribed into a circle of radius $ 1$. Given three vertices of this polygon such that they form an isosceles triangle, let $X$ be the expected area of the isosceles triangle they create. $X$ can be written as $\frac{1}{m \tan((2\pi)/n)}$ where $m$ and $n$ are integers. Compute $m + n$.

Answer: 5049


Consider a $2021$-by-$2021$ board of unit squares. For some integer $k$, we say the board is tiled by $k$-by-$k$ squares if it is completely covered by (possibly overlapping) $k$-by-$k$ squares with their corners on the corners of the unit squares. What is the largest integer k such that the minimum number of $k$-by-$k$ squares needed to tile the $2021$-by-$2021$ board is exactly equal to $100$?

Answer: 224


There are n lilypads in a row labeled $1, 2, \dots, n$ from left to right. Fareniss the Frog picks a lilypad at random to start on, and every second she jumps to an adjacent lilypad; if there are two such lilypads, she is twice as likely to jump to the right as to the left. After some finite number of seconds, there exists two lilypads $A$ and $B$ such that Fareniss is more than $1000$ times as likely to be on $A$ as she is to be on $B$. What is the minimal number of lilypads $n$ such that this situation must occur?

Answer: 12


Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p) = h(x)(x-p)^2$ for all x \in R. Now, consider the polynomial $$f(x) = 12x^5 - 15x^4 - 40x^3 + 540x^2 - 2160x + 1,$$ and suppose that it’s double attractors are $a_1, a_2, ... , a_n$. If the sum $\sum^{n}_{i=1}|a_i|$ can be written as $\sqrt{a} +\sqrt{b}$, where $a, b$ are positive integers, find $a + b$.

Answer: 49


$ABC$ is a triangle where $AB = 10$, $BC = 14$, and $AC = 16$. Let $DEF$ be the triangle with smallest area so that $DE$ is parallel to $AB$, $EF$ is parallel to $BC$, $DF$ is parallel to $AC$, and the circumcircle of $ABC$ is $DEF$’s inscribed circle. Line $DA$ meets the circumcircle of $ABC$ again at a point $X$. Find $AX^2$
.

Answer: 196


Let $f(n) = \sum^n_{i=1}\frac{gcd(i,n)}{n}$. Find the sum of all positive integers $ n$ for which $f(n) = 6$.

Answer: 1192


The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$  has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$  for relatively prime positive integers $p, q$, find $p + q$.

Answer: 31


Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b  \times 5^c  \times 7^d  \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.

Answer: 31


Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for $n$ minutes once he arrives at the airport before going through security. What is the smallest $n$ they can select such that they meet with at least 50% probability? The answer will be of the form $a + b\sqrt{c}$ for integers $a$, $b$, and $c$, where $c$ has no perfect square factor other than $1$. Report $a + b + c.$

Answer: 67


A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.

Answer: 701


Alex is a content creator who spends 3 hours each day creating informative and educational videos on technology. During one week, Alex plans to create a special series of videos about the history of computers. Each video in the series takes twice as long to produce as a regular video. If Alex dedicates 2 days of the week exclusively to creating this special series, how many regular videos can Alex produce in the remaining days of the week? Note: Assume Alex works every day of the week and only works on one type of video per day.

Answer: 5


In the Maldives, a rival politician is organizing rallies to gather support for an upcoming election. The politician plans to visit 5 different islands, with each island requiring a boat trip. On the first island, he expects 150 people to attend his rally, and he plans to increase the number of attendees by 20% on each subsequent island. If the politician visits all 5 islands, how many people in total does he expect to attend his rallies?

Answer: 1116


Mr. Thompson, a parent who believes that teachers should focus solely on academic subjects, is organizing a math-focused weekend event for students. He decides to create a workbook with a series of math exercises. For every 5 math problems he writes, he takes a 10-minute break. If he spends a total of 3 hours working on the workbook, how many math problems does Mr. Thompson write, assuming he only takes breaks during this time?

Answer: 45


Alex is a customer service manager at an online money transferring platform. In one day, Alex received 40 customer complaints regarding failed transactions. On average, each complaint takes 15 minutes to handle. If Alex works an 8-hour shift, how many minutes will Alex have left after addressing all the complaints for that day?

Answer: -120


A best-selling author known for weaving political allegories into their novels has just released their latest book. In the first week, the book sold 7,500 copies. The author's previous book, which also contained political themes, sold 5,200 copies in its first week. By the end of the second week, the new book's sales increased by 60% compared to its first-week sales. How many more copies has the new book sold than the previous book by the end of the second week?

Answer: 6800


The artisan carpenter, known for his exquisite traditional woodwork, has a new project to create handcrafted wooden chairs for a local community center. He plans to make 15 chairs, and each chair requires 3 wooden planks. However, he discovered that 4 of his original wooden planks have knots that make them unusable. Fortunately, he found 2 additional planks from an old project. How many more wooden planks does the carpenter need to complete all 15 chairs without having to resort to using any PVC-U products, which he prefers to avoid?

Answer: 2


Melody is an up-and-coming R&B artist inspired by H.E.R. She has been practicing her guitar and songwriting skills. This week, she wrote 3 songs on Monday, 2 songs on Wednesday, and 4 songs on Friday. Each song takes her about 2 hours to write. She also practices her guitar for 1 hour every day. How many total hours did Melody spend on writing songs and practicing her guitar this week?

Answer: 25


An ambitious young woman named Aisha grew up in the small town of Allipur and recently moved to a large city to attend university. In Allipur, Aisha used to walk 2 kilometers to her local library every week. Now, in the city, the university library is 5 kilometers from her apartment, but she can use a bicycle to travel.

Aisha wants to know how much more distance she covers each month if she visits the library twice a week in the city compared to her weekly visits in Allipur. Calculate the additional distance Aisha travels in a month (consider a month to have 4 weeks) now that she is in the city.

How many more kilometers does Aisha travel each month in the city compared to Allipur?

Answer: 64


The lead actor in the school plays is rehearsing for a new performance. In preparation for the audition for a scholarship at a prestigious performing arts university, he decides to practice 4 different monologues. Each monologue takes 15 minutes to perform. After each monologue, he spends 5 minutes reviewing his performance notes. If he takes a 10-minute break after practicing two monologues, how long will it take him to complete his practice session, including all performances, reviews, and the break?

Answer: 90


The owner of a Cavalier King Charles Spaniel rescue organization recently took in 12 new dogs that need to be rehomed. Last month, they successfully found new homes for 8 dogs. This month, they have already found homes for 5 dogs and plan to continue their efforts. If they manage to find homes for 6 more dogs by the end of this month, how many Cavalier King Charles Spaniels will still be waiting to be adopted?

Answer: 1


As an active community member in Loudoun County, Virginia, you are organizing a fundraiser to support a local school. You decide to sell homemade apple pies at the community fair. You bake 45 pies, and each pie costs $12 to make. You plan to sell each pie for $20. If you sell 30 pies at the fair, how much profit do you make from selling these pies?

Answer: 240


Mrs. Thompson, a retired school teacher who has lived in the bustling city for over 30 years, is advising a new teacher, Ms. Lopez, on how to manage her time effectively in the urban environment. Mrs. Thompson suggests that Ms. Lopez should allocate her weekdays as follows: 2 hours for commuting, 6 hours for teaching, 1 hour for preparing lessons, 2 hours for grading assignments, and 1 hour for personal errands. On weekends, she recommends spending 3 hours on lesson planning and 2 hours on exploring the city to unwind.

If Ms. Lopez follows this schedule, how many hours in total will she spend on teaching-related activities and city exploration over the course of a week?

Answer: 55


10.238. The larger base of the trapezoid is twice as large as its smaller base. A line parallel to the bases is drawn through the point of intersection of the diagonals. Find the ratio of the height of each of the two resulting trapezoids to the height of the given trapezoid.

Answer: \dfrac{1}{3}


Problem 9.2. During the first half of the year, lazy Pasha forced himself to solve math problems. Every day he solved no more than 10 problems, and if on any day he solved more than 7 problems, then for the next two days he solved no more than 5 problems per day. What is the maximum number of problems Pasha could solve in 7 consecutive days?

Answer: 52


4. As shown in the figure: A table can seat 6 people, two tables put together can seat 10 people, three tables put together can seat 14 people. In this way, 10 tables arranged in two rows, with 5 tables in each row, can seat $\qquad$ people.

Answer: 44


[ Arithmetic. Mental calculation, etc. ]
$[$ Common fractions $\quad]$

After Natasha ate half of the peaches from the jar, the level of the compote decreased by one third. By what part (of the resulting level) will the level of the compote decrease if half of the remaining peaches are eaten?

Answer: \dfrac{1}{4}


In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called [b]uniform[/b] if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?

