Alex is a professional soccer player who dedicates 15% of their earnings every month to support organizations that raise awareness about human rights issues. This month, Alex earned $80,000 from playing soccer matches and another $20,000 from endorsements. Additionally, Alex decided to organize a charity event where they managed to raise $5,000. How much money in total will Alex donate to the human rights organizations this month?

Answer: 20000


A young California-based conceptual artist is planning an art installation that involves creating a large mural using recycled materials. The mural will be a rectangular piece with a width of 8 feet and a height of 15 feet. The artist wants to cover the entire surface of the mural with square tiles made of recycled plastic, each tile measuring 1 foot on each side. Additionally, the artist wants to use a special golden tile in the center of the mural as a focal point.

How many recycled plastic tiles will the artist need to complete the mural, including the special golden tile?

Answer: 120


A successful entrepreneur is searching for their dream home with luxurious amenities. They are looking at a mansion that has 5 bedrooms, each with its own bathroom, and a guest house that has 2 bedrooms with 1 shared bathroom. The entrepreneur wants to install a state-of-the-art entertainment system in every bedroom, costing $2,500 each, and one in the living room of the main house costing $5,000. Additionally, they want to have a luxury spa installed, which costs $12,000. How much will the entrepreneur spend in total on the entertainment systems and the spa?

Answer: 34500


Sipho, a South African fan of local television dramas, watches a show featuring Greteli Fincham every evening. Each episode is 45 minutes long. He also spends an additional 15 minutes online reading about the latest news and updates on the show and its cast. He does this every day for a week. On Saturdays, he attends a local fan club meeting for 2 hours where they discuss the show and plan events to support Greteli Fincham. How many hours in total does Sipho spend on activities related to the show in one week?

Answer: 9


In right triangle \(ABC\), \(BC = 5\), \(AC = 12\), and \(AM = x\); \(\overline{MN} \perp \overline{AC}\), \(\overline{NP} \perp \overline{BC}\); \(N\) is on \(AB\). If \(y = MN + NP\), one-half the perimeter of rectangle \(MCPN\), find the value of \(y\). The original answer is in the form \(\frac{k - mx}{n}\). Please provide the value of \(k + m + n\).

Answer: 163


How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$? Find the number of such coefficients.

Answer: 16


A man with mass $m$ jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the bungee cord is four times the weight of the man.

Find the maximum extension of the bungee cord $h$ in terms of $H$. The original answer is in the format \frac{k}{m}, please give the value of k + m.

Answer: 3


In how many ways can 10 and 50 fillér coins be placed side by side (all centers on a straight line) to cover a $1 \mathrm{~m}$ long segment (not more), if at least 50 coins are used, and the order of the two types of coins is taken into account. (Coins of the same value are not distinguished. The diameter of the 10 fillér is $19 \mathrm{~mm}$, and that of the 50 fillér is $22 \mathrm{~mm$).

Answer: 270725


B1. What is the smallest positive integer consisting of the digits 2, 4, and 8, where each of these digits appears at least twice and the number is not divisible by 4?

Answer: 244882


13. A rabbit and a hedgehog participated in a running race on a $550 \mathrm{~m}$ long circular track, both starting and finishing at the same point. The rabbit ran clockwise at a speed of $10 \mathrm{~m} / \mathrm{s}$ and the hedgehog ran anticlockwise at a speed of $1 \mathrm{~m} / \mathrm{s}$. When they met, the rabbit continued as before, but the hedgehog turned round and ran clockwise. How many seconds after the rabbit did the hedgehog reach the finish?

Answer: 45


[Inclusion-Exclusion Principle] [Arithmetic. Mental calculation, etc.]

In a group of 50 children, some know all the letters except "r", which they simply skip when writing, while others know all the letters except "k", which they also skip. One day, the teacher asked 10 students to write the word "кот" (cat), 18 other students to write the word "рот" (mouth), and the rest to write the word "крот" (mole). It turned out that the words "кот" and "рот" were each written 15 times. How many children wrote their word correctly?

Answer: 8


$5 \cdot 61$ A group of young people went to dance disco. Each dance costs 1 yuan, and each boy danced with each girl exactly once, then they went to another place to dance. Here, they paid with subsidiary coins, and they spent the same amount of money as before. The entry fee for each person is one subsidiary coin, and each dance costs one subsidiary coin. Each person danced with every other person exactly twice (regardless of gender), and there was 1 subsidiary coin left at the end. How many subsidiary coins does 1 yuan equal?

Answer: 5


5. We will call a number $\mathrm{X}$ "50-supporting" if for any 50 real numbers $a_{1}, \ldots, a_{50}$, the sum of which is an integer, there exists at least one for which $\left|a_{i}-\frac{1}{2}\right| \geq X$.

