A unit cube has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\), and \(P_4'\). Vertices \(P_2, P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\leq i\leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'\), and \(P_1'P_4'\). Find the octahedron's side length. The original answer is in the form \(\frac{a\sqrt{b}}{c}\), please give the value of \(a + b + c\).

Answer: 9


Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. Find the number of positive integers $k$ less than $2014$ for which $f(k)=8$.

Answer: 165


A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has.

Answer: 715


Two strips of width 1 overlap at an angle of \(\alpha\) as shown. Find the area of the overlap (shown shaded) in terms of \(\alpha\). The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\).

Answer: 2


Two boys start moving from the same point $A$ on a circular track but in opposite directions. Their speeds are $5$ ft. per second and $9$ ft. per second. If they start at the same time and finish when they first meet at the point $A$ again, find the number of times they meet, excluding the start and finish.

Answer: 13


Elara, a storyteller known for her ghost tours and urban legends in European cities, is planning a special event for Halloween. She will tell 5 unique ghost stories in each of 3 different cities: Prague, Edinburgh, and Rome. Each story takes 15 minutes to tell. After each story, she takes a 5-minute break.

Calculate the total time, in hours, Elara will spend telling stories and taking breaks during her Halloween event across all three cities.

Answer: 5


Alex is a staunch supporter of workers' rights and loves attending heavy metal concerts. He decides to organize a charity concert to support local workers. The concert venue can hold 150 people, and he plans to sell tickets for $20 each.

For each ticket sold, $5 will go towards supporting workers' rights initiatives, and the rest will cover the costs of the concert and band fees. The band he hired charges a flat fee of $1,200 for the night. If Alex sells out all 150 tickets, how much money will be left after covering the band's fee, assuming all the remaining money goes to support workers' rights?

Calculate the total amount raised for workers' rights after paying the band.

Answer: 1800


A travel blogger exploring Hull for the first time decided to visit 4 of the city's hidden gems in one day. In the morning, they visit the Hull Maritime Museum and the Wilberforce House, spending 2 hours at each location. For lunch, they stop at a local cafe for 1 hour. In the afternoon, they visit the Humber Bridge and the Streetlife Museum of Transport, spending 1.5 hours at each location. The blogger spends 30 minutes traveling between each location. How many hours does the blogger spend exploring Hull, including travel time?

Answer: 10


The owner of a large cleaning company is worried about the rise of small cleaning services that offer flexible scheduling and efficient communication channels. These smaller companies manage to complete 15 cleaning jobs per week. To stay competitive, the owner wants to ensure their company completes 50% more jobs than the smaller companies each week. If the larger company currently completes 20 cleaning jobs per week, how many additional jobs does the larger company need to complete each week to meet their goal?

Answer: 3


Juan, a local villager, loves helping the foreign volunteers who come to his village. Despite the language barrier, he has developed a simple system to guide them. Every day, he meets 5 new volunteers and spends 12 minutes with each one, showing them around the community. Afterward, he takes them to the local market, which is 15 minutes away on foot. At the market, he helps them for 18 minutes in buying fresh produce.

One day, a total of 8 volunteers arrive, and Juan wants to ensure he spends an equal amount of time with each group. How many minutes in total does Juan spend helping the volunteers, including both guiding them around the village and assisting them at the market?

Answer: 162


Sam is a die-hard Mets fan who has held season tickets for the last 20 years. Each season, Sam attends 81 home games. Over the years, Sam has collected a special Mets souvenir for every 3 games he attended. This year, the Mets introduced a new promotional item, and Sam decided to collect an additional item every 9 games. How many total souvenirs and promotional items has Sam collected over the past 20 years?

Answer: 549


In the dense jungle, a skilled poacher named Carlos competes with another poacher for prized exotic birds. Each day, Carlos manages to capture twice as many birds as his competitor. On Monday, Carlos captured 12 exotic birds. The following day, his competitor captured 5 more birds than he did on Monday. How many birds did Carlos and his competitor capture together on Tuesday?

Answer: 33


Alex is a tech enthusiast who loves using the Opera browser and often conducts searches on Google. One day, Alex decides to explore the performance of their browser activities. In the morning, Alex uses the Opera browser to open 5 tabs, each conducting a different Google search. Each tab uses 4 megabytes of data. In the afternoon, Alex closes 2 of the morning's tabs and opens 3 new tabs, each also using 4 megabytes of data. By the end of the day, Alex decides to calculate the total amount of data used for all these tabs. How many megabytes of data did Alex use in total for the day?

Answer: 32


An elderly local historian from Zavidovići is creating a timeline of significant events in the town's history. They plan to display this timeline in the local museum. The timeline consists of 5 panels. Each panel can display 12 events. The historian has already gathered information about 38 historical events but wants to include exactly 60 events in total.

How many more panels does the historian need to create in order to display all 60 events, assuming each panel still holds 12 events?

Answer: 0


At an exhibition showcasing the indigenous civilization, there are 5 cultural artifacts being displayed. Each artifact is accompanied by 3 informational panels that provide cultural insights and historical context. As a descendant of the indigenous civilization, you are supporting preservation efforts by volunteering to give guided tours. If you give a tour to 4 different groups each day, with each group consisting of 10 people, how many total informational panels will you have explained by the end of the day?

Answer: 60


Emily, a published poet, often writes about historical events that inspire her creativity. One day, she decides to write a poem about the ancient Roman Empire. She remembers from her history class that the Roman Empire lasted from 27 BC to 476 AD, a period of 503 years. Emily plans to write a poem with one stanza for every 50 years of the Roman Empire's history. Each stanza will have 8 lines. Additionally, she wants to include a special prologue and epilogue, each consisting of 4 lines. How many lines will Emily's complete poem have?

Answer: 96


Lisa is a proud manager at BlueGrace, a logistics company. She is organizing a team-building event for her department, which consists of 12 employees. She plans to provide snacks and drinks for everyone. She decides to buy 3 packs of juice boxes, each containing 8 juice boxes, and 4 packs of snack bags, each containing 10 snack bags.

If Lisa wants to ensure that each employee receives an equal number of juice boxes and snack bags, how many complete sets of juice boxes and snack bags can she distribute to each of her employees?

Answer: 2


A multilingual film critic is attending an international film festival organized by an event coordinator who appreciates international filmmakers. The film festival features 5 films from France, 4 films from Japan, 6 films from Brazil, and 3 films from South Korea. During the festival, each film is shown twice. The critic plans to watch all the films once and then select 6 films to watch a second time. If the critic speaks the languages of the films from France and Brazil, how many different combinations of films can the critic choose to watch twice, given that at least 2 of them must be from a country whose language the critic speaks?

Answer: 18326


The high-tech CEO has a collection of simple gray t-shirts, each of which costs $15. He likes to buy t-shirts in bulk every quarter to ensure he always has enough to wear. If the CEO wants to buy 5 t-shirts for each day of the week and an extra set of 10 t-shirts for emergencies, how much will he spend on t-shirts this quarter?

Answer: 675


The sets $A$ and $B$ are subsets of the positive integers. The sum of any two different elements from $A$ is an element of $B$. The quotient of any two different elements from $B$ (where we divide the largest by the smallest) is an element of $A$. Determine the maximum number of elements in $A \cup B$.

Answer: 5


Given the quadratic function \( f(x) = ax^2 + bx + c \), for any \( x \in [-1, 1] \), it always holds that \(-1 \leq f(x) \leq 1\). Prove that when \( x \in [-2, 2] \), we have \(-7 \leq f(x) \leq 7\).

Answer: -7 \leq f(x) \leq 7


4. The cities A, B, and C are connected by straight roads. Along the road A-B, there is a square field with a side length of $0.5 \overline{A B}$, and along the road B-C, there is a square field with a side length of $\overline{B C}$; along the road A-C, there is a rectangular forest with a length of $\overline{A C}$ and a width of 4 kilometers. Find the area of the forest, if it is 20 square kilometers larger than the sum of the areas of the square fields.

Answer: 40


15. A mathematics interest group at a school consists of 14 students, who form $n$ different project teams. Each project team has 6 students, each student participates in at least 2 project teams, and any two project teams have at most 2 students in common. Find the maximum value of $n$.

Answer: 7


5. On the blackboard are the numbers $1,2, \cdots, 1987$. Perform the following transformation: erase some of the numbers on the blackboard and add the remainder when the sum of the erased numbers is divided by 7. After several such transformations, only two numbers remain on the blackboard, one of which is 987. Find the other number.

Answer: 0


2. 103 Use 94 large and small bricks, each with dimensions $4 \times 10 \times 19$, to stack into a tower 94 bricks high, where each brick can be placed to provide a height of 4, 10, or 19. If all 94 bricks are used, how many different heights can the tower be (height unit in inches)?

Answer: 465


Given an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\) with left focus \(F\). Points \(P_{1}, P_{2}, \cdots, P_{n}\) are \(n\) points arranged in a clockwise order on the ellipse such that \(\angle P_{1}FP_{2} = \angle P_{2}FP_{3} = \angle P_{3}FP_{4} = \cdots = \angle P_{n}FP_{1}\). Find the sum of the reciprocals of the distances of these \(n\) points to the left directrix \(x = -\frac{a^{2}}{c} \ (c=\sqrt{a^{2}-b^{2}})\) expressed in terms of \(a, b, c, n\).

Answer: \dfrac{n c}{b^{2}}


In a certain year, Monday occurs more frequently than Friday, and Sunday occurs more frequently than Wednesday. Please, what day of the week was March 1st in that year?

Answer: Thursday


For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation
 \[
        S(n) = (S(S(n)))^2.
 \]
Find the remainder when $K$ is divided by $1000$.

Answer: 632


1. Find all natural numbers $a, b$, and $c$ that satisfy the equation

$$
\frac{a+b}{a+c}=\frac{b+c}{b+a}
$$

and for which $a b+a c+b c$ is a prime number.

Answer: 1


In a regular quadrilateral pyramid, the slant height is equal to the side length of the base. Inside the pyramid, there are two spheres: a sphere with radius \( r \) that touches all lateral faces, and a sphere with radius \( 2r \) that touches the base and two adjacent lateral faces. Both spheres touch each other externally. Find the slant height of this pyramid.

Answer: \dfrac{2r}{5} \left(8\sqrt{3} + \sqrt{37}\right)


A point $D$ is chosen on side $B C$ of triangle $A B C$. The line tangent to the incircles of triangles $A B D$ and $A D C$ and different from $B C$ and $A D$ intersects segment $A D$ at $T$. If $A B=40 \text{ cm}, A C=50 \text{ cm}$, and $B C=60 \text{ cm}$, determine the value of the length of $A T$.

![](https://cdn.mathpix.com/cropped/2024_05_01_4a19aea8e08094ae376fg-07.jpg?height=539&width=1108&top_left_y=1079&top_left_x=546)

Answer: 15


9. Given $n$ positive integers (not necessarily distinct), their sum is 100, and the sum of any seven of them is less than 15. Then the minimum value of $n$ is $\qquad$ .