Answer: 40


5. To the Mechanics and Mathematics Faculty! Several identical books and identical albums were bought. The books cost 10 rubles 56 kopecks. How many books were bought if the price of one book is more than one ruble higher than the price of an album, and 6 more books were bought than albums?

Answer: 8


\section*{Problem 3 - 151033}

When printing a math problem, instead of \(\left(1+a^{2} x^{2}\right): x^{2}=b\) (with given numbers \(a, b\)), the equation \(\left(1+a^{2} x^{2}\right) \cdot x^{2}=b\) (with the same numbers \(a, b\)) was printed by mistake.

Despite this, the resulting equation had the same non-empty solution set as the originally intended equation.

Determine this solution set!

Answer: \{-1, 1\}


7th Iberoamerican 1992 Problem B1 Let a n , b n be two sequences of integers such that: (1) a 0 = 0, b 0 = 8; (2) a n+2 = 2 a n+1 - a n + 2, b n+2 = 2 b n+1 - b n , (3) a n 2 + b n 2 is a square for n > 0. Find at least two possible values for (a 1992 , b 1992 ).

Answer: (3976032, 7976)


## Task 3 - 080623

Above the side $C D$ of a square $A B C D$ with $A B=4 \mathrm{~cm}$, an equilateral triangle $\triangle D C E$ is to be constructed such that the square and the triangle share the side $C D$.

The point $E$ of the triangle $\triangle D C E$ is to be located outside the square $A B C D$. Connect $E$ with $A$ and with $B$!

Calculate the size of the angle $\angle A E B$!

Answer: 30^\circ


1665. It is known that the density of the distribution of a random variable $X$ has the form

$$
p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{x^{2}}{2 \sigma^{2}}}
$$

Find the density of the distribution of the random variable $Y=X^{2}$.

Answer: \dfrac{1}{\sigma \sqrt{2 \pi y}} e^{-\dfrac{y}{2 \sigma^2}}


All angles of $ABC$ are in $(30,90)$. Circumcenter of $ABC$ is $O$ and circumradius is $R$. Point $K$ is projection of $O$ to angle bisector of $\angle B$, point $M$ is midpoint $AC$. It is known, that $2KM=R$. Find $\angle B$

Answer: 60^\circ


## Task A-3.8. (10 points)

A right trapezoid is given, in which a circle can be inscribed. If the distances from the center of the inscribed circle to the ends of the longer leg are $15 \mathrm{~cm}$ and $20 \mathrm{~cm}$, what is the area of the trapezoid?

Answer: 588


A magician has a hundred cards, numbered from 1 to 100. He places each of them into one of three boxes - a red box, a white box, and a blue box - in such a way that each box contains at least one card.

A member of the audience selects two of the three boxes and draws one card from each, then announces the sum of the numbers on the drawn cards. Knowing this sum, the magician can tell which is the box from which no card was drawn.

In how many ways can the cards be placed in the boxes so that this trick always works? (Two arrangements are considered different if there is at least one card that ends up in a different box.)

Answer: 12


Example 7 A backpack contains 12 cards labeled $1,1,2,2, \cdots, 6,6$. A person takes one card at a time from it, without putting it back; if he draws a card with the same label as one he has already drawn, he discards both cards; if he has 3 single cards in his hand or all the cards in the backpack have been taken, the operation ends. Find the probability that all the cards in the backpack are taken.

Answer: \dfrac{9}{385}


10. Convex quadrilateral $M A T H$ is given with $H M / M T=3 / 4$, and $\angle A T M=\angle M A T=$ $\angle A H M=60^{\circ}$. $N$ is the midpoint of $M A$, and $O$ is a point on $T H$ such that lines $M T, A H, N O$ are concurrent. Find the ratio $H O / O T$.

Answer: \dfrac{9}{16}


3. The area of triangle $\triangle A B C$ is 10 cm $^{2}$. What is the smallest value in centimeters that the circumference of the circle circumscribed around triangle $\triangle A B C$ can take, given that the midpoints of the heights of this triangle lie on the same line? If the answer is not an integer, round it to the nearest integer.

Answer: 20


Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the number of joke sequences in terms of $n$.

Answer: 4n^3


## Task B-1.1.

Determine all integers $a$ for which $4 a^{2}-24 a-45$ is a prime number.

Answer: -2


For positive integer $n$ , let $s(n)$ denote the sum of the digits of $n$ .  Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$ .
Please give the answer directly without any intermediate steps.

Answer: 695


158*. Using the digits from 1 to 9 once each, form the smallest nine-digit number that is divisible by 11.

Answer: 123475869


II. (50 points)
Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$. Determine the maximum value of $P = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$.

Answer: \dfrac{10}{3}


 Let $ a_1,a_2, \cdots ,a_{2015} $ be $2015$-tuples of positive integers (not necessary distinct) and let $ k $ be a positive integers. Denote $\displaystyle f(i)=a_i+\frac{a_1a_2 \cdots a_{2015}}{a_i} $.

a) Prove that if $ k=2015^{2015} $, there exist $ a_1, a_2, \cdots , a_{2015} $ such that $ f(i)= k $ for all $1\le i\le 2015 $.\\
b) Find the maximum $k_0$ so that for $k\le k_0$, there are no $k$ such that there are at least $ 2 $ different $2015$-tuples which fulfill the above condition.

Answer: 2


The sum of Mária's and Anna's ages is 44 years. Mária is twice as old as Anna was when Mária was half as old as Anna will be when Anna is three times as old as Mária was when Mária was three times as old as Anna. How old are Mária and Anna?

Answer: 27.5


A six-digit square number, when divided into two-digit segments, reveals that the last two-digit number is the same as the middle two-digit number, while the first two-digit number completes the middle two-digit number to 100. Which is this square number?

Answer: 316969


6. Find the largest ten-digit number of the form $\overline{a_{9} a_{8} a_{7} a_{6} a_{5} a_{4} a_{3} a_{2} a_{1} a_{0}}$, possessing the following property: the digit equal to $\mathrm{a}_{\mathrm{i}}$ appears in its representation exactly $\mathrm{a}_{9-\mathrm{i}}$ times (for example, the digit equal to $\mathrm{a}_{2}$ appears exactly $\mathrm{a}_{7}$ times).

ANSWER 8888228888.

Answer: 8888228888


## Task 2 - 201232

Let $f$ be the function defined by

$$
f(x)=x^{4}-(x+1)^{4}-(x+2)^{4}+(x+3)^{4}
$$

where the domain of $f$ is

a) the set of all integers,

b) the set of all real numbers.

Investigate, for both case a) and case b), whether the function $f$ attains a minimum value, and determine, if applicable, this minimum value in each case.

Answer: 16


Given a trapezoid \( ABCD \) with bases \( AD \) and \( BC \). The angle bisectors at vertices \( A \) and \( B \) intersect at point \( M \), and the angle bisectors at vertices \( C \) and \( D \) intersect at point \( N \). Find \( MN \), given that \( AB = a \), \( BC = b \), \( CD = c \), and \( AD = d \).

Answer: \dfrac{|b + d - a - c|}{2}


6. 105 Let $S$ be the unit circle in the complex plane (i.e., the set of complex numbers with modulus 1), and $f$ be a mapping from $S$ to $S$. For any $z \in S$, define
$$\begin{array}{l}
f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \cdots, \\
f^{(k)}(z)=\underbrace{f(f(\cdots(f}_{k \uparrow f}(z) \cdots))),
\end{array}$$

If $c \in S$, such that
$$\begin{array}{l}
f^{(1)}(C) \neq C, f^{(2)}(C) \neq C, \cdots \\
f^{(n-1)}(C) \neq C, f^{(n)}(C)=C .
\end{array}$$

we say that $C$ is an $n$-periodic point of $f$.
Let $m$ be a natural number greater than 1, and $f$ is defined as follows:
$$f(z)=z^{m}(z \in S)$$

Try to calculate the total number of 1989-periodic points of $f$.

Answer: m^{1989} - m^{663} - m^{153} - m^{117} + m^{51} + m^{39} + m^9 - m^3


5. As shown in Figure 1, the vertices of $\triangle O A B$ are $O(0,0)$, $A(2,1)$, and $B(10,1)$. The line $C D \perp x$-axis and bisects the area of $\triangle O A B$. If the coordinates of point $D$ are $(x, 0)$, then the value of $x$ is . $\qquad$

Answer: 10 - 2\sqrt{10}


4. On graph paper, a stepped right-angled triangle with legs equal to 6 cells (see fig.) was drawn. Then all the grid lines inside the triangle were outlined. What is the maximum number of rectangles that can be found in this drawing?

Answer: 126


In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more.
Find maximum $d$ for which we can guaranteed paint at least $d$ points.