In your answer, specify the largest 50-supporting $X$, rounded to the hundredths according to standard mathematical rules.

Answer: 0.01


6.1. A cube lies on a plane. Each face is marked with 1 to 6 points such that the sum of points on opposite faces is always 7. The cube is rolled over the plane as shown in the picture. How many points will be on the top face of the cube when it lands on the last cell?

![](https://cdn.mathpix.com/cropped/2024_05_06_ffa720cb9bc207f8b509g-06.jpg?height=854&width=1816&top_left_y=284&top_left_x=217)

Answer: 5


$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.

Answer: k = 2n


Question 15. Three $12 \mathrm{~cm} \times 12 \mathrm{~cm}$ squares are each divided into two pieces, $A$ and $B$, by a line connecting the midpoints of two adjacent sides, as shown in the first figure. The six pieces are then attached to the outside of a regular hexagon, as shown in the second figure. The pieces are then folded to form a polyhedron. Find the volume of this polyhedron (in $\mathrm{cm}^3$).

Answer: 864


On the board after the class, there was a note left:

"Calculate $t(0)-t(\pi / 5)+t\left((\pi / 5)-t(3 \pi / 5)+\ldots+t\left({ }^{8 \pi} / 5\right)-t(9 \pi / 5)\right.$, where $t(x)=\cos 5 x+* \cos 4 x+* \cos 3 x+* \cos 2 x+$ $*^{\cos x}+* "$.

Seeing it, a math student told a friend that he could calculate this sum even without knowing the values of the coefficients erased from the board (denoted by * in our notation). Is he mistaken?

Answer: 10


3. For a rational number $r$, its period is the length of the smallest repeating block in its decimal expansion. For example, the number $r=0.123123123 \cdots$ has period 3 . If $S$ denotes the set of all rational numbers $r$ of the form $r=0 . \overline{a b c d e f g h}$ having period 8 , find the sum of all the elements of $S$.

Answer: 49995000


1. On a certain day, there were several nuts in the bag. The next day, as many nuts were added to the bag as there already were, but eight nuts were taken away. On the third day, again as many nuts were added as there already were, but eight were taken away. The same thing happened on the fourth day, and after that, there were no nuts left in the bag. How many nuts were there in the bag at the beginning?

Answer: 7


12. If the sum of the digits of a natural number $a$ equals 7, then $a$ is called a "lucky number". Arrange all lucky numbers in ascending order as $a_{1}, a_{2}, a_{3}, \cdots$, if $a_{n}=$ 2005, then $a_{5 n}=$ $\qquad$

Answer: 52000


10. The area of square $A B C D$ is 9 square centimeters, and the area of square $E F G H$ is 64 square centimeters. As shown in the figure, side $B C$ lies on $E H$. Given that the area of triangle $A C G$ is 6.75 square centimeters, then the area of triangle $A B E$ is square centimeters.

Answer: 2.25


13. In a store, there are large, medium, and small boxes of marbles, containing 13, 11, and 7 marbles respectively. If someone wants to buy 20 marbles, they don't have to break open the boxes (1 large box plus 1 small box). If someone wants to buy 23 marbles, they must break open the boxes. Please find the smallest number such that anyone buying more than this number of marbles definitely doesn't have to break open the boxes. This smallest number is ( ).

Answer: 30


Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$, then $N = 7$.

Answer: 19


Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A $3\times 4$ rectangle is cut into $n$ pieces by making straight line cuts. What is the minimum value of $n$ so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least $2021$?

Answer: 202


Billy the baker makes a bunch of loaves of bread every day, and sells them in bundles of size $1, 2$, or $3$. On one particular day, there are $375$ orders, $125$ for each bundle type. As such, Billy goes ahead and makes just enough loaves of bread to meet all the orders. Whenever Billy makes loaves, some get burned, and are not sellable. For nonnegative i less than or equal to the total number of loaves, the probability that exactly i loaves are sellable to customers is inversely proportional to $2^i$ (otherwise, it’s $0$). Once he makes the loaves, he distributes out all of the sellable loaves of bread to some subset of these customers (each of whom will only accept their desired bundle of bread), without worrying about the order in which he gives them out. If the expected number of ways Billy can distribute the bread is of the form $\frac{a^b}{2^c-1}$, find $a + b + c$.

Answer: 1011


Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?

Answer: 64


Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number $i \in \{0, 1, . . . , 5\}$, some coin has exactly $i$ dots on it.

Answer: 79200


Answer: 18


Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and radius $10$. Let sides $AB$, $BC$, $CD$, and $DA$ have midpoints $M, N, P$, and $Q$, respectively. If $MP = NQ$ and $OM + OP = 16$, then what is the area of triangle $\vartriangle OAB$?