Answer: 50


Three. (20 points) Let the edge length of the cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$ be 1. Find a point $P$ on the diagonal $B D_{1}$ and a point $Q$ on the base quadrilateral $A B C D$ such that the value of $P Q+P C_{1}$ is maximized.

---

The translation retains the original text's line breaks and formatting.

Answer: 2\sqrt{2}


6. Let $x+2 y \leqslant 5, 2 x+y \leqslant 4, x \geqslant 0, y \geqslant 0$, when $(x, y)=$ $\qquad$, the maximum value of $3 x+4 y$ is $\qquad$.

Answer: (1, 2)


2. What is the maximum number of different numbers from 1 to 1000 that can be chosen so that the difference between any two chosen numbers is not equal to any of the numbers 4, 5, 6.

Answer: 400


A square is inscribed in a circle of radius  $6$ . A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is  $n\pi$  for some positive integer  $n$ , what is  $n$ ?
[asy]
        size(5 cm);
        draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
        draw(circle((1,1),1.41));
        draw(arc((0,0),2,0,90));[/asy]

*2021 CCA Math Bonanza Lightning Round #1.2*

Answer: 18


Triangle  $ABC$  lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
![Image](https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png)

Answer: \frac{1}{5}


We have  $10$  points on a line  $A_1,A_2\ldots A_{10}$  in that order. Initially there are  $n$  chips on point  $A_1$ . Now we are allowed to perform two types of moves. Take two chips on  $A_i$ , remove them and place one chip on  $A_{i+1}$ , or take two chips on  $A_{i+1}$ , remove them, and place a chip on  $A_{i+2}$  and  $A_i$  . Find the minimum possible value of  $n$  such that it is possible to get a chip on  $A_{10}$  through a sequence of moves.

Answer: 46


Answer: \begin{pmatrix}0&7&9&14\\11&12&2&5\\6&1&15&8\\13&10&4&3\end{pmatrix}


The diagram below shows nine points on a circle where  $AB=BC=CD=DE=EF=FG=GH$ .  Given that  $\angle GHJ=117^\circ$  and  $\overline{BH}$  is perpendicular to  $\overline{EJ}$ , there are relatively prime positive integers  $m$  and  $n$  so that the degree measure of  $\angle AJB$  is  $\textstyle\frac mn$ .  Find  $m+n$ .

[asy]
size(175);
defaultpen(linewidth(0.6));
draw(unitcircle,linewidth(0.9));
string labels[] = {"A","B","C","D","E","F","G"};
int start=110,increment=20;
pair J=dir(210),x[],H=dir(start-7*increment);
for(int i=0;i<=6;i=i+1)
{
x[i]=dir(start-increment*i);
draw(J--x[i]--H);
dot(x[i]);
label(" $"+labels[i]+"$ ",x[i],dir(origin--x[i]));
}
draw(J--H);
dot(H^^J);
label(" $H$ ",H,dir(origin--H));
label(" $J$ ",J,dir(origin--J));
[/asy]

Answer: 29


In triangle  $ABC$  lines  $CE$  and  $AD$  are drawn so that  $\dfrac{CD}{DB}=\dfrac{3}{1}$  and  $\dfrac{AE}{EB}=\dfrac{3}{2}$ . Let  $r=\dfrac{CP}{PE}$

where  $P$  is the intersection point of  $CE$  and  $AD$ . Then  $r$  equals:

[asy]
size(8cm);
pair A = (0, 0), B = (9, 0), C = (3, 6);
pair D = (7.5, 1.5), E = (6.5, 0);
pair P = intersectionpoints(A--D, C--E)[0];
draw(A--B--C--cycle);
draw(A--D);
draw(C--E);
label(" $A$ ", A, SW);
label(" $B$ ", B, SE);
label(" $C$ ", C, N);
label(" $D$ ", D, NE);
label(" $E$ ", E, S);
label(" $P$ ", P, S);
//Credit to MSTang for the asymptote
[/asy] $\textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \dfrac{3}{2}\qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ \dfrac{5}{2}$

Answer: 5


Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly  $ 5\over 4$  times greater than  Vasya's result. What minimum number of portraits can be on the wall?

*Author: V. Frank*

Answer: 5


For integers  $a, b$ , call the lattice point with coordinates  $(a,b)$  **basic** if  $gcd(a,b)=1$ . A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between  $(a_1,b_1)$  and  $(a_2,b_2)$  if and only if  $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$  or  $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$ . Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?

Answer: 1


Let  $\{ z_n \}_{n \ge 1}$  be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer  $k$ ,  $|z_k z_{k+1}|=2^k$ . Denote  $f_n=|z_1+z_2+\cdots+z_n|,$  for  $n=1,2,\cdots$ (1) Find the minimum of  $f_{2020}$ .
(2) Find the minimum of  $f_{2020} \cdot f_{2021}$ .

Answer: 2^{1011}


There is a set of 1000 switches, each of which has four positions, called  $A, B, C,$  and  $D.$   When the position of any switch changes, it is only from  $A$  to  $B,$  from  $B$  to  $C,$  from  $C$  to  $D,$  or from  $D$  to  $A.$   Initially each switch is in position  $A.$   The switches are labeled with the 1000 different integers  $2^x3^y5^z,$  where  $x, y,$  and  $z$  take on the values  $0, 1, \ldots, 9.$   At step  $i$  of a 1000-step process, the  $i$ th switch is advanced one step, and so are all the other switches whose labels divide the label on the  $i$ th switch.  After step 1000 has been completed, how many switches will be in position  $A$ ?

Answer: 500


Let  $m, n, a, k$  be positive integers and  $k>1$  such that the equality   $$ 5^m+63n+49=a^k $$  holds. Find the minimum value of  $k$ .

Answer: 5


A man named Juan has three rectangular solids, each having volume  $128$ . Two of the faces of one solid have areas  $4$  and  $32$ . Two faces of another solid have areas  $64$  and  $16$ .  Finally, two faces of the last solid have areas  $8$  and  $32$ . What is the minimum possible exposed surface area of the tallest tower Juan can construct by stacking his solids one on top of the other, face to face? (Assume that the base of the tower is not exposed.)

Answer: 688


Let  $A_1$ ,  $\ldots$ ,  $A_{2022}$  be the vertices of a regular  $2022$ -gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)

In each round, Bob chooses a point in the plane (not necessarily among  $A_1, \ldots, A_{2022}$ ) and Alice responds truthfully with the color of that point. What is the smallest number  $Q$  for which Bob has a strategy to always determine the colors of points  $A_1, \ldots, A_{2022}$  in  $Q$  rounds?

Answer: 22


Let  $\Omega$  and  $\omega$  be circles with radii  $123$  and  $61$ , respectively, such that the center of  $\Omega$  lies on  $\omega$ . A chord of  $\Omega$  is cut by  $\omega$  into three segments, whose lengths are in the ratio  $1 : 2 : 3$  in that order. Given that this chord is not a diameter of  $\Omega$ , compute the length of this chord.

Answer: 42


Four. (20 points) Let $n$ be a positive integer greater than 1. If there exists a positive integer $i (i < n)$ such that
$$
\sum_{a=1}^{i} a^{2}-\sum_{b=i+1}^{n} b^{2}
$$

is a perfect square, then $n$ is called a "TI number." For example, by $\left(1^{2}+2^{2}+\cdots+9^{2}\right)-\left(10^{2}+11^{2}\right)=8^{2}$, we know that 11 is a TI number. Question: Are 46, 36, and 26 TI numbers? Why?

Answer: 46


6. $A$ is the foot of the mountain, $B$ is the peak, and $C$ is a point on the slope, with $A C=\frac{1}{3} A B$. Person A and Person B start from the foot of the mountain at the same time, reach the peak, and return to the foot, repeating this back-and-forth movement. The speed ratio of A to B is $6: 5$, and the downhill speed of both A and B is 1.5 times their respective uphill speeds. After some time, A first sees B climbing on the $A C$ segment from the peak; after some more time, A sees B climbing on the $A C$ segment from the peak for the second time. How many times has A reached the peak by the time A sees B climbing on the $A C$ segment for the second time (including this time)?

Answer: 9


3. Two trains leave from two cities at the same time. The first one travels at 40 km/h, while the second one travels at 48 km/h. How far apart will these trains be from each other after 8 hours, if they are moving in the same direction and the distance between the cities is 892 km?

Answer: 956


## Task A-3.5.

For a pair of numbers $\{a, b\}$, we say it has a weight of $|a-b|$. In how many ways can the set $\{1,2, \ldots, 12\}$ be divided into six pairs such that the total sum of the weights of these pairs is 30?

Answer: 1104


What is the last non-zero digit of the product of the first hundred natural numbers?

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

Answer: 4


10 From a point $O$ on a plane, two rays $O A$ and $O B$ are drawn. $P$ moves on $O A$, and $Q$ moves on $O B$ while maintaining $|\overrightarrow{P Q}|$ as a constant $a\left(P, Q\right.$ do not coincide with $O$). Given $\angle A O B=60^{\circ}$, $a=\sqrt{7}$, then the range of $\frac{\overrightarrow{P Q} \cdot \overrightarrow{P O}}{|\overrightarrow{P O}|}+\frac{3 \overrightarrow{Q P} \cdot \overrightarrow{Q O}}{|\overrightarrow{Q O}|}$ is $\qquad$.

Answer: \left( -\dfrac{\sqrt{7}}{2}, 7 \right]


9. Let $A B C D$ be a trapezoid with $A B \| C D$ and $A D=B D$. Let $M$ be the midpoint of $A B$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle B C D$ and the diagonal $A C$. Suppose that $B C=27, C D=25$, and $A P=10$. If $M P=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Answer: 2705


4、A row of 20 seats, some of which are already occupied. If one more person sits down, there will be two adjacent seats occupied. The minimum number of people already seated is
$\qquad$ people

Answer: 7


3. Simplify the fraction $\frac{2 x^{6}+5 x^{4}-3 x^{3}+2 x^{2}-12 x-14}{4 x^{6}-4 x^{4}-6 x^{3}-3 x^{2}+25 x-28}$. After simplification, the degrees of the polynomials in the numerator and the denominator should be reduced.

#

Answer: \dfrac{x^{3} + 2x + 2}{2x^{3} - 3x + 4}


Task solved by StrgAltEntf

# Task 4 - 120924

A convex tangential quadrilateral $A B C D$ (a quadrilateral in which a circle can be inscribed such that it touches each of the four sides of the quadrilateral at one point) has the perimeter $u$, and the radius of its incircle is $r$.

Calculate the area $F$ of this tangential quadrilateral!

Answer: \dfrac{u r}{2}


ABC is a triangle with ∠C = 90° and CA ≠ CB. CH is an altitude and CL is an angle bisector. Show that for X ≠ C on the line CL, we have ∠XAC ≠ ∠XBC. Show that for Y ≠ C on the line CH we have ∠YAC≠ ∠YBC.