Answer: 9744


[ Problems on percentages and ratios ]

Little Man and Karlson together ate a jar of jam. In this case, Karlson ate $40 \%$ fewer spoons of jam than Little Man, but on the other hand, his spoon held $150 \%$ more jam than Little Man's spoon. What part of the jar of jam did Karlson eat?

Answer: \dfrac{3}{5}


6. (15 points) Every day, Ivan Ivanovich is taken to work by a company car. One day, Ivan Ivanovich decided to walk and left the house an hour earlier than usual. On the way, he met the company car and finished the journey in it. As a result, he arrived at work 10 minutes earlier than the usual time. How long did Ivan Ivanovich walk?

Answer: 55


A sequence of numbers starts with $1$ $2$ , and $3$ . The fourth number of the sequence is the sum of the previous three numbers in the sequence: $1+2+3=6$ . In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
$\textbf{(A)}\  11  \qquad \textbf{(B)}\   20  \qquad \textbf{(C)}\   37  \qquad \textbf{(D)}\   68  \qquad \textbf{(E)}\    99$

Answer: 68


3. Set $A=\left\{x \mid \sqrt{a^{2}+4 x-x^{2}}>a(x+1)\right\}, B=$ $\left\{\left.x|| x-\frac{1}{2} \right\rvert\,<\frac{1}{2}\right\}$. When $A \subseteq B$, the range of values for $a$ is

Answer: [1, \sqrt{2}]


## Task 5 - 240735

In the schema 43_1_5_, each of the blanks _ is to be filled with a digit so that the resulting seven-digit number is divisible by 75.

State how many seven-digit numbers can arise in this way!

Answer: 33


4. In Rt $\triangle A B C$, $\angle A=90^{\circ}, A D \perp B C$ at $D$, the angle bisector of $\angle B$ intersects $A D$ at $M$, and intersects $A C$ at $N$, and $A B-A M=\sqrt{3}-1, B M \cdot B N=2$. Then $\frac{B C-A B}{C N}=$ $\qquad$

Answer: \dfrac{\sqrt{3}}{2}


2. Given that when $0 \leqslant \theta \leqslant \frac{\pi}{2}$, it always holds that $\cos ^{2} \theta+2 m \sin \theta-2 m-2<0$. Then the range of real number $m$ is $\qquad$.

Answer: (-\frac{1}{2}, +\infty)


Consider the set $\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$ .

Prove that there is no function in this function space that gives us the minimum of  $S=\int_{-1}^1x^2f'(x)^2dx$ . What is the infimum of  $S$  for the functions of this space?

Answer: 0


Pлотников M.

Let $T_{1}, T_{2}$ be the points of tangency of the excircles of triangle $ABC$ with sides $BC$ and $AC$ respectively. It turns out that the point symmetric to the center of the inscribed circle of the triangle with respect to the midpoint of $AB$ lies on the circumcircle of triangle $C T_{1} T_{2}$. Find the angle $BCA$.

#

Answer: 90^\circ


Let $T$ be the answer to question $18$. Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$. Point $B$ lies on segment $ZO$, $O'$ lies on segment $OM$, and $E$ lies on segment $RM$ such that $BR = BE = EO'$, and $\angle BEO' = 90^o$. Compute $2(ZO + O'M + ER)$.

PS. You had better calculate it in terms of $T$.

Answer: 7T


For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.

Answer: 5


1. How many irreducible fractions exist where the sum of the numerator and denominator is 100?

Answer: 20


Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. What could be the perimeter of the new figure?

Answer: 18


To make the statement $(6\ ?\ 3) + 4 - (2 - 1) = 5$ true, replace the question mark with the correct arithmetic operation. Convert the correct arithmetic operation to its ASCII code value and provide this value as your answer.

Answer: 47


ABC is a triangle. Show that, if the point P inside the triangle is such that the triangles PAB, PBC, PCA have equal area, then P must be the centroid.

Answer: P \text{ is the centroid of } \triangle ABC


12. There are two two-digit numbers, their difference is 58, and the last two digits of their square numbers are the same. Then these two two-digit numbers are $\qquad$ .

Answer: 21


1. A butcher cuts a sausage into pieces. First, he cuts off a third of the whole sausage. The two resulting pieces he also cuts: from the larger piece, he cuts off a fifth, and from the smaller piece, he cuts off a quarter. What is the length of the smallest piece of the sausage if the length of the largest piece is $48 \mathrm{~cm}$?

Answer: \dfrac{15}{2}


We shuffle a deck of 52 French playing cards, and then draw one by one from the pile until we find an ace of black color. In what position is it most likely for the first black ace to appear?

Answer: 1


## 38. How old is the eldest brother?

Determine this yourself, if it is known that the age of the middle brother is equal to the product of the ages of his two brothers, that the sum of the ages of all three brothers is 35, while the sum of the decimal logarithms of their ages is 3.

Answer: 20


10. (20 points) Let the line $y=2 \sqrt{2} x+2 \sqrt{3}$ intersect the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at points $A$ and $B$, with the origin of coordinates being $O$. If $\angle A O B=90^{\circ}$ and $A B=\frac{12 \sqrt{11}}{17}$, find the equation of the ellipse.

Answer: \dfrac{x^2}{4} + \dfrac{y^2}{2} = 1


6 Given $\odot O_{1}\left(r_{1}\right), \odot O_{2}\left(r_{2}\right)$ are externally tangent to $\odot O(r)$ at $A_{1}, A_{2}$. When $r_{1}=1, r_{2}=2$, $r=3, A_{1} A_{2}=4$, find the length of the external common tangent $T_{1} T_{2}$ of $\odot O_{1}$ and $\odot O_{2}$.

Answer: \dfrac{8\sqrt{5}}{3}


2. Find the largest positive integer $n$, such that $n^{2}+2000 n$ is a perfect square.

Answer: 249001


8. Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}, F_{2}$, a line $l$ passing through the right focus intersects the ellipse at points $P, Q$. Then the maximum value of the area of the incircle of $\triangle F_{1} P Q$ is $\qquad$

Answer: \dfrac{9}{16}\pi


[

The ratio in which the bisector divides the side. [The ratio of the areas of triangles with a common base or common height]

In triangle $ABC$, the median $AD$ and the bisector $BE$ are perpendicular and intersect at point $F$. It is known that $S_{DEF}=5$. Find $S_{ABC}$.

#

Answer: 60


Example 11 (1988 National High School League Question) Team A and Team B each send out 7 players to participate in a Go chess competition in a pre-arranged order. Both sides start with the No. 1 player competing. The loser is eliminated, and the winner then competes with the No. 2 player of the losing side... until all players of one side are eliminated, and the other side wins, forming a competition process. The number of all possible competition processes is

Answer: 3432


3. Let $x_{1}$ and $x_{2}$ be the roots of the equation

$$
p^{2} x^{2}+p^{3} x+1=0
$$

Determine $p$ such that the expression $x_{1}^{4}+x_{2}^{4}$ has the smallest value.

Answer: \sqrt[8]{2}


You start with a big pile of \(3^{2004}\) cards, with the numbers \(1, 2, 3, \ldots, 3^{2004}\) written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these \(3^{2003}\) remaining cards into groups of three any way you like, and in each group, keep the card with the smallest number and discard the other two. You now have \(3^{2002}\) cards, and you again arrange these into groups of three and keep the largest number in each group. You proceed in this manner, alternating between keeping the largest number and keeping the smallest number in each group, until you have just one card left.

How many different values are possible for the number on this final card?

Answer: 3^{2004} - 2^{2004}


8. Let $a$ and $b$ be constants, and $b<0$. The graph of the parabola $y=a x^{2}+b x+a^{2}+\sqrt{2} a-4$ is one of the four graphs in Figure 1. Then $a=$ $\qquad$ .

Answer: \sqrt{2}


4. Among the natural numbers from 2021 to 9998, the numbers with the same tens and units digits are $\qquad$ in total.

Answer: 797


Non-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\angle ABC = \angle BCD = \angle CDA$. Given that $AB = 3$ and $B \neq D$, determine the number of possible lengths for $BC$.

Answer: 8


Call an ordered triple $(a, b, c)$ of integers feral if $b - a$, $c - a$, and $c - b$ are all prime numbers. Find the number of feral triples where $1 \leq a < b < c \leq 20$.

Answer: 72


There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?

Answer: 15


Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.

Answer: 118


Answer: 6723


Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, find the area of $\triangle DBE$.

Answer: 200


Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3 \times 3 \times 3$ cube such that each $1 \times 1 \times 3$ row or column contains exactly one white cube. Note that rotations are considered \textit{distinct}.