Answer: \angle XAC \neq \angle XBC


In rectangle $ABCD$, $AB=1, BC=m, O$ is its center, $EO \perp$ plane $ABCD, EO=n$, and there exists a unique point $F$ on side $BC$ such that $EF \perp FD$. What conditions must $m, n$ satisfy for the angle between plane $DEF$ and plane $ABCD$ to be $60^{\circ}$?

Answer: m = 2\sqrt{2} \text{ and } n = \dfrac{3}{2}


12. At 12 o'clock, the three hands of the clock coincide. After $x$ minutes, the second hand first bisects the acute angle between the minute and hour hands. Then the value of $x$ is $\qquad$ . .

Answer: \dfrac{1440}{1427}


2. Given vectors $\boldsymbol{a} 、 \boldsymbol{b}$ satisfy $|\boldsymbol{a}|=1,|\boldsymbol{b}|=\sqrt{3}$, and $(3 a-2 b) \perp a$. Then the angle between $a 、 b$ is $\qquad$

Answer: 30^\circ


9. Let set $A$ consist entirely of positive integers, and for any $x, y \in A (x \neq y)$, we have $|x-y| \geqslant \frac{1}{25} x y$. How many numbers can $A$ contain at most?

Answer: 9


1. a) (3p) Determine the minimum of the expression $E(a)=(a+2)\left(a^{2}-6 a+16\right)$, where $a \in[0,+\infty)$.

b) (4p) Prove that for any positive real numbers $a, b, c \in[0,+\infty)$, with $a+b+c=6$, the inequality holds: $\frac{1}{a^{2}-6 a+16}+\frac{1}{b^{2}-6 b+16}+\frac{1}{c^{2}-6 c+16} \leq \frac{3}{8}$. Specify the case of equality.

Dan Popescu, Suceava

Answer: 32


11. As shown in the figure, square $\mathrm{ABCD}$ and square $\mathbf{EFGH}$ have their four pairs of sides parallel to each other. Connecting $\mathbf{CG}$ and extending it to intersect BD at point $I_{0}$. Given $BD=10, S_{\triangle BFC}=3, S_{\triangle CHD}=5$, then the length of $\mathbf{BI}$ is?

Answer: \dfrac{15}{4}


12. The parabolas $y=x^{2}-b x$, $y=-x^{2}+b x$ and the line $y=k x$ can divide the plane into at most
$\qquad$ parts.

Answer: 9


1. Three non-zero real numbers form an arithmetic progression, and their squares taken in the same order form a geometric progression. Find all possible common ratios of the geometric progression.

Answer: 1


6.4. There is a 36-digit number, in which the digits $1,2, \cdots, 9$ each appear 4 times, and except for 9, all other digits are less than the digit that follows them. It is known that the first digit of the number is 9. What is the last digit of the number? Please provide all possible answers and prove that there are no other answers.

Answer: 8


Answer: \dfrac{1005}{3017}


365. Railway Switch. How can two trains pass each other using the switch depicted here and continue moving forward with their locomotives? The small side spur is only sufficient to accommodate either a locomotive or one car at a time. No tricks with ropes or flying are allowed. Each change of direction made by one locomotive counts as one move. What is the minimum number of moves?

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-118.jpg?height=207&width=1154&top_left_y=1227&top_left_x=451)

For a more convenient solution, draw the railway tracks on a piece of paper and place a hryvnia coin and three two-kopek coins (heads up) to represent the left train, and a hryvnia coin with two two-kopek coins (tails up) to represent the right train.

Answer: 14


Consider a set of \( n \) lines drawn in the plane, no two of which are parallel and no three of which are concurrent. These lines divide the plane into a certain number of regions, some finite and some infinite. Show that it is possible to color at least \( \sqrt{\frac{n}{2}} \) of these lines in blue in such a way that there is no finite region whose perimeter is entirely blue.

Answer: \sqrt{\frac{n}{2}}


\( f : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) (where \( \mathbb{R} \) is the real line) is defined by \( f(x, y) = \left( -\frac{y}{x^2 + 4y^2}, \frac{x}{x^2 + 4y^2}, 0 \right) \). Can we find \( F : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \), such that:
1. If \( F = (F_1, F_2, F_3) \), then \( F_i \) all have continuous partial derivatives for all \( (x, y, z) \neq (0, 0, 0) \);
2. \( \nabla \times F = 0 \) for all \( (x, y, z) \neq 0 \);
3. \( F(x, y, 0) = f(x, y) \)?

Answer: \text{No}


In the diagram, the area of triangle $\triangle ABC$ is 100 square centimeters, and the area of triangle $\triangle ABD$ is 72 square centimeters. Point $M$ is the midpoint of side $CD$, and $\angle MHB = 90^{\circ}$. Given that $AB = 20$ centimeters, find the length of $MH$ in centimeters.

Answer: 8.6


In figure 1, \( \triangle ABC \) and \( \triangle EBC \) are two right-angled triangles with \( \angle BAC = \angle BEC = 90^\circ \). Given that \( AB = AC \) and \( EDB \) is the angle bisector of \( \angle ABC \), find the value of \( \frac{BD}{CE} \).

Answer: 2


Let the sequences \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) satisfy \( x_{n} + i y_{n} = \left(\frac{\sqrt{3}i - 1}{2}\right)^{n} \). Find the sum of the first 1994 terms of the sequence \(\left\{x_{n}\right\}\), denoted by \(S_{1994}\).

Answer: -1


Given natural numbers \( k \) and \( n \) such that \( 1 < k \leq n \). What is the smallest \( m \) for which the following statement holds: for any placement of \( m \) rooks on an \( n \times n \) board, it is always possible to select \( k \) rooks from these \( m \) such that no two of the selected rooks can attack each other?

Answer: n(k-1) + 1


The median $AM$ of triangle $ABC$ divides the segment $PR$, parallel to the side $AC$, with endpoints on sides $AB$ and $BC$, into segments of lengths 5 cm and 3 cm, measured from side $AB$. What is the length of side $AC$?

Answer: 13 \ \text{cm}


Let \( n \) be a positive integer. Define a sequence by \( a_{0}=1 \), \( a_{2i+1}=a_{i} \), and \( a_{2i+2}=a_{i} + a_{i+1} \) for each \( i \geq 0 \). Determine, with proof, the value of \( a_{0} + a_{1} + a_{2} + \cdots + a_{2^{n}-1} \).

Answer: \frac{3^n + 1}{2}


Given a triangle \(ABC\) with a perimeter of 1. A circle is inscribed in angle \(BAC\), lying outside the triangle \(ABC\) and touching side \(BC\) (and the extensions of sides \(AB\) and \(AC\)). Find the length of segment \(AM\), where \(M\) is the point where the circle touches line \(AC\).

Answer: 1/2


A dandelion blooms in the morning, remains yellow for three days, turns white on the fourth morning, and sheds its seeds by the evening of the fifth day. On Monday afternoon, there were 20 yellow and 14 white dandelions in the meadow. On Wednesday, there were 15 yellow and 11 white dandelions. How many white dandelions will be in the meadow on Saturday?

Answer: 6


Tiana is a young woman from the Cook Islands who dreams of becoming a published author. She has decided to write a book about the beautiful islands and their rich culture. Tiana plans to write a total of 120 pages. She writes 3 pages each day during the week and doubles her effort by writing 6 pages each day during the weekend. How many days will it take for Tiana to complete her book?

Answer: 32


5. The border of a square with a side of 9, cut out of white cardboard, is painted red. It is necessary to cut the square into 6 equal-area parts, the boundaries of which contain segments painted red with the same total length.

## Solutions

Option 1

Problem 1

Answer: 6


Answer: 2


Example 1 Let $P(x)$ be a polynomial of degree $2n$ such that
$$
\begin{array}{l}
P(0)=P(2)=\cdots=P(2 n)=0, \\
P(1)=P(3)=\cdots=P(2 n-1)=2, \\
P(2 n+1)=-6 .
\end{array}
$$

Determine $n$ and $P(x)$.
(Adapted from the 13th United States of America Mathematical Olympiad)
[Analysis] Notice that, in the given conditions, several consecutive values of the polynomial are involved, which can be associated with differences.
Thus, by Newton's formula (2), let $x=0$, we have
$$
\Delta^{k} f(0)=\sum_{i=0}^{k}(-1)^{k-i} \mathrm{C}_{k}^{i} f(i)=0 .
$$

This gives the values of $P(0), P(1), \cdots, P(2 n+1)$ and $\Delta^{2 n+1} P(0)$.

Answer: 1


Example 5 In coin tossing, if Z represents heads and F represents tails, then the sequence of coin tosses is represented by a string composed of Z and F. We can count the number of occurrences of heads followed by tails (ZF), heads followed by heads (ZZ)...... For example, the sequence ZZFFZZZZFZZFFFF is the result of 15 coin tosses, which includes 5 ZZ, 3 ZF, 2 FZ, and 4 FF. How many sequences of 15 coin tosses have exactly 2 ZZ, 3 ZF, 4 FZ, and 5 FF?
(4th American Invitational Mathematics Examination (AIME))

Answer: 560


9. Let $[x]$ denote the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then the solution to the equation
$$
[x]^{4}+\{x\}^{4}+x^{4}=2048
$$

is

Answer: -3 - \sqrt{5}


Consider coins with positive real denominations not exceeding 1 . Find the smallest $C>0$ such that the following holds: if we are given any 100 such coins with total value 50 , then we can always split them into two stacks of 50 coins each such that the absolute difference between the total values of the two stacks is at most $C$.

Answer: \dfrac{50}{51}


In triangle $\vartriangle ABC$ with orthocenter $H$, the internal angle bisector of $\angle BAC$ intersects $\overline{BC}$ at $Y$ . Given that $AH = 4$, $AY = 6$, and the distance from $Y$ to $\overline{AC}$ is $\sqrt{15}$, compute $BC$.

Answer: 4\sqrt{35}


3. We will call a pair of numbers magical if the numbers in the pair add up to a multiple of 7. What is the maximum number of magical pairs of adjacent numbers that can be obtained by writing down all the numbers from 1 to 30 in a row in some order?

Answer: 26


Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

[i]Proposed by Ray Li[/i]

Answer: 213


B1. Krogu s ploščino $100 \pi \mathrm{cm}^{2}$ včrtamo pravokotnik, katerega dolžini stranic se razlikujeta za $4 \mathrm{~cm}$. Izračunaj ploščino pravokotnika.

Answer: 192


In parallelogram \(ABCD\), the angle \(BAD\) is acute and the side \(AD\) is shorter than the side \(AB\). The bisector of the angle \(\angle BAD\) intersects the side \(CD\) at \(E\). From \(D\), a perpendicular to \(AE\) is drawn, which intersects \(AE\) at \(P\) and \(AB\) at \(F\). A perpendicular from \(E\) to \(AE\) is drawn, intersecting the side \(BC\) at \(Q\). Additionally, the segment \(PQ\) is parallel to \(AB\) and the length of \(AB\) is \(20 \text{ cm}\).

a) Find the value of \(\frac{CQ}{AD}\).

b) Find the length of the side \(AD\).

Answer: \dfrac{1}{2}


2. (8 points) Rongrong transferred from Class 1 to Class 2, and Leilei transferred from Class 2 to Class 1. As a result, the average height of students in Class 1 increased by 2 cm, and the average height of students in Class 2 decreased by 3 cm. If Leilei's height is 158 cm and Rongrong's height is 140 cm, then the total number of students in the two classes is $\qquad$ people.

Answer: 15


Magozinov A.

A straight stick 2 meters long was sawn into $N$ sticks, the length of each of which is expressed as an integer number of centimeters. For what smallest $N$ can it be guaranteed that, using all the resulting sticks, one can, without breaking them, form the contour of some rectangle?

Answer: 102


The lateral sides of a trapezoid lie on perpendicular lines. Find the area of the quadrilateral with vertices at the midpoints of the diagonals and at the midpoints of the bases of the trapezoid, given that its lateral sides are equal to \(a\) and \(b\).

Answer: \dfrac{ab}{4}


3. The polar equation of curve $C$ is $\rho=1+\cos \theta$, and the polar coordinates of point $A$ are $(2,0)$. When curve $C$ rotates once around $A$ in its plane, the area of the figure it sweeps through is $\qquad$.

Answer: \dfrac{16}{3}\pi


Example 2. Given $a_{\mathrm{n}}=3 n-1$, find $S_{\mathrm{n}}$.

Answer: \dfrac{3n^2 + n}{2}


86. Chen Gang and Zhang Qiang started from their respective homes at the same time and walked towards each other. They first met at a point 5 kilometers away from Chen Gang's home. After meeting, they continued to walk at their original speeds. Chen Gang reached Zhang Qiang's home, and Zhang Qiang reached Chen Gang's home, and both immediately returned. As a result, they met again at a point 3 kilometers away from Zhang Qiang's home. Therefore, the distance between Chen Gang's home and Zhang Qiang's home is $\qquad$ kilometers.

Answer: 12


4. Let natural numbers $p, q(p<q)$ be given. Find the least natural number $m$ with the following property: the sum of all fractions whose denominators (in lowest terms) are equal to $m$ and whose values lie in the open interval $(p, q)$ is at least $56\left(q^{2}-p^{2}\right)$.

Answer: 113


Example 2. A chord with slope $t$ is drawn through the endpoint $(-a, 0)$ of the real axis of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Find the equation of the line passing through the other endpoint of the chord and the endpoint $(a, 0)$ of the real axis.

Answer: y = \dfrac{b^2}{a^2 t}(x - a)


Example 2. Given a rectangular cuboid $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with length, width, and height being $a, b, c(a>b)$ respectively. Find the angle formed by $A C^{\prime}$ and $B D$. (81 College Entrance Examination for Science Mathematics Supplementary Paper, Question 7)

Answer: \arccos \dfrac{a^2 - b^2}{\sqrt{(a^2 + b^2)(a^2 + b^2 + c^2)}}


Answer: \dfrac{2}{3}


7. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
\cdots \cdots \cdots \cdots \cdots \\
a_{n 1} a_{n 2} a_{n 3} a_{n 4} \cdots a_{n n}
\end{array}
$$

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+a_{23}+\cdots+a_{n n}$.

Answer: 2 - \dfrac{n + 2}{2^n}


Isosceles triangles $A B C(A B=B C)$ and $A_{1} B_{1} C_{1} \quad\left(A_{1} B_{1}=B_{1} C_{1}\right)$ are similar and $A C: A_{1} C_{1}=5: \sqrt{3}$.

Vertices $A_{1}$ and $B_{1}$ are located on sides $A C$ and $B C$, respectively, and vertex $C_{1}$ is on the extension of side

$A B$ beyond point $B$, and $A_{1} B_{1} \perp B C$. Find the angle $B$.

Answer: 120^\circ


Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$ .
Please give the answer directly without any intermediate steps.

Answer: 401


Let \( A B C D \) be a unit square, and let \( P \) be a point inside triangle \( A B C \). Let \( E \) and \( F \) be the points on \( A B \) and \( A D \), respectively, such that \( A E P F \) is a rectangle. Let \( x \) and \( y \) denote the lengths of \( A E \) and \( A F \), respectively.

Determine a two-variable polynomial \( f(x, y) \) with the property that the area of \( A E P F \) is greater than the area of each of the quadrilaterals \( E B C P \) and \( F P C D \) if and only if \( f(x, y)>0 \).

Answer: 3xy - x + y - 1


Exercise 7. An integer $k>1$ is said to be superb if there exist $m, n$, and $a$, three strictly positive integers such that

$$
5^{\mathrm{m}}+63 \mathrm{n}+49=\mathrm{a}^{\mathrm{k}}
$$

Determine the smallest superb integer.

Answer: 5


5. As shown in Figure 9, in $\triangle A B C$, $\angle C=$ $90^{\circ}, I$ is the intersection of the angle bisectors $A D$ and $B E$ of $\angle A$ and $\angle B$. Given that the area of $\triangle A B I$ is 12. Then the area of quadrilateral $A B D E$ is $\qquad$

Answer: 24


Two squares are arranged as shown in the figure, and the marked segments are equal. Prove that triangle $BDG$ is isosceles.

Answer: \triangle BDG \text{ is isosceles}


10.2. Let $f(n)$ be equal to the product of the even digits of the natural number $\mathrm{n}$ or be zero if there are no even digits. Find the sum $f(1)+f(2)+\cdots+f(100)$.

Answer: 620


Inside an acute-angled triangle \(ABC\), a point \(O\) is taken. Likewise, inside an acute-angled triangle \(A^{\prime} B^{\prime} C^{\prime}\), a point \(O^{\prime}\) is taken. From point \(O\), perpendiculars are dropped: \(OA_1\) to side \(BC\), \(OB_1\) to side \(CA\), and \(OC_1\) to side \(AB\). Similarly, from point \(O^{\prime}\), perpendiculars are dropped: \(O^{\prime}A_1^{\prime}\) to \(B^{\prime}C^{\prime}\), \(O^{\prime}B_1^{\prime}\) to \(C^{\prime}A^{\prime}\), and \(O^{\prime}C_1^{\prime}\) to \(A^{\prime}B^{\prime}\). It turns out that \(OA_1 \parallel O^{\prime}A^{\prime}\), \(OB_1 \parallel O^{\prime}B^{\prime}\), \(OC_1 \parallel O^{\prime}C^{\prime}\), and \(OA_1 \cdot O^{\prime}A^{\prime} = OB_1 \cdot O^{\prime}B^{\prime} = OC_1 \cdot O^{\prime}C^{\prime}\).

Prove that \(O^{\prime}A_1^{\prime} \parallel OA\), \(O^{\prime}B_1^{\prime} \parallel OB\), \(O^{\prime}C_1^{\prime} \parallel OC\), and \(O^{\prime}A_1^{\prime} \cdot OA = O^{\prime}B_1^{\prime} \cdot OB = O^{\prime}C_1^{\prime} \cdot OC\).

Answer: O^{\prime}A_1^{\prime} \cdot OA = O^{\prime}B_1^{\prime} \cdot OB = O^{\prime}C_1^{\prime} \cdot OC


2. Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute ($n=1,2, \ldots$), each of them chooses whether to move north, east, south, or west, and then travels a distance of $q^n$ meters. After an integer number of minutes, they are again at the same point in the plane (not necessarily point $X$), and the paths they have traveled are not entirely identical. Determine all possible values of the number $q$.

(United Kingdom)

Answer: 1


A quadrilateral is divided by its diagonals into four triangles. If the square of the area of one of these triangles is equal to the product of the areas of the adjacent triangles, then the quadrilateral is either a trapezoid or a parallelogram.

Answer: \text{The quadrilateral is either a trapezoid or a parallelogram.}


7. Given a fixed point $A(4, \sqrt{7})$. If a moving point $P$ is on the parabola $y^{2}=4 x$, and the projection of point $P$ on the $y$-axis is point $M$, then the maximum value of $|P A|-|P M|$ is $\qquad$.

Answer: 5


Let $n$ be a non-zero natural number, and let $m$ be the LCM (least common multiple) of all integers less than or equal to $n$. Let $\sigma$ be a permutation of the set $\{1, 2, \ldots, n\}$. Show that $\sigma$ has a finite order, and that the order of $\sigma$ divides $m$.

Answer: \text{The order of } \sigma \text{ divides } m


2. A right cylinder with radius $r=\sqrt{39}$ and a right cone with radius $R=9$ have the same height $H=15$. At what non-zero height $x$, should the cylinder and the cone be intersected so that the newly formed cylinder with height $x$ and the intersected cone with height $x$ have equal volumes?

Answer: 10


Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

Answer: 2^{n-1}


10. As shown in the figure, $ABCD$ is a square piece of paper. The paper is folded along $CE$, with point $D$ folding to point $G$. Then, the paper is folded along $CF$, with point $B$ folding to point $G$. If $DE=18$, $BF=6$, then the area of $\triangle AEF$ is $\qquad$ .

Answer: 108


45. Find the largest real number $\alpha$, such that there exists an infinite sequence of positive integers $\left\{a_{n}\right\}_{n=1}^{+\infty}$, satisfying:
(1) For any $n \in \mathbf{N}^{*}$, we have $a_{n}>2008^{n}$;
(2) For any $n \in \mathbf{N}^{*}, n \geqslant 2$, the number $a_{n}^{\alpha}$ does not exceed the greatest common divisor of all numbers in the set $\left\{a_{i}+a_{j} \mid i+j=n, i, j \in \mathbf{N}^{*}\right\}$.

Answer: \dfrac{1}{2}


Example 1 For any $n(n \geqslant 2)$ points given in space. If the midpoint of each pair of points connected by a line segment is painted red, find the minimum number of red points.

Answer: 2n - 3


12. A. In the Cartesian coordinate system $x O y$, points with integer abscissas and ordinate values that are perfect squares are called "good points". Find the coordinates of all good points on the graph of the quadratic function $y=(x-90)^{2}-4907$.

Answer: (2544, 6017209)


Bakayev E.V.

Forty children were holding hands in a circle. Out of them, 22 were holding hands with a boy and 30 were holding hands with a girl. How many girls were in the circle?

#

Answer: 24


5. A5 (KOR) Let $\mathbb{R}^{+}$be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$that satisfy the following conditions:
(i) $f(x y z)+f(x)+f(y)+f(z)=f(\sqrt{x y}) f(\sqrt{y z}) f(\sqrt{z x})$ for all $x, y, z \in$ $\mathbb{R}^{+}$.
(ii) $f(x)<f(y)$ for all $1 \leq x<y$.

Answer: f(x) = x^k + \dfrac{1}{x^k}


1. Let $A B C$ be a triangle with $\angle B A C>90^{\circ}$. Let $D$ be a point on the segment $B C$ and $E$ be a point on the line $A D$ such that $A B$ is tangent to the circumcircle of triangle $A C D$ at $A$ and $B E$ is perpendicular to $A D$. Given that $C A=C D$ and $A E=C E$, determine $\angle B C A$ in degrees.