Answer: 12


Ali defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } | \text{ THEN } | \text{ A}$, $\text{ATH } | \text{ E } | \text{ NA}$, $\text{ATHENA}$, and $\text{AT } | \text{ HEN } | \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.)

Answer: 45


Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not intersect circle $R$ intersects circle $P$ at point $P_Q$ and circle $Q$ at point $Q_P$. Points $Q_R$, $R_Q$, $R_P$, and $P_R$ are defined similarly.

If the radius of circle $Q$ is $4$ and the line segment $\overline{Q_PP_Q}$ is parallel to $\overline{R_QQ_R}$, compute the length of $R_PP_R$.

Answer: 8


In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
 label("$A$", (0,23), NW);
 label("$B$", (23, 23), NE);
 label("$C$", (23,0), SE);
 label("$D$", (0,0), SW);
 draw((0,6)--(23,6));
 draw((0,19)--(23,19));
 draw((5,0)--(5,23));
 draw((12,0)--(12,23));
 label("13", (17/2, 21));
 label("111",(35/2,25/2));
 label("37",(17/2,3));
 label("123",(2.5,12.5));
[/asy]

Answer: 180


A set of  $8$  problems was prepared for an examination. Each student was given  $3$  of them. No two students received more than one common problem. What is the largest possible number of students?

Answer: 8


There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers.

Find the smallest positive integer  $n{}$  such that regardless of the values of the initial 100 numbers, after  $n{}$  steps Alex can arrange to have on the board only pairwise coprime numbers.

Answer: 99


There are 12 people such that for every person A and person B there exists a person C that is a friend to both of them.  Determine the minimum number of pairs of friends and construct a graph where the edges represent friendships.

Answer: 20


- A power grid with the shape of a  $3\times 3$  lattice with  $16$  nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
- Previous problem for the grid of  $5\times 5$  lattice.

Answer: 35


Let  $l,\ m$  be the tangent lines passing through the point  $A(a,\ a-1)$  on the line  $y=x-1$  and touch the parabola  $y=x^2$ .
Note that the slope of  $l$  is greater than that of  $m$ .

(1) Exress the slope of  $l$  in terms of  $a$ .

(2) Denote  $P,\ Q$  be the points of tangency of the lines  $l,\ m$  and the parabola  $y=x^2$ .
Find the minimum area of the part bounded by the line segment  $PQ$  and the parabola  $y=x^2$ .

(3) Find the minimum distance between the parabola  $y=x^2$  and the line  $y=x-1$ .

Answer: \frac{3}{8}\sqrt{2}


For finite sets  $A,M$  such that  $A \subseteq M \subset \mathbb{Z}^+$ , we define   $$ f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}. $$   Given a positive integer  $k$ , we call  $M$  *k-colorable* if it is possible to color the subsets of  $M$  with  $k$  colors so that for any  $A \subseteq M$ , if  $f_M(A)\neq A$  then  $f_M(A)$  and  $A$  have different colors.
Determine the least positive integer  $k$  such that every finite set  $M \subset\mathbb{Z}^+$  is k-colorable.

Answer: 2


A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture).  Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is  $r$ . Find  $100r^2$ .
[asy]
// dragon96, replacing
// ![Image](http://i.imgur.com/08FbQs.png)
size(140); defaultpen(linewidth(.7));
real alpha=10, x=-0.12, y=0.025, r=1/sqrt(3);
path hex=rotate(alpha)*polygon(6);
pair A = shift(x,y)*(r*dir(330+alpha)), B = shift(x,y)*(r*dir(90+alpha)), C = shift(x,y)*(r*dir(210+alpha));
pair X = (-A.x, -A.y), Y = (-B.x, -B.y), Z = (-C.x, -C.y);
int i;
pair[] H;
for(i=0; i<6; i=i+1) {
H[i] = dir(alpha+60*i);}
fill(X--Y--Z--cycle, rgb(204,255,255));
fill(H[5]--Y--Z--H[0]--cycle^^H[2]--H[3]--X--cycle, rgb(203,153,255));
fill(H[1]--Z--X--H[2]--cycle^^H[4]--H[5]--Y--cycle, rgb(255,203,153));
fill(H[3]--X--Y--H[4]--cycle^^H[0]--H[1]--Z--cycle, rgb(153,203,255));
draw(hex^^X--Y--Z--cycle);
draw(H[1]--B--H[2]^^H[3]--C--H[4]^^H[5]--A--H[0]^^A--B--C--cycle, linewidth(0.6)+linetype("5 5"));
draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy]

Answer: 4


The diagram below is made up of a rectangle AGHB, an equilateral triangle AFG, a rectangle ADEF, and a parallelogram ABCD. Find the degree measure of ∠ABC. For diagram go to http://www.purplecomet.org/welcome/practice, the 2015 middle school contest, and go to #2

Answer: 60


Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.

Answer: 97


A rectangular storage bin measures  $10$  feet by  $12$  feet, is  $3$  feet tall, and sits on a flat plane.  A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground  $4$  feet away from the base of the storage bin as shown.  The number of cubic feet of dirt needed to form the pile can be written as   $m + n \pi$  where   $m$  and  $n$  are positive integers.  Find  $m + n.$

Answer: 280


**(a)**Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity.**(b)**What is the smallest area possible of pentagons with integral coordinates.

Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

Answer: \frac{3}{2}


Inside a circle with radius  $6$  lie four smaller circles with centres  $A,B,C$  and  $D$ . The circles touch each other as shown. The point where the circles with centres  $A$  and  $C$  touch each other is the centre of the big circle. Calculate the area of quadrilateral  $ABCD$ .
![Image](https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png)

Answer: 24


The kid cut out of grid paper with the side of the cell  $1$  rectangle along the grid  lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle.
- My rectangle ... - kid sobbed. - There is something strange about this figure!
- Nonsense, do not mention it -  Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of the rectangle was, and the area is the same as was the perimeter!
What size square did Carlson cut out?

Answer: 2


Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than  $11$ .
P.s. the current year in the problem is  $2010$ .

Answer: 24


Let  $M$  be the intersection point of the medians  $AD$  and  $BE$  of a right triangle  $ABC$  ( $\angle C=90^\circ$ ). It is known that the circumcircles of triangles  $AEM$  and  $CDM$  are tangent. Find the angle  $\angle BMC.$

Answer: 90


On an $11 × 11$ checker-board 22 squares are marked so that exactly two of the marked squares lie in each column and each row. Two arrangements of marked squares are considered equivalent if in any number of permutations of the columns and/or (independent) permutations of rows one arrangement can be obtained from the other. How many nonequivalent arrangements of marked squares are there?

Answer: 14


In the city of Fuchs, Ostap Bender organized a distribution of elephants to people. There were present 28 trades union members and 37 non-members. Ostap distributed the elephants so that the shares of all trades union members were equal, and the shares of non-members were also equal. It turned out that there existed only one such distribution (of all elephants). What greatest number of elephants could O. Bender have had?

Answer: 2072


Consider integers with the sum of their digits divisible by 7. What is the greatest difference between two consecutive such integers?

Answer: 13


12 people took part in a chess tournament. After the end of the tournament every participant made 12 lists. The first list consisted of the author; the second list – of the author and of those (s)he has beaten; and so on; the 12-th list consisted of all the people on the 11-th list and those they have beaten. It is known that the 12-th list of every participant contains a person who is not on the participant’s 11-th list. How many games ended in a draw?

Answer: 54


We mark point $O_1$ on a unit circle and, using $O_1$ as the center, we mark (by means of a compass) we mark point $O_2$ on the circle (clockwise starting with $O_1$). Using point $O_2$ as a new center, we repeat the procedure in the same direction with the same radius; and so on. After we had marked point $O_{1968}$ a circle is cut through each mark so we get 1968 arcs. How many different arc lengths can we thus procure?

Answer: 3


A straight line cuts a square piece of paper into two parts. Another straight line cuts one of the parts into two parts. One of the three pieces of paper obtained is again cut into two parts along a line, and so on. What least number of lines must be drawn in order to obtain 100 various (perhaps, (perhaps, identical) 20-gons?

Answer: 1699


We multiply the number $1234567 . . . 1000$ (juxtaposed are all natural numbers 1 to 1000) by a number from 1 to 9, and strike out all 1’s in the product. We multiply the number obtained by a nonzero one-digit number once again, and strike out the 1’s, and so on, many times over. What is the least number one can obtain in this manner?

Answer: 0


To guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either a day, or a night shift, or work around the clock. In these cases the guard is given a leave of absence of not less than $1, 1.5$ or 2.5 full days, respectively. What is the least number of guards necessary to ensure the security of the installation?