Answer: 45


1.58. Perpendiculars are drawn from the vertex of the acute angle of a rhombus to the lines containing the sides of the rhombus to which this vertex does not belong. The length of each perpendicular is 3 cm, and the distance between their bases is $3 \sqrt{3}$ cm. Calculate the lengths of the diagonals of the rhombus.

Answer: 6


1. Alice and Ben play the game on a board with 72 cells around a circle. First, Ben chooses some cells and places one chip on each of them. Each round, Alice first chooses one empty cell and then Ben moves a chip from one of the adjacent cell onto the chosen one. If Ben fails to do so, the game ends; otherwise, another round follows. Determine the smallest number of chips for which Ben can guarantee that the game will last for at least 2023 rounds.

(Václav Blažej)

Answer: 36


11. The selling price of a commodity is an integer number of yuan. 100 yuan can buy at most 3 pieces. Person A and Person B each brought several hundred-yuan bills. The money A brought can buy at most 7 pieces of this commodity, and the money B brought can buy at most 14 pieces. When the two people's money is combined, they can buy 1 more piece. The price of each piece of this commodity is yuan.

Answer: 27


27. The number of five-digit numbers that are multiples of 4 and whose sum of digits is 40 is $\qquad$.

 untranslated part: $\qwen$ (Note: This part is not translatable as it is a placeholder or variable in the original text.)

For the purpose of a complete translation, the final answer should look like this:

27. The number of five-digit numbers that are multiples of 4 and whose sum of digits is 40 is $\qquad$.

Answer: 20


5. For a set, the difference between the maximum and minimum elements is called the "capacity" of the set. Let $2 \leqslant r \leqslant n$, and let $F(n, r)$ denote the arithmetic mean of the capacities of all $r$-element subsets of the set $M=\{1,2, \cdots, n\}$. Then $F(n, r)=$

Answer: \dfrac{(n + 1)(r - 1)}{r + 1}


There are some positive integers with more than two digits, such that each pair of adjacent digits forms a perfect square. Then the sum of all positive integers satisfying the above conditions is ?

Answer: 97104


12.170. In triangle $A B C$, the acute angles $\alpha$ and $\gamma (\alpha>\gamma)$ at the base $A C$ are given. From vertex $B$, the altitude $B D$ and the median $B E$ are drawn.

Find the area of triangle $B D E$, if the area of triangle $A C B$ is $S$.

Answer: \dfrac{S \sin(\alpha - \gamma)}{2 \sin(\alpha + \gamma)}


One, (20 points) A certain musical instrument has 10 holes, sequentially labeled as hole 1, hole 2, $\cdots \cdots$, hole 10. When played, the sound quality index $D$ of the $n$-th hole is given by the relationship
$$
D=n^{2}+k n+90 .
$$

The lowest sound quality index of the instrument is $4 k+106$. Find the range of values for the constant $k$.

Answer: [-9, -7]


Each face of a cube is painted either red or blue, each with probability \( \frac{1}{2} \). The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide the value of k + m.

Answer: 21


The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. If the number of possible values of $n$ is denoted as $k$, find the value of $k$.

Answer: 250500


In \(\triangle ABC\), \(AB = BC\), and \(\overline{BD}\) is an altitude. Point \(E\) is on the extension of \(\overline{AC}\) such that \(BE = 10\). The values of \(\tan \angle CBE\), \(\tan \angle DBE\), and \(\tan \angle ABE\) form a geometric progression, and the values of \(\cot \angle DBE,\) \(\cot \angle CBE,\) \(\cot \angle DBC\) form an arithmetic progression. The area of \(\triangle ABC\) is in the form \(\frac{m}{n}\). Please calculate the value of \(m+n\).

Answer: 53


Two \(4 \times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \(k - m\pi\), please provide the value of \(k + m\).

Answer: 30


Let $f(t) = \frac{t}{1-t}$, where $t \neq 1$. If $y = f(x)$, express $x$ in terms of $y$. The original answer is in the form \frac{m}{n}, please give the value of m + n.

Answer: 2


A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between them. Determine all the values of $ n$ not exceeding 10 for which this arrangement is possible. For which values of $ n$ is it impossible?

Answer: 3


p1. Determine what is the smallest positive integer by which the number $2010$ must be multiplied so that it is a perfect square.

p2. Consider a chessboard of $8\times 8$. An [i]Alba trajectory[/i] is defined as any movement that contains $ 8$ white squares, one by one, that touch in a vertex. For example, the white square diagonal is an Alba trajectory. Determine all possible Alba trajectories that can be built on the board.

p3. Find all prime numbers $p$ such that $p + 2$ and $2p + 5$ are also prime.

p4. Find all natural numbers $n$ such that $1 / n$ has a decimal representation finite.

p5. Prove that all numbers of the form $5^n$, with a positive integer $n$, can be written as the sum of two squares of integer numbers.

p6. Let $ABCD$ be a square with side $a$, $N$ be the midpoint of side $BC$ and $M$ be a point on $CD$ such that $MC = 2 MD$. Let $P$ be the intersection point of lines $AM$ and $DB$, $Q$ be the intersection point of lines $AN$ and $BD$. Calculate the area of the pentagon $MPQNC$ in terms of $a$.

PS. Juniors P2 was also proposed as [url=https://artofproblemsolving.com/community/c4h2692797p23378693]Seniors [/url] P2.

Answer: 2010


Example 2. It is known that the equation $a x^{2}-(a-3) x+a-2=0$ has at least one integer solution, find the integer $a$.

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

(Note: The note is for you, the assistant, and should not be included in the output.)

Example 2. It is known that the equation $a x^{2}-(a-3) x+a-2=0$ has at least one integer solution, find the integer $a$.

Answer: -1


4. From $0, 1, 2, 3, 4, 5, 6, 7$, select 6 digits to form a six-digit number without repeated digits, which can be divisible by 63. The largest six-digit number is $\qquad$ .

Answer: 765324


Tokorev S.i.

In each cell of a $4 \times 4$ table, there is a sign "+" or "-". It is allowed to simultaneously change the signs to the opposite in any cell and in all cells that share a side with it. How many different tables can be obtained by repeatedly applying such operations?

Answer: 4096


11. As shown in the figure, in $\triangle \mathrm{ABC}$, $\mathrm{M}$ is the midpoint of side $\mathrm{AB}$, $\mathrm{N}$ is a trisection point on side $\mathrm{AC}$, and $\mathrm{CM}$ intersects $\mathrm{BN}$ at point K. If the area of $\triangle B C K$ is 1, then the area of $\triangle A B C$ is $\qquad$ .

Answer: 4


$(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$. If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take.

Answer: [2, 5]


In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.

Answer: n


1.200 people stand in a circle, some of whom are honest people, and some are liars. Liars always tell lies, while honest people tell the truth depending on the situation. If both of his neighbors are honest people, he will definitely tell the truth; if at least one of his neighbors is a liar, he may sometimes tell the truth and sometimes lie. It is known that among these 200 people, 100 people say they are liars, and 100 people say they are honest. How many honest people can there be at most in these 200 people?

Answer: 150


21. Among the 100 integers from $1 \sim 100$, arbitrarily select three different numbers to form an ordered triplet $(x, y, z)$. Find the number of triplets that satisfy the equation $x+y=3z+10$.

Answer: 3194


LIX OM - I - Task 7

In an $ n $-person association, there are $ 2n-1 $ committees (any non-empty set of association members
forms a committee). A chairperson must be selected in each committee. The following condition must be met: If
committee $ C $ is the union $ C = A\cup B $ of two committees $ A $ and $ B $, then the chairperson of committee $ C $ is also
the chairperson of at least one of the committees $ A $, $ B $. Determine the number of possible selections of chairpersons.

Answer: n!


A passenger train traveling from $A$ to $B$ has a speed of $60 \mathrm{~km} / \mathrm{h}$. The train, which departs later, is a fast train with a speed of $120 \mathrm{~km} / \mathrm{h}$ and must catch up to the passenger train at $B$. After covering $2 / 3$ of its journey, the passenger train continues at half its original speed and is caught by the fast train 80 kilometers before $B$. How far is $A$ from $B$?

Answer: 360


1. (17 points) Masha's tablet, which she needed for a presentation at school, was completely drained. Using additional equipment, the tablet can be fully charged in 2 hours and 40 minutes, without it in 8 hours. Masha first put the discharged tablet on regular charging, and when she found the equipment, she switched to fast charging and fully charged the tablet. It is known that the tablet charges uniformly both with fast and regular charging. Find the total charging time of the tablet, if it was on fast charging for a third of the total charging time. Give your answer in minutes.

Answer: 288


Problem 5.4. Arrange the digits from 1 to 6 (each must be used exactly once) so that the sum of the three numbers located on each of the 7 lines is equal to 15. In your answer, indicate which digits should be placed at positions $A-F$.

![](https://cdn.mathpix.com/cropped/2024_05_06_6da73bfd3e09e8b55e3fg-09.jpg?height=363&width=456&top_left_y=612&top_left_x=499)

(a) Instead of the letter $A$

(b) Instead of the letter $B$

(c) Instead of the letter $C$

(d) Instead of the letter $D$

(e) Instead of the letter $E$

(f) Instead of the letter $F$
(1) the digit 1 should be placed

(2) the digit 2 should be placed

(3) the digit 3 should be placed

(4) the digit 4 should be placed

(5) the digit 5 should be placed

(6) the digit 6 should be placed

Answer: 6


There are some positive integers on a blackboard, some of which are red and some are blue. A attempts to erase some (but not all) of the numbers, such that the remaining red numbers are a multiple of 3 and the sum of all remaining numbers is a multiple of 2013. Despite A's intention, it is impossible to achieve this. What is the maximum possible number of numbers on the blackboard?

Answer: 2014


5. Let $x, y$ be real numbers, and $x^{2}+x y+y^{2}=3$. Find the maximum and minimum values of $x^{2}-$ $x y+y^{2}$.
(1994, Huanggang City, Hubei Province Junior High School Mathematics Competition)

Answer: 9


Baba Yaga in her hut on chicken legs has kept fairy tale animals. All of them, except for two, are Talking Cats; all, except for two, are Wise Owls; the rest are Whiskered Cockroaches. How many inhabitants are there in Baba Yaga's hut?

Answer: 3


Find the length of the diagonal of a square, two vertices of which lie on the circle of radius \( R \), and the other two vertices lie on the tangent to this circle.

Given: \(\omega(0,|OA|)\) is the circle, \(|OA|=R\), \(ABCD\) is the square, \(A \in \omega\), \(D \in \omega\), \((BC)\) is tangent to \(\omega\), \([AC]\) is the diagonal of the square.

Find: \(|AC|\).

Answer: \dfrac{8\sqrt{2}}{5}R


The segment $MN$, which is parallel to side $CD$ of quadrilateral $ABCD$, divides its area in half (points $M$ and $N$ lie on sides $BC$ and $AD$). The lengths of the segments drawn from points $A$ and $B$ parallel to $CD$ to the intersection with lines $BC$ and $AD$ are $a$ and $b$. Prove that $MN^{2}=\left(a b+c^{2}\right) / 2$, where $c=CD$.