Answer: 4


A Parliament of $2000 MP$s decided to ratify the state budget of 200 expenditure items. Each $MP$ prepared a draft budget with what (s)he thinks the maximum possible allocation for each item so that the total expenditure does not exceed a given ceiling, S. For each item, the Parliament approves the maximum expenditure approved by not less than k $MP$s. What is the least value of k to ensure that the approved total does not exceed S?

Answer: 1991


In a $9 × 9$ square table, 9 cells are marked: those at the intersection of the 2-nd, 5-th and 8-th rows with the 2-nd, 5-th and 8-th column. In how many ways can one get from the lower left cell to the upper right one moving only upwards and to the right without entering marked cells?

Answer: 678


Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.
Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.

Answer: 50


Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$
Hint: Telescoping product.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 79


Find the maximum value of
$$
\int_{0}^{y} \sqrt{x^{4}+\left(y-y^{2}\right)^{2}} d x
$$
for $0 \leq y \leq 1$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 4


Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$  The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 79


The pairs of numbers $(a, b)$ such that $\left|a+b t+t^{2}\right| \leq 1$ for $0 \leq t \leq 1$ fill a certain region in the $(a, b)$-plane. What is the area of this region?The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 29


Express

$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$

as a rational number.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11


Let exp(t) denote e' and

$F( x) = \frac {x^{* }}{\exp ( x^{3}) }\int _{0}^{x}\int _{0}^{x- u}\exp ( u^{3}+ v^{3})$ $dv$ $du.$

Find lim $_{x\to\infty}F(x)$ or prove that it does not exist.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11


Let $E(n)$ denote the largest integer $k$ such that $5^k$ is an integral divisor of the product $1^12^23^3\cdots n^n.$ Calculate
$$\lim_{n\to\infty}\frac{E(n)}{n^{2}}$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 9


Let $C$ be a fixed unit circle in the Cartesian plane. For any convex polygon $P$ each of whose sides is tangent to $C$. let $N(P, h, k)$ be the number of points common to $P$ and the unit circle with center at ( $h, k$ ). Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P)$. Find the smallest number $u$ with
$$
\frac{1}{F(P)} \iint N(P, x, y) d x d y<u
$$
for all polygons $P$, where the double integral is taken over $H(P)$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11


[The ratio of the areas of triangles with a common base or common height $]$ Complex Trapezoids (other)

The area of trapezoid $A B C D$ with bases $A D$ and $B C (A D > B C)$ is 48, and the area of triangle $A O B$, where $O-$ is the point of intersection of the diagonals of the trapezoid, is 9. Find the ratio of the bases of the trapezoid $A D: B C$.

Answer: 3


31. For each number from 1 to 1000, all its natural divisors were listed (as a result, some numbers were listed multiple times). Determine which is greater: the sum of all listed numbers or one million?

Answer: 1000000


B6. Sixteen counters, which are black on one side and white on the other, are arranged in a 4 by 4 square. Initially all the counters are facing black side up. In one 'move', you must choose a 2 by 2 square within the square and turn all four counters over once.
Describe a sequence of 'moves' of minimum length that finishes with the colours of the counters of the 4 by 4 square alternating (as shown in the diagram).

Answer: 6


10.314. Given a rhombus $A B C D$, the diagonals of which are 3 and 4 cm. From the vertex of the obtuse angle $B$, two heights $B E$ and $B F$ are drawn. Calculate the area of the quadrilateral $B F D E$.

Answer: \dfrac{108}{25}


3. Let three positive integers $a$, $x$, $y$ greater than 100 satisfy $y^{2}-1=a^{2}\left(x^{2}-1\right)$.
Find the minimum value of $\frac{a}{x}$.

Answer: 2


# Problem 3.

In triangle $A B C$, the bisector $B E$ and the median $A D$ are equal and perpendicular. Find the area of triangle $A B C$, if $A B=\sqrt{13}$.

Answer: 12


3. Over the trapezoid $A B C D$, a circle is described with center on the base $A B$ and radius $R$. Express the length of the leg $A D$ in terms of $R$, if

$$
\overline{A D}^{2}+3 \overline{C D}^{2}=2 R^{2}
$$

Answer: \dfrac{R\sqrt{15}}{3}


Problem 16. (5 points)

Maxim Sergeevich works as a financial manager in a large industrial company. He has long dreamed of acquiring a large plot of land with a house on it. Maxim Sergeevich saved money for a down payment on a mortgage for about two years, and finally, when a sufficient amount was in his account, he came across exactly the option he had been looking for. The bank manager, to whom Maxim Sergeevich turned for a mortgage, offered a sufficient amount for the purchase of the house for 5 years at 12.5% per annum. In the evening, Maxim Sergeevich's wife, Arina Anatolievna, told her husband that taking on debt was too risky, and their family owned a large package of shares in the company where Maxim Sergeevich works. The shares are preferred, and dividends are paid annually at a rate of 17%. The value of the share package is sufficient to fully cover the cost of the house, taking into account the existing savings and avoiding debt. Maxim Sergeevich disagreed with his wife. Who do you think is right in this situation: Maxim Sergeevich or Arina Anatolievna?

Answer: Maxim Sergeevich


12. (3 points) A large rectangular block of wood is painted red on all its surfaces, then divided into several equally sized smaller rectangular blocks. Among these, the number of smaller blocks that have exactly two red faces is exactly 12. What is the minimum number of smaller rectangular blocks into which the large rectangular block must be divided?

Answer: 20


Evdokimov M.A.

Petya has a deck of 36 cards (4 suits with 9 cards each). He chooses half of the cards (whichever he wants) and gives them to Vasya, while keeping the other half for himself. Each turn, the players take turns laying one card on the table (of their choice, face up); Petya goes first. If, in response to Petya's move, Vasya can lay a card of the same suit or rank, Vasya earns

1 point. What is the maximum number of points Vasya can guarantee to earn?

Answer: 15


34. The digits of the number 123456789 can be rearranged to form a number that is divisible by 11. For example, 123475869, 459267831 and 987453126 . How many such numbers are there?

Answer: 31680


## SUBJECT I

Let the sets $\mathrm{A}=\left\{\left.\frac{a}{b} \right\rvert\, a, b \in \mathrm{N}^{*}, a+b=2015\right\}$ and $\mathrm{B}=\left\{\mathrm{x} \in \mathbf{Q} \left\lvert\, \mathrm{x}=\frac{\mathrm{a}}{\mathrm{b}}\right., \mathrm{a} \in \mathrm{D}_{12}, \mathrm{~b} \in \mathrm{D}_{6}\right\}$. Calculate:

a) the product of all elements of set A;

b) the product of all elements of set B.

Answer: 1


4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $60^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.

Answer: \sqrt{3} : 1


Let $O$ be the centre of the square $ABCD$. Let $P,Q,R$ be respectively on the segments $OA,OB,OC$ such that $OP=3,OQ=5,OR=4$. Suppose $S$ is on $OD$ such that $X=AB\cap PQ,Y=BC\cap QR$ and $Z=CD\cap RS$ are collinear. Find $OS$.

Answer: \dfrac{60}{23}


## Task 2 - 330522

Rolf is looking for four-digit numbers in which no two digits are the same. The difference between the tens and the hundreds digit should be 3, and the difference between the hundreds and the thousands digit should be 4.

When calculating these differences, the order of the two digits involved should not matter.

How many four-digit numbers of the desired type are there in total?

Explain why there cannot be more than the number you have specified!

Answer: 112


Circle $B$ has radius $6\sqrt{7}$.  Circle $A$, centered at point $C$, has radius $\sqrt{7}$ and is contained in $B$.  Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circle $B$ again at $X$ and $Y$, then $XY$ is also tangent to $A$.  Find the area contained by the boundary of $L$.

Answer: 168\pi


Example 10. Let the sequence be $u_{1}=3$, $s_{n+1}=\frac{2 u_{n}+3}{u_{n}+4}$, find its general term.

Answer: \dfrac{3(5^{n-1} + 1)}{3 \cdot 5^{n-1} - 1}


Example 3. As shown in Figure 2, given the line $\mathrm{y}=2 \mathrm{x}$ and point $\mathrm{M}(6, 2)$, try to find points $A$ and $B$ on the $x$-axis and the line $y=2 x$ respectively, such that the perimeter of $\triangle A \bar{B} M$ is minimized, and find the minimum perimeter.

Answer: 8\sqrt{2}


10.372 The bisector of angle $A$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$. Find the length of the chord $D C$, if the center of the circle inscribed in the given triangle is at a distance $\boldsymbol{n}$ from point $D$.