Answer: MN^2 = \dfrac{ab + c^2}{2}


Arun and Disha have some numbered discs to share out between them. They want to end up with one pile each, not necessarily of the same size, where Arun's pile contains exactly one disc numbered with a multiple of 2 and Disha's pile contains exactly one disc numbered with a multiple of 3.

For each case below, either count the number of ways of sharing the discs, or explain why it is impossible to share them in this way.
(a) They start with ten discs numbered from 1 to 10.
(b) They start with twenty discs numbered from 1 to 20.

Answer: 48


## Task B-1.3.

Ships Bonaca and Galeb are sailing straight and at a constant speed towards the same port. At noon, the positions of the port and the ships form the vertices of an equilateral triangle. At the moment when ship Galeb has traveled $30 \mathrm{~km}$, the positions of the port and the ships form the vertices of a right triangle. When ship Bonaca entered the port, Galeb still had $12.5 \mathrm{~km}$ to go to the port. How far apart were the ships at noon?

Answer: 50


On each of the four cards, a natural number is written. Two cards are randomly taken, and the numbers on them are added. With equal probability, this sum can be less than 9, equal to 9, and greater than 9. What numbers can be written on the cards?

#

Answer: \{1, 2, 7, 8\}


1. Find the natural number $n$ if the tenth term in the expansion of the binomial $\left(\frac{1}{5} x+\frac{2}{5}\right)^{n}$ has the largest coefficient.

Answer: 13


On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$?

[i]Proposed by S. Berlov[/i]

Answer: 2


Let $a_{n}$ be the first (most significant) digit in the decimal representation of $n^{2}$ for $n=1,2,3, \ldots$ (e.g., $a_{1}=1, a_{2}=4, a_{3}=9, a_{4}=1, a_{5}=2, \ldots$). Prove that this sequence is not periodic.

Answer: \text{The sequence is not periodic.}


8,9

In an isosceles right triangle, the radius of the inscribed circle is 2.

Find the distance from the vertex of the acute angle to the point where the inscribed circle touches the opposite leg of this angle.

Answer: 2\sqrt{7 + 4\sqrt{2}}


8. There is an unlimited number of test tubes of three types - A, B, and C. Each test tube contains one gram of a solution of the same substance. Test tubes of type A contain a $10\%$ solution of this substance, type B $-20\%$ solution, and type C $-90\%$ solution. Sequentially, one after another, the contents of the test tubes are poured into a certain container. In this process, two consecutive pourings cannot use test tubes of the same type. It is known that a $20.17\%$ solution was obtained in the container, performing the minimum number of pourings. What is the maximum number of test tubes of type C that can be used in this process?

Answer: 73


G is a group, not necessarily abelian, with the operation denoted by juxtaposition and the identity by 1. There is a function φ : G → G such that if \(abc = def = 1\), then \(\phi(a)\phi(b)\phi(c) = \phi(d)\phi(e)\phi(f)\). Prove that there exists an element \(k \in G\) such that \(kφ(x)\) is a homomorphism.

Answer: k = \phi(1)^{-1}


Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.

Answer: 84


35th Putnam 1974 Problem A5 The parabola y = x 2 rolls around the fixed parabola y = -x 2 . Find the locus of its focus (initially at x = 0, y = 1/4). Solution

Answer: y = \dfrac{1}{4}


Suggestion: Show that triangle BEC is isosceles.

Facts that Help: The sum of the measures of the interior angles of a triangle is equal to $180^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_64d7a736a938f072abd8g-4.jpg?height=454&width=402&top_left_y=1903&top_left_x=193)

Inside a triangle $ABC$, a point $E$ is taken such that $AE = BE$ and $AB = EC$. If $A \hat{B E} = \alpha = E \hat{C A}$, $E \hat{A} C = 2 \alpha$ and $E \hat{B C} = 5 \alpha$, determine $\alpha$.

##

Answer: 15^\circ


For each integer $n\geq3$ , let $f(n)$ be the number of $3$ -element subsets of the vertices of a regular $n$ -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$ .
Please give the answer directly without any intermediate steps.

Answer: 245


. Decompose 2008 into a sum of natural numbers whose product is maximal.

Answer: 3^{668} \times 2^2


## Task A-4.1.

Determine all prime numbers $p$ and $q$ such that $p^{q}+1$ is also prime.

Answer: 2


6. Given that the odd number $n$ is a three-digit number, and the sum of the last digits of all its factors (including 1 and $n$) is 33. Then $n$ $=$ . $\qquad$

Answer: 729


6. (i) Consider two positive integers $a$ and $b$ which are such that $\boldsymbol{a}^{\boldsymbol{a}} \boldsymbol{b}^{\boldsymbol{b}}$ is divisible by 2000 . What is the least possible value of the product $a b$ ?

(ii) Consider two positive integers $a$ and $b$ which are such that $\boldsymbol{a}^{\boldsymbol{b}} \boldsymbol{b}^{\boldsymbol{a}}$ is divisible by 2000 . What is the least possible value of the product $a b$ ?

Answer: 10


Example 15. Let $a, b \in N$, when $a^{2}+b^{2}$ is divided by $a+b$, the quotient is $q$, and the remainder is $r$. Find all pairs of numbers $(a, b)$ such that $q^{2}+r=1977$. (19th 1MO problem)

Answer: (50, 7)


3. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=B C=4, A A_{1}=2, P$ is a point on the plane $A_{1} B C$, and $\overrightarrow{D P} \cdot \overrightarrow{P B}=0$. Then, the area of the plane region enclosed by all points $P$ that satisfy the above conditions is $\qquad$.

Answer: \dfrac{36}{5}\pi


Let $\xi_{1}, \ldots, \xi_{2n}$ be independent Bernoulli random variables with $\mathrm{P}\left\{\xi_{k}=1\right\}=\mathrm{P}\left\{\xi_{k}=-1\right\}=1/2$ for $k \leqslant 2n$. Define $S_{0}=0$, $S_{k}=\xi_{1}+\ldots+\xi_{k}$, and let

$$
g_{2n}=\max \left\{0<2k \leqslant 2n: S_{2k}=0\right\}
$$

be the last zero moment in the sequence $(S_{2}, S_{4}, \ldots, S_{2n})$; if such a moment does not exist, then set $g_{2n}=0$.

Show that

$$
\mathrm{P}\left\{g_{2n}=2k\right\}=u_{2n} u_{2(n-k)}, \quad 1 \leqslant k \leqslant n
$$

where $u_{2k}=\mathrm{P}\left\{S_{2k}=0\right\}=2^{-2k} \binom{2k}{k}$.

Additionally, relate the distribution of $g_{2n}$ to the probability $P_{2k, 2n}$ that a "particle" spends $2k$ units of time on the positive axis within the interval $[0, 2n]$ using the relation:

$$
\sum_{\left\{k: 0<\frac{k}{n} \leqslant x\right\}} \mathrm{P}\left\{g_{2n}=2k\right\} \rightarrow \frac{2}{\pi} \arcsin \sqrt{x}, \quad n \rightarrow \infty,
$$

indicating that asymptotically, the distribution of the last zero position complies with the arcsine law.

Answer: \mathrm{P}\left\{g_{2n}=2k\right\} = u_{2k} u_{2(n - k)}


Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$, each with $3$ elements, so that no two of them are disjoint.

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


3. $N$ is the set of positive integers. Define the function $f$ on $N$ as follows:
$f(1)=1, f(3)=3$, and for $n \in N$ we have $f(2 n)=f(n)$,
$f(4 n+1)=2 f(2 n+1)-f(n)$,
$f(4 n+3)=3 f(2 n+1)-2 f(n)$.

Question: How many $n \in N$, and $n \leqslant 1988$ such that $f(n) = n$?

Answer: 92


Spivak A.V.

If for a number $x$ we calculate the sum of its digits and repeat this process with the resulting number two more times, we will get three more numbers. Find the smallest $x$ for which all four numbers are different, and the last of them is 2.

Answer: 2999


Exercise 16. Determine the number of integers divisible by 11, whose decimal representation is of the form $N=a b c d a b c d \cdots a b c d$, the pattern $a b c d$ being repeated 2016 times, and $a, b, c, d$ being digits such that $a \neq 0$.

Answer: 819


Chibi Maruko went to the mall to buy pencils. The mall has many different types of pencils, with prices of 1 yuan, 3 yuan, 5 yuan, and 8 yuan. Maruko wants to buy two pencils, and there are $\qquad$ different total prices.

Answer: 9


6. (8 points) Solve the equation

$$
\sqrt{a^{2}+c-1-2 x-x^{2}}+\sqrt{b^{2}+c-1-2 x-x^{2}}=\sqrt{x^{2}+2 x+1-c+(a+b)^{2}}
$$

In your answer, write the sum of the squares of all possible pairwise differences of the real roots of the equation. Numbers $a, b>0$.

Answer: 4c


[Law of Cosines] [Law of Sines]

In an acute triangle $ABC$, the angle $\angle ACB = 75^{\circ}$, and the altitude dropped from this vertex is 1.

Find the radius of the circumscribed circle if it is known that the perimeter of triangle $ABC$ is $4 + \sqrt{6} - \sqrt{2}$.

Answer: \sqrt{6} - \sqrt{2}


One. (20 points) As shown in Figure 5, $P$ is a point outside circle $\odot O$, $PA$ is tangent to $\odot O$ at point $A$, and $PBC$ is a secant of $\odot O$. $AD \perp PO$ at point $D$. If $PB=4$, $CD=5$, $BC=6$, find the length of $BD$.

Answer: 2


19. Different letters represent different numbers, for the five-digit number $\overline{A B C D E}$, the differences between adjacent digits are $F 、 G 、 H 、 I$, then the maximum value of $\overline{A B C D E}$ is $\qquad$.

Answer: 98274


Three. (20 points) As shown in Figure 2, the distance from the center O to a chord of circle $\odot O$ is $O H=2008$. In the two segments, two squares $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ are constructed, where points $A, D, A_{1}, B_{1}$ are on the arc, and points $B, C, C_{1}, D_{1}$ are on the chord. Find the difference in the side lengths of these two squares (the larger side minus the smaller side).

Answer: \dfrac{16064}{5}


Problem 2. On each of the two gardens, Grandpa planted the same number of turnips. If Granddaughter enters the garden, she pulls out exactly $1 / 3$ of the turnips present at that moment. If Doggy enters, she pulls out $1 / 7$ of the turnips, and if Mousey enters, she pulls out only $1 / 12$ of the turnips. By the end of the week, 7 turnips remained in the first garden, and 4 in the second. Did Doggy enter the second garden?

Answer: Yes


3. In the Cartesian coordinate system, if a line is at a distance of 1 from point $A(2,2)$, and at a distance of 3 from point $B(m, 0)$, then the set of real values for $m$ for which exactly three such lines exist is $\qquad$.

Answer: \{2 + 2\sqrt{3}, 2 - 2\sqrt{3}\}


For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $\frac{b_{36}}{b_{25}}$.

Answer: 5394


In an acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$, and $\tan C = \frac{4\sqrt{21}}{17}$, the value of $EF$ may be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Answer: 267


Let $D$, $E$, and $F$ be the feet of the altitudes from vertices $A$, $B$, and $C$ of an acute triangle $\triangle ABC$, respectively. Given that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be a point on segment $BE$ such that $AP = 42$. Determine the length of $CP$.

Answer: 14


fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part 1 of the contest, go outside for a break, and come back to take part 2 of the contest. fantasticbobob sits among the contestants during part 1, also goes outside during the break, but when he returns, he finds that his seat has been taken. Furthermore, each of the $840$ contestants now sit in a chair horizontally or vertically adjacent to their original chair. How many seats could fantasticbobob have started in?

Answer: 421


Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?

Answer: 33


We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25\%$ have sent a message to $A$. How many possible two-digit values of $n$ are there?

Answer: 26


The numbers $1, 2, \ldots, 49, 50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a, b, c$ from the board, replaces them by their sum $a + b + c$, and writes $(a + b)(b + c)(c + a)$ in her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all 24 numbers written in the notebook. Let $A$ and $B$ be the maximum and minimum possible sums that Ann can obtain. Find the value of $\frac{A}{B}$.

Answer: 4


Balls of $3$ colours — red, blue, and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.

Answer: 9


King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 65


There are $14$ players participating in a chess tournament, each playing one game with every other player. After the tournament, players are ranked in descending order based on their points. The sum of the points of the first three players is equal to the sum of the points of the last nine players. What is the highest possible number of draws in the tournament? (For a victory, a player gets $1$ point, for a loss $0$ points, and in a draw, both players get $0.5$ points.)

Answer: 40


Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Answer: 101


Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30\% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet.

Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs, $T$. Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then, $T$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime. Find $m+n$.

Answer: 101


Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?

Answer: 23


Every cell of a $3 \times 3$ board is colored either red or blue. Find the number of all possible colorings such that there are no $2 \times 2$ squares where all cells are red.

Answer: 417


Let $A$ be a set of numbers chosen from $1, 2, \ldots, 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non-equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

Answer: 10


The picture below shows how Juan wants to divide a square field into three regions, all sharing a well at vertex $B$. If the side length of the field is $60$ meters, and each of the three regions has the same area, how far must the points $M$ and $N$ be from $D$? Note: The area of each region includes the area occupied by the well.

```latex
\begin{asy}
pair A=(0,0), B=(60,0), C=(60,-60), D=(0,-60), M=(0,-40), N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));
\end{asy}
```

Unrelated information has been removed for clarity.

Answer: 20


Ana, Beatriz, Carlos, Diego, and Emilia play a chess tournament. Each player faces each of the other four only once. Each player gets $2$ points if they win the match, $1$ point if they draw, and $0$ points if they lose. At the end of the tournament, it turns out that the scores of the $5$ players are all different. Find the maximum number of ties there could be in the tournament and justify why there could not be a higher number of ties.

Answer: 4


Suppose rectangle $FOLK$ and square $LORE$ are on the plane such that $RL = 12$ and $RK = 11$. Compute the product of all possible areas of triangle $RKL$.

Answer: 414


On Lineland, there are 2018 bus stations numbered from 1 to 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers).

Define any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u < j_v$ for all $u,v \in \{1,2\}$ as a good group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$.

Answer: 1009


627. Calculate the derivative of the function $y=x^{3} \log _{5} x$.

Answer: 3x^2 \log_5 x + \dfrac{x^2}{\ln 5}


Outside of the equilateral triangle \(ABC\) but inside the angle \(BAC\), a point \(M\) is taken such that the angle \(CMA\) equals \(30^\circ\) and the angle \(BMA\) equals \(\alpha\). Find the angle \(ABM\).

Answer: 180^\circ - 2\alpha


Example 3 The radius of the circle is both the mean proportional between two chords and equal to the difference of these two chords. Find the degree measures of the central angles subtended by the two chords.

---

The translation maintains the original text's line breaks and format.

Answer: 108^\circ


## Task 4 - 120834

Given a circle with center $M$. A diameter of this circle is $A B$. Two points $P_{1}$ and $P_{2}$ start moving uniformly from the same time on each of the two semicircular arcs from $A$ to $B$, with the movement of point $P_{1}$ being four times as fast as that of point $P_{2}$.

Is there a time between the start and the arrival of $P_{1}$ (at $B$) when the triangles $\triangle A B P_{1}$ and $\triangle A B P_{2}$ have the same area?

If yes, determine for each such time the size of the angle $\angle A M P_{2}$!

Answer: 36^\circ


[ $\left.\frac{\text { angles between bisectors }}{\text { Dihedral angle }}\right]$

The base of the pyramid $S A B C$ is a right triangle $A B C$ ( $C$ - the vertex of the right angle). All lateral faces of the pyramid are inclined to its base at the same angle, equal to $\arcsin \frac{5}{13}$. Find the area of the lateral surface of the pyramid, if $S O-$ is the height of the pyramid, $A O=1, B O=3 \sqrt{2}$.

Answer: \dfrac{91}{25}


# Task № 3.4

## Condition:

Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.

The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, diamond, and agate, and clasps: classic, butterfly, and buckle. Anton is only satisfied when three watches are laid out in a row on the display according to the following rules:

- There must be steel watches with a classic clasp and a zircon stone
- Next to the watches with the classic clasp, there must be gold and silver watches;
- The three watches in a row must have different bracelets, precious stones, and clasps.

How many ways are there to make Anton happy?

Answer: 48


Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1\leq x,y\leq1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit goes to the point $(0,0)$ strictly before Ayush does can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.

[i]2020 CCA Math Bonanza Lightning Round #3.2[/i]

Answer: 7


6. In tetrahedron $ABCD$, $\angle ABC = \angle BAD = 90^{\circ}$, $AB = 1$, $CD = 2$, and the angle between $AD$ and $BC$ is $30^{\circ}$. Then the volume of the circumscribed sphere of the tetrahedron is $\qquad$.

Answer: \dfrac{13\sqrt{13}}{6} \pi


We attached three rods (whose thickness can be neglected) pairwise perpendicularly to each other at a common point at one of their ends. The lengths of the rods are 1, 2, and 3. We placed the resulting structure on the table so that the free ends of the rods touch the table surface. Determine the exact height at which the attachment point is above the table.

We attached three rods (whose thickness can be neglected) pairwise perpendicularly to each other at a common point at one of their ends. The lengths of the rods are 1, 2, and 3. We placed the resulting structure on the table so that the free ends of the rods touch the table surface. Determine the exact height at which the attachment point is above the table.

Answer: \dfrac{6}{7}


## Task A-3.5.

In a scalene triangle $ABC$, the lengths of two altitudes are equal to the lengths of two medians. What is the ratio of the lengths of the remaining altitude and the remaining median?

Answer: \dfrac{2}{7}


14. First group (from 8th grade. 3 points). The sequence consists of 19 ones and 49 zeros, arranged in a random order. We will call a group the maximal subsequence of identical symbols. For example, in the sequence 110001001111 there are five groups: two ones, then three zeros, then one one, then two zeros, and finally four ones. Find the expected value of the length of the first group.

Answer: \dfrac{283}{100}


## Task 4 - 040514

In Mücheln (Geisel Valley, District of Halle), the longest railway bridge in the GDR was completed. It consists of 7 identical bridge arches.

For one bridge arch, $147 \mathrm{~m}^{3}$ of concrete was used. $1 \mathrm{~m}^{3}$ of concrete has a mass of $24 \mathrm{dt}$. How many tons of concrete were needed for the bridge construction? (Round to the nearest whole ton!)

Answer: 2470


[Example 3.3.7] In the plane, there are 100 lines. Can there be 1985 intersection points among them.

Answer: 1985


Let the set \( S_{n} = \{1, 2, \cdots, n\} \). If \( X \) is a subset of \( S_{n} \), the sum of all numbers in \( X \) is called the "capacity" of \( X \) (the capacity of the empty set is defined as 0). If the capacity of \( X \) is odd (even), then \( X \) is called an odd (even) subset of \( S_{n} \).
1. Prove that the number of odd subsets of \( S_{n} \) is equal to the number of even subsets.
2. Prove that for \( n \geq 3 \), the total capacity of all odd subsets of \( S_{n} \) is equal to the total capacity of all even subsets of \( S_{n} \).
3. For \( n \geq 3 \), find the total capacity of all odd subsets of \( S_{n} \).

Answer: 2^{n-3}n(n+1)


668. Determine the mass of a sphere of radius $r$, if the density at each of its points is proportional to the distance from the center of the sphere.

Answer: k\pi r^4


63. A number that can be expressed as the difference of squares of two non-zero natural numbers is called a "wise number", for example, $16=5^{2}-3^{2}, 16$ is a "wise number". In the sequence of natural numbers starting from 1, the 2008th "wise number" is $\qquad$.

Answer: 2680


85. 50 chess pieces are arranged in a circle, numbered $1,2,3, \cdots \cdots, 50$ in a clockwise direction. Then, starting in a clockwise direction, every other piece is removed until only one piece remains. If the remaining piece is number 42, then the first piece removed is $\qquad$ number.

Answer: 7


The Jackson High School Soccer Team has 24 players on the team, including four goalkeepers. They are conducting a drill where each goalkeeper practices saving penalty kicks. Each player, including the other goalkeepers, takes a turn to shoot against each goalkeeper. How many penalty kicks must be taken to ensure that every player has shot against each of the goalkeepers?

Answer: 92


Suppose that \( f(x)=\frac{2}{3x+c} \). For what value of \( c \) does \( f^{-1}(x) = \frac{3 - 6x}{x} \)?

Answer: 18


A hexagon $ABCDEF$ has internal angles where $\angle A = 100^\circ$ and $\angle B = 120^\circ$. Angles $C$ and $D$ are equal, and angle $E$ is $30^\circ$ more than the average of angles $C$, $D$, and $F$. Find the measure of the largest angle in the hexagon.

Answer: 147.5^\circ


Four points $C, D, F, M$ are on a straight line. The point $H$ is not on the line such that $\angle CDH = 130^\circ$ and $\angle HFM = 70^\circ.$ If the reflex angle at $H$ is $y^\circ,$ what is the value of $y$?

Answer: 340^\circ


Wei decides to modify the logo for his new company by using a larger square and additional circles. The new design has six circles, each tangent to two sides of the square and adjacent circles, set within a square frame. If the side length of the new square is 24 inches, determine how many square inches will be shaded.

Answer: 576 - 96\pi


The value of $y$ varies inversely as $x^2$. If $y=8$ when $x=1$, find the value of $x$ when $y=0.5$.

Answer: 4


At Linda's Local Produce, 4 bananas cost the same as 3 pears, and 8 pears cost the same as 5 mangoes. How many mangoes cost the same as 20 bananas?