Answer: n


4. A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice?

Answer: 0


There are $2n$ barrels, each containing a unique reactant. These reactants form $n$ pairs of conflicting reactants, but it is unknown which reactant conflicts with which. An engineer needs to determine these conflicting pairs. They have $n$ empty test tubes. In one action, the engineer can add the reactant from any barrel to any test tube (whether it is empty or not). As long as no conflicting reactants are in the same test tube, nothing happens. As soon as conflicting reactants are in the same test tube, it will burst and can no longer be used. The engineer cannot remove anything from the test tubes. How can the engineer achieve their goal?

Answer: n


$AL$, $BM$, and $CN$ are the medians of $\triangle ABC$. $K$ is the intersection of the medians. If $C, K, L, M$ are concyclic and $AB = \sqrt{3}$, find the length of the median $CN$. The original answer is in the form \(\frac{k\sqrt{3}}{n}\), where k and n are integers. Please find the value of k + n.

Answer: 5


In triangle $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$, and $EC$ are $2$, $3$, and $6$, respectively. Find the length of the shortest side of $\triangle ABC$. The original answer is in the form $x\sqrt{y}$, please give the value of $x + y$.

Answer: 12


Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, what number could be obtained?

Answer: 119


A regular hexagon has side length \(6\). Congruent arcs with radius \(3\) are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. The original answer is in the form \(a\sqrt{b}-c\pi\). Please find the value of a + b + c.

Answer: 75


Equilateral \(\triangle ABC\) has side length \(1\), and squares \(ABDE\), \(BCHI\), \(CAFG\) lie outside the triangle. Find the area of hexagon \(DEFGHI\). The answer is in the form \(k+\sqrt{m}\), please give the value of \(k+m\).

Answer: 6


Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? Provide your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers. Please find the value of a + b.

Answer: 6


In figure 1, \( AB \) is parallel to \( DC \), \(\angle ACB\) is a right angle, \( AC = CB \), and \( AB = BD \). If \(\angle CBD = b^{\circ}\), find the value of \( b \).

Answer: 15


The number 1 is written at one vertex of a cube, while the other vertices contain zeros. You can add one to the numbers at both ends of any edge.

Is it possible to make all the numbers divisible by: a) 2; b) 3?

Answer: \text{No}


An equilateral triangle is divided into smaller equilateral triangles with side length 1 by lines parallel to its sides, dividing each side into $n$ parts (in the given figure, $n=5$).

What is the maximum number of segments of length 1 with endpoints at the vertices of these triangles that can be marked in such a way that no triangle has all its sides consisting of marked segments?

Answer: n(n+1)


From the vertex of the obtuse angle \( A \) of triangle \( ABC \), a perpendicular \( AD \) is drawn. From point \( D \), a circle is drawn with radius equal to \( AD \), intersecting the sides \( AB \) and \( AC \) at points \( M \) and \( N \), respectively. Find the side \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).

Answer: \frac{mc}{n}


In the expression below, each letter has replaced a certain digit in the base-6 numbering system:

$$
F A R E S = (F E E)^{2}
$$

(identical letters represent identical digits). Restore the original digits.

Answer: 15324


Is it possible to find two lucky tickets among ten consecutive tickets? A ticket is considered lucky if the sum of the first three digits is equal to the sum of the last three digits.

Answer: \text{Yes}


Let $k$ be a real number. In the Cartesian coordinate plane $xOy$, there are two sets of points:
\[ A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \]
\[ B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \]
If the intersection of $A$ and $B$ is a single-element set, what is the value of $k$?

Answer: -2 - \sqrt{3}


Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow $\mathbb{R}$ such that

$$
f(x^{2}+f(y))=y+(f(x))^{2}
$$

holds for all $x, y$ in $\mathbb{R}$.

Answer: f(x) = x


Consider a 99-sided polygon where its edges are initially colored in the sequence red, blue, red, blue, ..., red, blue, yellow, with one color per edge. Given that no two adjacent edges can have the same color, is it possible, through a series of operations where each operation consists of changing the color of exactly one edge, to transform the coloring into the sequence red, blue, red, blue, ..., red, yellow, blue?

Answer: \text{No}


A $5 \times 5$ square contains a light bulb in each of its cells. Initially, only one light bulb is on. You can change the state of the light bulbs in any $k \times k$ square, with $k > 1$. What are the initial positions of the light bulb that is on that allow turning off all the light bulbs?

Answer: (3,3)


The legs of a compass are positioned at the nodes of an infinite sheet of ruled paper, where the cells are squares with side length 1. It is allowed, without changing the compass opening, to move one leg to another node on the sheet by rotating the compass around the other leg. Is it possible, in a few such steps, to swap the legs of the compass?

Answer: \text{No}


There are several points on a plane, all pairwise distances between which are different. Each of these points is connected to the nearest one. Can a closed polygonal chain result from this?

Answer: \text{No}


Let \( n \) (\( \geqslant 3 \)) be a positive integer, and \( M \) be an \( n \)-element set. Find the largest positive integer \( k \) that satisfies the following condition:

There exists a family \( \psi \) of \( k \) three-element subsets of \( M \) such that the intersection of any two elements in \( \psi \) (note: the elements of \( \psi \) are all three-element sets) is nonempty.

Answer: \binom{n-1}{2}


Let $2 \leq r \leq \frac{n}{2}$, and let $\mathscr{A}$ be a family of $r$-element subsets of $Z = \{1, 2, \ldots, n\}$. If $\mathscr{A} = \{A \mid A$ is an $r$-element subset of $Z$ that contains a fixed element $x \in Z\}$, then equality holds.

Answer: \binom{n-1}{r-1}


A material point moves in such a way that its velocity is proportional to the distance traveled. At the initial moment, the point was at a distance of 1 meter from the origin, and after 2 seconds, it was at a distance of $e$ meters. Find the equation of motion for the material point.

Answer: s = e^{\frac{t}{2}}


Point $M$ lies on the leg $AC$ of the right triangle $ABC$ with a right angle at $C$, such that $AM=2$ and $MC=6$. Segment $MH$ is the altitude of triangle $AMB$. Point $D$ is located on the line $MH$ such that the angle $ADB$ is $90^{\circ}$, and points $C$ and $D$ lie on the same side of line $AB$. Find the length of segment $DC$ if the tangent of the angle $ACH$ is $1 / 7$. (16 points)

Answer: 4


9. Two people, A and B, are playing a table tennis match, agreeing that the winner of each game gets 1 point, and the loser gets 0 points. The match stops when one person is 2 points ahead or after 6 games have been played. Suppose the probability of A winning each game is $\frac{2}{3}$, and the probability of B winning each game is $\frac{1}{3}$, and the outcomes of each game are independent. Then the mathematical expectation $\mathrm{E} \xi$ of the number of games $\xi$ played when the match stops is $\qquad$

Answer: \dfrac{266}{81}


Question 64: Given $a, b \in [1, \sqrt{3}]$, find the maximum and minimum values of $\frac{a^{2}+b^{2}-1}{a b}$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

Answer: \sqrt{3}


A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?

Answer: 0


5. (10 points) Teacher Hu originally has 100 red cards and 100 blue cards. Teacher Hu can exchange 2 red cards for 1 blue card and 1 purple card; or exchange 3 blue cards for 1 red card and 1 purple card. After several exchanges, what is the maximum number of purple cards that Teacher Hu can hold? $\qquad$

Answer: 138


3. What is the minimum number of cells that need to be colored on a $6 \times 6$ board so that, for any placement (including rotations and flips) of a figure consisting of 4 cells in the shape of the letter Г on the board, at least one colored cell is found?

Answer: 12


## Task 23/87

There are prime numbers $p_{i}$ that have the following properties:

1. They are (in decimal notation) truly four-digit.
2. Their digit sum is $Q\left(p_{i}\right)=25$.
3. If you add 4 to them, the result is a "mirror number".

A "mirror number" is a number whose sequence of digits is symmetric with respect to an imaginary central line. Determine all such prime numbers $p_{i}$!

Answer: 1987


22. The figure below shows a circle with diameter $A B . C$ and $D$ are points on the circle on the same side of $A B$ such that $B D$ bisects $\angle C B A$. The chords $A C$ and $B D$ intersect at $E$. It is given that $A E=169 \mathrm{~cm}$ and $E C=119 \mathrm{~cm}$, If $E D=x \mathrm{~cm}$, find the value of $x$.

Answer: 65


[ Transfer of side, diagonal, etc.] The Law of Sines

In trapezoid $ABCD (AB \| CD)$, $AB + CD = b$, diagonals $AC$ and $BD$ are related by $5AC = 3BD$. Find the area of the trapezoid if $\angle BAC = 2 \angle DBA$.

Answer: \dfrac{5\sqrt{11}}{64} b^2


Answer: (1, -2, 5)


13.273. The ball is rolling perpendicular to the sideline of a football field. Suppose, moving uniformly decelerated, the ball rolled 4 m in the first second, and 0.75 m less in the next second. A football player, initially 10 m away from the ball, started running in the direction of the ball's movement to catch it. Moving uniformly accelerated, the player ran 3.5 m in the first second, and 0.5 m more in the next second. How long will it take for the player to catch the ball, and will he be able to catch it before the ball goes out of bounds, if the player needs to run 23 m to reach the sideline?

Answer: 5


2. Color the positive integers $1,2, \cdots, 15$ in blue or red. Find the number of coloring methods that satisfy the following conditions:
(1) The positive integer 15 is red;
(2) If the positive integers $x$ and $y$ are of different colors, and $x+y \leqslant 15$, then $x+y$ is colored blue;
(3) If the positive integers $x$ and $y$ are of different colors, and $x y \leqslant 15$, then $xy$ is colored red.

Answer: 4


In an isosceles trapezoid, there are two touching circles of radii $R$, each of which touches both bases and one of the lateral sides, and the centers of the circles lie on the diagonals. Find the sides of the trapezoid.

Answer: 2R\sqrt{2}


A number is written on the board. At each step, the largest of its digits is added to it (for example, if 142 is written, the next number will be 146). What is the maximum number of consecutive odd numbers that can be written by following this method?

Answer: 5


403. Determine the force of water pressure on a vertical parabolic segment, the base of which is $4 \mathrm{m}$ and is located on the water surface, while the vertex is at a depth of $4 \mathrm{m}$ (Fig. 204).

Answer: 167424


Khachaturyan A.V.

Mom baked pies - three with rice, three with cabbage, and one with cherries - and laid them out in a circle on a plate (see fig.). Then she put the plate in the microwave to warm it up. All the pies look the same. Masha knows how they were arranged, but she doesn’t know how the plate turned. She wants to eat the cherry pie, and considers the others tasteless. How can Masha definitely achieve this, biting into as few tasteless pies as possible?

![](https://cdn.mathpix.com/cropped/2024_05_06_61f459f9e45cb4d6ceb6g-10.jpg?height=352&width=352&top_left_y=1808&top_left_x=859)

Answer: 2


Kozhevnikov P.A

On each of two parallel lines $a$ and $b$, 50 points were marked.

What is the maximum possible number of acute-angled triangles with vertices at these points?

Answer: 41650


Circles  $\omega_1$  and  $\omega_2$  are externally tangent to each other at  $P$ . A random line  $\ell$  cuts  $\omega_1$  at  $A$  and  $C$  and  $\omega_2$  at  $B$  and  $D$  (points  $A,C,B,D$  are in this order on  $\ell$ ). Line  $AP$  meets  $\omega_2$  again at  $E$  and line  $BP$  meets  $\omega_1$  again at  $F$ . Prove that the radical axis of circles  $(PCD)$  and  $(PEF)$  is parallel to  $\ell$ .

*(Vlad Robu)*

Answer: \text{The radical axis of circles } (PCD) \text{ and } (PEF) \text{ is parallel to } \ell


We choose $n$ points on a circle and draw all the associated chords (we make sure that 3 chords are never concurrent). Into how many parts is the circle divided?

Answer: 1 + \dfrac{n(n - 1)}{2} + \dfrac{n(n - 1)(n - 2)(n - 3)}{24}


## Task 2 - 340712

For a triangle $A B C$, it is required: The angle bisector through $A$ and the perpendicular bisector of $A B$ intersect at a point $D$, which lies on the side $B C$.

a) What size must the angle $\angle A C B$ have in a triangle that meets this requirement and in which the angle $\angle A B C$ has a size of $35^{\circ}$? Draw such a triangle!

b) Draw a triangle that also meets the above requirement and in which the angle $\angle A B C$ has a size of $50^{\circ}$!

c) Is there a triangle that also meets the above requirement and in which the angle $\angle A B C$ has a size of $60^{\circ}$?

Answer: 75^\circ


7. Determine the number of 8-digit positive integers such that after deleting any one digit, the remaining 7-digit number is divisible by 7 .

Answer: 64


## Task 2.

Let $n \geqslant 3$ be a natural number. Determine the minimum number of points that need to be marked inside any convex $n$-gon so that every triangle whose vertices are also vertices of that $n$-gon contains at least one marked point in its interior.

Answer: n - 2


(17) (This sub-question is worth 20 points)
Given the function $f(x)=\frac{a x^{2}}{e^{x}}$, and the line $y=\frac{1}{e} x$ is a tangent line to the curve $y=f(x)$.
(1) Find the value of the real number $a$;
(II) Let $\min \{m, n\}$ denote the minimum of $m$ and $n$. Define the function $g(x)=\min \left\{f(x), x-\frac{1}{x}\right\}(x>0)$. If the function $h(x)=g(x)-c x^{2}$ is an increasing function, find the range of the real number $c$.

Answer: 1


256. Palindromic Squares. A certain square is written in the base-6 numeral system using 5 different non-zero digits. If the digit in the units place is moved from the end of this number to the beginning, the square root of the resulting number matches the square root of the original number, written in reverse order. Find the original number.

Answer: 53241


## Task 1

In men's rowing, 7 different boats will start from the GDR:

the single scull, two doubles (without coxswain), a double with coxswain, the four without coxswain, the four with coxswain, the eight with coxswain.

How many rowers will start?

(Note: In a double without coxswain, two rowers sit. In a double with coxswain, three rowers sit. Pay close attention to all designations!)

Answer: 26


In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that
[b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i];
[b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i].
Find the least possible number of [i]central[/i] vertices in $G$.

Answer: 2k - n


146. Count how many different $p$-arithmetic fractional-linear functions there are.

The function $f(f(x))$ is denoted by $f_{2}(x)$. Similarly, we introduce fractional-linear functions $f_{\mathrm{B}}(x), f_{1}(x), \ldots, f_{2}(x)$, assuming

$$
\begin{aligned}
f_{3}(x) & =f\left(f_{2}(x)\right), \\
f_{4}(x) & =f\left(f_{\mathrm{B}}(x)\right), \\
\cdots & \cdot \\
f_{n}(x) & =f\left(f_{n-1}(x)\right) .
\end{aligned}
$$

For example,

$$
\begin{aligned}
& f(x)=\frac{x+1}{x} \\
& f_{2}(x)=\frac{\frac{x+1}{x}+1}{\frac{x+1}{x}}=\frac{2 x+1}{x+1} \\
& f_{3}(x)=\frac{\frac{2 x+1}{x+1}+1}{\frac{2 x+1}{x+1}}=\frac{3 x+1}{2 x+1}
\end{aligned}
$$

The action of all functions $f_{n}(x)$ can be traced on a diagram constructed for the function $f(x)$. The action of the function $f(x)$ is such that each point of its diagram moves to the next one, i.e., makes one step in the direction of the arrows. When applying the function $f_{2}(x)$, each point on the diagram of $f(x)$ already makes two steps, landing in the second point. Generally, when applying $f_{n}(x)$, each point on the diagram of $f(x)$ makes $n$ steps in the direction of the arrows, jumping over $n-1$ points to the $n$-th. This allows constructing the diagrams of the functions $f_{2}(x), f_{3}(x), \ldots$ from the diagram of the function $f(x)$ without calculations.

Answer: p(p^2 - 1)


1. Find the particular integral of the equation \( y \, dx + \cot x \, dy = 0 \) that satisfies the initial condition \( y\left(\frac{\pi}{3}\right) = -1 \).
2. Solve the differential equation \( s = s' \cos^2 t \ln s \) with the initial condition \( s(\pi) = 1 \).

Answer: y = -2 \cos x


66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.

Answer: 2^{22}


For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.
Find the maximum value of $ s(P)$ over all such sets $ P$.

Answer: 7


27. [15] Let $\left\{a_{i}\right\}_{i \geq 0}$ be a sequence of real numbers defined by
$$
a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}
$$
for $n \geq 0$. Determine the largest value for $a_{0}$ such that $\left\{a_{i}\right\}_{i \geq 0}$ is bounded.

Answer: 1 + \dfrac{1}{2^{2020}}


Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$.  Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral.  Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$.  Find $\tfrac{BX}{XC}$.

Answer: \dfrac{8}{13}


5. Let $n$ and $k$ be positive integers, $k \geqslant n$, and $k-n$ is an even number. $2 n$ lamps are sequentially numbered as $1,2, \cdots, 2 n$, and each lamp can be either "on" or "off". Initially, all lamps are off. An operation can be performed on these lamps: each operation changes the state of one lamp (i.e., on becomes off, off becomes on). Consider a sequence of length $k$, where the $i$-th term of the sequence is the number of the lamp whose state is changed during the $i$-th operation.