Answer: 9.375


In a circle, $AB$ is a diameter, and $CD$ is a chord parallel to $AB$. Point $E$ is the intersection of lines $AC$ and $BD$, and $\angle AED = \beta$, but now $\beta$ is such that $\angle AED + \angle BED = 180^\circ - 2\beta$. The ratio of the area of triangle $\triangle CDE$ to that of triangle $\triangle ABE$ is required.

Answer: \cos^2 \beta


A right triangle has legs of length 12 and 9. It is similar to another right triangle with one leg of length 6. What is the length of the other leg in the second triangle, called $y$?

Answer: 4.5


For a certain weekend, the weatherman predicts that it will rain with a $60\%$ probability on Saturday and a $40\%$ probability on Sunday. If it rains on Saturday, the probability that it will also rain on Sunday increases to $80\%$. Calculate the probability that it rains on at least one day over the weekend. Express your answer as a percentage.

Answer: 76\%


Let $x_1, x_2, x_3$ be the roots of the equation $\sqrt{2023}x^3 - 4047x^2 + 2 = 0$ with $x_1 < x_2 < x_3$. Find the value of $x_2(x_1 + x_3)$.

Answer: 2


William is participating in a 30-question multiple choice exam. Each question has five answer choices. He randomly guesses the answers for the last six questions. What is the probability that he will get at least two of these six questions correct?

Answer: \frac{5385}{15625}


Janine is creating a design using a large square and inside it are placed six small circles. Each circle is tangent to one side of the square and the three circles adjacent to it. The side length of the square in Janine's design is 30 inches. How many square inches will be shaded in her design?

[asy]
size(120);
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);

fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,grey);

draw(circle((1,1),1));
fill(circle((1,1),1),white);
draw(circle((5,1),1));
fill(circle((5,1),1),white);
draw(circle((1,5),1));
fill(circle((1,5),1),white);
draw(circle((5,5),1));
fill(circle((5,5),1),white);

draw(circle((3,2),1));
fill(circle((3,2),1),white);
draw(circle((3,4),1));
fill(circle((3,4),1),white);
[/asy]

Answer: 900 - 150\pi \text{ square inches}


In rolling five standard six-sided dice, it is found that two of the dice form a pair, another die shows a different number distinct from the pair, and the other two dice each show different numbers as well. The last two dice which show different numbers are set aside for rerolling. What is the probability that, after rerolling these two dice, at least three of the five dice show the same value?

Answer: \frac{1}{3}


For all positive integers $n$, let $g(n)=\log_{3003} n^2$. Calculate $g(7) + g(11) + g(13)$.

Answer: 2


Define $\$N$ by the formula $\$N = 0.3N + 2$. Calculate $\$(\$(\$60))$.

Answer: 4.4


A right triangle has sides 15 cm and 20 cm. This triangle is similar to another triangle which has one of its sides as twice the length of a rectangle's shorter side. If the rectangle's shorter side is 30 cm and the longer side is 60 cm, what is the perimeter of the larger triangle?

Answer: 240 \text{ cm}


A rectangle has an area of 576 $\text{cm}^2$. If the length of the rectangle is twice the width, what is its perimeter?

Answer: 72\sqrt{2} \ \text{cm}


For all real numbers $r$ and $s$, define the mathematical operation $\#$ such that: $r \# 0 = r$, $r \# s = s \# r$, and $(r + 1) \# s = (r \# s) + s + 2$. Compute the value of $12 \# 6$.

Answer: 96


In a right trapezoid $ABCD$ where $AB \parallel CD$, the angle at vertex $B$ measures $75^\circ$. Point $H$ is the foot of the perpendicular from point $A$ to the line $BC$. If $BH = DC$ and $AD + AH = 8$, find the area of $ABCD$.

Answer: 8


Dilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$, $f_{B,C}$, $f_{C,A}$, and $f_{C,B}$. Each function $f_{X,Y}$ takes an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his 6 functions, without repeats, in such a way that the resulting expression is well-typed. This means an object can be taken in by the first function, and the resulting output can then be taken in by the second function, and so on. Determine the number of possible orders in which he can compose his 6 functions to satisfy this constraint.

Answer: 18


A $2008 \times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with $1 \times 2$ dominoes?

Answer: 4068434225296


What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$.

Answer: 987654301


In triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\triangle ABC$ is centered at $I$ and touches $AB$, $AC$, and $BC$ at points $M$, $N$, and $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $\triangle MAN$ and $\triangle KID$ intersects lines $AB$ and $AC$ at points $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.

Answer: 720


Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$, and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$.) What is the expected length of the string Clayton wrote?

Answer: 16


Let $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmost position (the units position), the result is $\frac{n}{3}$. Find the smallest possible value of the sum of the digits of $n$.

Answer: 126


Given two non-negative integers $m > n$, let's say that $m$ ends in $n$ if we can obtain $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.

Determine how many three-digit numbers end in the product of their digits.

Answer: 103


Let $M$ be the set of all polynomials $P(x)$ with pairwise distinct integer roots, integer coefficients, and all absolute values of the coefficients less than $2007$. What is the highest degree among all the polynomials in the set $M$?

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.)

[i](Birgit Vera Schmidt)[/i]

[hide=original wording]Bei 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.)[/hide]

Answer: 1011


4. Point $K$ lies on side $B C$ of parallelogram $A B C D$, and point $M$ lies on its side $A D$. Segments $C M$ and $D K$ intersect at point $L$, and segments $A K$ and $B M$ intersect at point $N$. Find the maximum value of the ratio of the areas of quadrilaterals $K L M N$ and $A B C D$.

Answer: \dfrac{1}{4}


In the plane, $n$ lines have been drawn in such a way that no two lines are parallel and no three lines are concurrent. How many regions are delimited by these lines?

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


12. Place 1 sphere with a radius of 2 between two inclined planes whose lines of intersection are on a horizontal plane, with the angles between these planes and the horizontal plane being $60^{\circ}$ and $30^{\circ}$, respectively. Then the distance from the center of the sphere to the horizontal plane is $\qquad$

Answer: 1 + \sqrt{3}


Problem 11.4. Given a regular pentagon $A B C D E$. On side $A E$ there is a point $K$, and on side $C D$ there is a point $L$. It is known that $\angle L A E + \angle K C D = 108^{\circ}$, and $A K: K E = 3: 7$. Find $C L: A B$.

A regular pentagon is a pentagon in which all sides are equal and all angles are equal.

#

Answer: \dfrac{7}{10}


Problem 2. Let $n$ three-digit numbers satisfy the following properties:

(1) No number contains the digit 0 .

(2) The sum of the digits of each number is 9 .

(3) The units digits of any two numbers are different.

(4) The tens digits of any two numbers are different.

(5) The hundreds digits of any two numbers are different.

Find the largest possible value of $n$.

Answer: 5


5. Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies for any real numbers $x, y$
$$
f(x+y)+f(x-y)=2 f(x) \cos y,
$$

and $f(0)=0, f\left(\frac{\pi}{2}\right)=1$.
Consider the following conclusions:
(1) $f\left(\frac{\pi}{4}\right)=\frac{1}{2}$; (2) $f(x)$ is an odd function;
(3) $f(x)$ is a periodic function;
(4) $f(x)$ is a monotonic function in $(0, \pi)$.
Among these, the correct conclusions are $\qquad$

Answer: 2,3


106. Easy Division. By dividing the number 8101265822784 by 8, you will see that the answer can be obtained simply by moving the 8 from the beginning to the end of the number!

Could you find a number starting with 7 that can be divided by 7 in such a simple way?


B3. We must choose a seven-character password that contains at least one letter and at least one digit. We can choose from the characters $A, B, C, G, J, M, R, Z$ and 3.

a) How many possible choices are there for the password if the characters cannot be repeated?

b) How many possible choices are there for the password if the characters can be repeated?

## 16th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 16, 2016

## Solutions to Problems and Scoring Guide

(April 3, 2016, $20:53$ )

A competitor who arrives at the solution by any correct method (even if the scoring guide does not provide for it) receives all possible points.

A correct method is considered any procedure that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points provided. If, however, the solution is guessed (it is not possible to arrive at it without calculation), it is scored with 0 points.

A competitor who has only partially solved the problem, from otherwise correct solving procedures but without a visible path to the final solution of the problem, cannot receive more than half of the possible points.

The asterisk '*' next to the points means that the point or points can be awarded for a correct procedure, even if the calculation is incorrect.

## First Year

| $\mathrm{A} 1$ | $\mathrm{~A} 2$ | $\mathrm{~A} 3$ |
| :---: | :---: | :---: |
| $\mathrm{D}$ | $\mathrm{E}$ | $\mathrm{E}$ |

Answer: 141120


Example 12. Given $x=19^{94}-1, y=2^{m} \cdot 3^{n} \cdot 5^{l}$ $(m, n, l$ are non-negative integers, and $m+n+l \neq 0)$. Find the sum $S$ of all divisors of $x$ that are of the form $y$.

Answer: 1169


4. As shown in the figure, the area of parallelogram $A B C D$ is $60, D P$ intersects $A C$ at $M, A M=\frac{1}{4} A C$. The area of $\triangle A P D$ is _. $\qquad$

Answer: 10


Problem 2. Along the path between the houses of Nезнайка (Nезнayka) and Синеглазка (Sineglazka), there were 15 peonies and 15 tulips growing in a row, mixed together.

Setting out from home to visit Nезнайка, Синеглазка watered all the flowers in a row. After the 10th tulip, the water ran out, and 10 flowers remained unwatered.

The next day, setting out from home to visit Синеглазка, Незнайка picked flowers for her one by one. After picking the 6th tulip, he decided that it was enough for the bouquet. How many flowers remained growing along the path?

Answer: 19


5. In the country, there are 100 cities, and several non-stop air routes are in operation between them, such that one can travel from any city to any other, possibly with layovers. For each pair of cities, the minimum number of flights required to travel from one to the other was calculated. The transportation difficulty of the country is defined as the sum of the squares of these 4950 numbers. What is the maximum value that the transportation difficulty can take? The answer should be given as a number (in decimal notation).

Answer: 8332500


6. (10 points) As shown in the figure, the area of $\triangle A B C$ is 100 square centimeters, and the area of $\triangle A B D$ is 72 square centimeters. $M$ is the midpoint of side $C D$, $\angle M H B=90^{\circ}$, and it is known that $A B=20$ centimeters. Then the length of $M H$ is $\qquad$ centimeters.

Answer: \dfrac{43}{5}


In triangle \( \triangle ABC \), a smaller, similar triangle \( \triangle A_{1} B_{1} C_{1} \) is drawn inside such that \( \triangle ABC \sim \triangle A_{1} B_{1} C_{1} \). Construct perpendiculars \( B_{1}D \perp AC \) at point \( D \), \( C_{1}E \perp AB \) at point \( E \), and \( A_{1}F \perp BC \) at point \( F \). Prove that:
\[ A_{1}F \cdot BC + B_{1}D \cdot AC + C_{1}E \cdot AB = 2S_{\triangle ABC}. \]

Answer: 2S_{\triangle ABC}