Let $N$ be the number of different operation sequences of length $k$ that result in lamps $1,2, \cdots, n$ being on and lamps $n+1, n+2, \cdots, 2 n$ being off after $k$ operations;

Let $M$ be the number of different operation sequences of length $k$ that result in lamps $1,2, \cdots, n$ being on and lamps $n+1, n+2, \cdots, 2 n$ being off, with the condition that lamps $n+1, n+2, \cdots, 2 n$ are never turned on during the operations.
Find the ratio $\frac{N}{M}$.
(France provided)

Answer: 2^{k - n}


6. Let $n>2$ be a natural number. Determine the largest integer $d$ for which the following statement holds: From any $n$-element set of integers, one can select three different non-empty subsets such that the sum of the elements in each of them is an integer multiple of $d$. (The selected subsets may have common elements.)

(Jaromír Šimša)

Answer: n-1


3. In a circle of radius 3, triangles $A B C$ and $A M N$ are inscribed. The line $A M$ passes through the midpoint $E$ of segment $B C$, and the line $B N$ passes through the midpoint $F$ of segment $A C$. Find the perimeter of triangle $A B C$, if $A M: A E=2: 1$ and $B N: B F=17: 13$.

Answer: \dfrac{72}{5}


13.171. In a family, there is a father, a mother, and three daughters; together they are 90 years old. The age difference between the girls is 2 years. The mother's age is 10 years more than the sum of the daughters' ages. The difference in age between the father and the mother is equal to the age of the middle daughter. How old is each member of the family?

Answer: 38


6.3. How many natural numbers $\mathrm{N}$ greater than 700 exist such that among the numbers $3 \mathrm{~N}, \mathrm{~N}-$ $700, N+35,2 N$ exactly two are four-digit numbers?

Answer: 5229


6. As shown in the figure, the area of rectangle $A B C D$ is $36, F$ is the midpoint of $D E$. If the area of $\triangle B C E$ is 6, then the area of $\triangle B C F$ is $\qquad$ .

Answer: 6


Answer: \sqrt[3]{12}


Example 11 Four spheres, each with a radius of 1, are mutually tangent, three on the floor, and the fourth above them. A tetrahedron with side length $S$ has faces that are tangent to all these spheres. Find $S$.
(IMO - 29 Shortlist)

Answer: 2 + 2\sqrt{6}


On a plane, there are 5 points $A, B, C, U, V$ representing the complex numbers $a, b, c, u, v$ respectively. If the triangles $\triangle A U V$, $\triangle V B U$, and $\triangle U V C$ are directly similar to each other, prove that $\triangle A B C$ is also directly similar to them.

Answer: \triangle ABC \text{ is directly similar to } \triangle AUV, \triangle VBU, \text{ and } \triangle UVC


4. (10 points) As shown in the figure, 5 identical small rectangular prisms are exactly assembled into a rectangular prism with a volume of 22.5. Then, the sum of all edge lengths of one small rectangular prism is.

Answer: 22


## Task A-3.3.

In triangle $A B C$, $|A B|=|A C|$. On side $\overline{A C}$, there is a point $D$ such that $|A D|<|C D|$, and on segment $\overline{B D}$, there is a point $P$ such that $\varangle A P C$ is a right angle. If $\varangle A B P=\varangle B C P$, determine $|A D|:|C D|$.

Answer: 1:2


Example 4. Find the general solution of the equation $y^{\prime \prime}+3 y^{\prime}=0$.

Answer: y = C_1 + C_2 e^{-3x}


## 246. Math Puzzle $11 / 85$

A hollow round ceiling support in a large bakery is loaded with 28 tons. The outer diameter of the support is 145 millimeters, and the clear width, i.e., the inner diameter, is 115 millimeters.

What is the load per square centimeter of the cross-sectional area?

Answer: 457


[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part Rectangles and squares. Properties and criteria

In a square $A B C D$ with side $a$, a circle is inscribed, touching side $C D$ at point $E$.

Find the chord connecting the points where the circle intersects the line $A E$.

#

Answer: \dfrac{2\sqrt{5}}{5}a


Question 4. No figure, divide each side of a single square into $\mathrm{n}$ equal parts, then connect each vertex to the nearest division point of the opposite vertex, thus creating a smaller square inside the square. If the area of the smaller square (shaded in the figure) is exactly $\frac{1}{1985}$, find the value of $\mathrm{n}$.

Answer: 32


Consider all possible tetrahedrons \(A B C D\) where \(A B = 2\), \(A C = C B = 5\), and \(A D = D B = 7\). Each such tetrahedron is inscribed in a cylinder such that all vertices lie on the lateral surface of the cylinder and the edge \(C D\) is parallel to the cylinder's axis. Choose the tetrahedron for which the cylinder's radius is the smallest among the obtained ones. What values can the length of \(C D\) take in such a tetrahedron?

Answer: \sqrt{47} - \sqrt{23}


## 57. The Speed of a Car.

- I was walking down the road at a speed of $3 \frac{1}{2}$ km/h,- said Mr. Pipkins,- when suddenly a car sped past me, nearly knocking me off my feet ${ }^{2}$.
- How fast was it going? - asked his friend.
- I'll tell you. From the moment it sped past me until it disappeared around the bend, I took 27 steps and then, without stopping, I walked another 135 steps to reach the bend.[^4]- Then we can easily determine the speed of the car, assuming that our speeds were constant.

Answer: 21


Points \(M\) and \(N\) lie on edges \(BC\) and \(AA_1\) of the parallelepiped \(ABCD A_1 B_1 C_1 D_1\). Construct the intersection point of line \(MN\) with the base plane \(A_1 B_1 C_1 D_1\).

Answer: P


Let \( ABC \) be an equilateral triangle with side length \( a \), and let \( P \) be a point inside the triangle such that the distances from \( P \) to vertices \( A \), \( B \), and \( C \) are \( PA = u \), \( PB = v \), and \( PC = w \), respectively. Given that \( u^2 + v^2 = w^2 \), prove that \( w^2 + \sqrt{3} uv = a^2 \).

Answer: w^2 + \sqrt{3} uv = a^2


Answer: 3\sqrt{3}


1. As shown in Figure 3, in trapezoid $A B C D$, $A B / / C D$, $A B=a, C D=b(a$ $<b), \angle B C D=$ $60^{\circ}, \angle A D C=45^{\circ}$. The area of trapezoid $A B C D$ is equal to $\qquad$ (express the area in terms of $a$ and $b$).

Answer: \dfrac{(b^2 - a^2)(3 - \sqrt{3})}{4}


On two perpendicular axes \(OX\) and \(OY\), we choose two points, \(A\) and \(B\), such that \(OA + OB\) equals a constant \(k\). From point \(O\), we draw a line parallel to \(AB\), which intersects the circumcircle of triangle \(AOB\) at point \(M\). What is the locus of point \(M\)?

Answer: \left(x - \frac{k}{2}\right)^2 + \left(y - \frac{k}{2}\right)^2 = \frac{k^2}{2}


6. A rectangular piece of land enclosed by fences has a length and width of $52 \mathrm{~m}$ and $24 \mathrm{~m}$, respectively. An agricultural science technician wants to divide this land into several congruent square test plots. The land must be fully divided, and the sides of the squares must be parallel to the boundaries of the land. There are $2002 \mathrm{~m}$ of fencing available. The land can be divided into a maximum of $\qquad$ square test plots.

Answer: 702


13. Find the smallest positive integer $n$ such that: if for any finite set of points $A$ in the plane, any $n$ points can be covered by two lines, then there exist two lines that can cover all points in $A$.

Answer: 6


4.41. Find all natural numbers $m$ for which: a) $3^{m}-1$ is divisible by $2^{m}$; b) $31^{m}-1$ is divisible by $2^{m}$.

## 4.8. Remainders of Division

Answer: 1


6. Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying
$$
\begin{array}{l}
3 a+5 b \equiv 19 \quad(\bmod n+1) \\
4 a+2 b \equiv 25 \quad(\bmod n+1)
\end{array}
$$

Find $2 a+6 b$.
Proposed by: David Vulakh

Answer: 96


## Task A-2.5.

Ivica made a large cube with a side length of $n$ from $n^3$ unit cubes and then painted some of the six faces of the large cube, while leaving others unpainted. When he disassembled the large cube, he found that exactly 1000 unit cubes had no painted faces. Show that this is indeed possible and determine the number of faces of the large cube that Ivica painted.

Answer: 3


12 chess players participated in a round-robin tournament. Afterwards, each player wrote 12 lists. The first list contains only himself, the $(k+1)$-th list contains the players from the $k$-th list and those who were defeated by any of the players in the $k$-th list. It turned out that the 12th list of each player is different from their 11th list. How many draws were there?

Answer: 54