Answer: 78


We say that a positive integer $n$ is [i]divable [/i] if there exist positive integers $1 < a < b < n$ such that, if the base-$a$ representation of $n$ is $\sum_{i=0}^{k_1} a_ia^i$ , and the base-$b$ representation of $n$ is $\sum_{i=0}^{k_2}  b_ib^i$ , then for all positive integers $c > b$, we have that $\sum_{i=0}^{k_2}  b_ic^i$ divides $\sum_{i=0}^{k_1}  a_ic^i$. Find the number of non-divable $n$ such that $1 \le n \le  100$.

Answer: 27


Let there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random direction, and so forth until he leaps $15$ units in a random direction to celebrate PUMaC’s 15th year.
There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly $1/2$ after the $15$ leaps. The area of that circle can be written as $A\pi$. What is $A$?

Answer: 1240


Given a sequence $a_0, a_1, a_2, ... , a_n$, let its [i]arithmetic approximant[/i] be the arithmetic sequence $b_0, b_1, ... , b_n$ that minimizes the quantity $\sum_{i=0}^{n}(b_i -a_i)^2$, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence $4$, $8$, $12$, $16$ and whose anti-arithmeticity is at most $20$.

Answer: 15


Let $\{x\} = x- \lfloor x \rfloor$ . Consider a function f from the set $\{1, 2, . . . , 2020\}$ to the half-open interval $[0, 1)$. Suppose that for all $x, y,$ there exists a $z$ so that $\{f(x) + f(y)\} = f(z)$. We say that a pair of integers $m, n$ is valid if $1 \le m, n \le 2020$ and there exists a function $f$ satisfying the above so $f(1) = \frac{m}{n}$. Determine the sum over all valid pairs $m, n$ of ${m}{n}$.

Answer: 1019595


Gary is baking cakes, one at a time. However, Gary’s not been having much success, and each failed cake will cause him to slowly lose his patience, until eventually he gives up. Initially, a failed cake has a probability of $0$ of making him give up. Each cake has a $1/2$ of turning out well, with each cake independent of every other cake. If two consecutive cakes turn out well, the probability resets to $0$ immediately after the second cake. On the other hand, if the cake fails, assuming that he doesn’t give up at this cake, his probability of breaking on the next failed cake goes from p to $p + 0.5$. If the expected number of successful cakes Gary will bake until he gives up is$ p/q$, for relatively prime $p, q$, find $p + q$.

Answer: 86


p1. Determine the smallest natural number that has the property that it's cube ends in  $888$ .

p2. Triangle  $ABC$  is isosceles with  $AC = BC$ . The angle bisector at  $A$  intercepts side  $BC$  at the point  $D$  and the bisector of the angle at  $C$  intercepts side  $AB$  at  $E$ . If  $AD = 2CE$ , find the measure of the angles of triangle  $ABC$ .

p3.In the accompanying figure, the circle  $C$  is tangent to the quadrant  $AOB$  at point  $S$ . AP is tangent to  $C$  at  $P$  and  $OQ$  is tangent at  $Q$ . Calculate the length  $AP$  as a function of the radius $ R$  of the quadrant  $AOB$ .
![Image](https://cdn.artofproblemsolving.com/attachments/3/c/d34774c3c6c33d351316574ca3f7ade54e6441.png)

p4.  Let  $A$  be a set with seven or more natural numbers. Show that there must be two numbers in  $A$  with the property that either its sum or its difference is divisible by ten. Show that the property can fail if set  $A$  has fewer than seven elements.

p5. An  $n$ -sided convex polygon is such that there is no common point for any three of its diagonals. Determine the number of triangles that are formed such that two of their vertices are vertices of the polygon and the third is an intersection of two diagonals.

Answer: 192


A *calendar* is a (finite) rectangular grid. A calendar is *valid* if it satisfies the following conditions:

(i) Each square of the calendar is colored white or red, and there are exactly 10 red squares.

(ii) Suppose that there are  $N$  columns of squares in the calendar. Then if we fill in the numbers  $1,2,\ldots$  from the top row to the bottom row, and within each row from left to right, there do not exist  $N$  consecutive numbers such that the squares they are in are all white.

(iii) Suppose that there are  $M$  rows of squares in the calendar. Then if we fill in the numbers  $1,2,\ldots$  from the left-most column to the right-most column, and within each column from bottom to top, there do not exist  $M$  consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by  $90^{\circ}$ , the resulting calendar still satisfies (ii).

How many different kinds of valid calendars are there?

(Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the  $10$  red squares are at different locations.)

Answer: 10!


p1. The next game is played between two players with a pile of peanuts. The game starts with the man pile being divided into two piles (not necessarily the same size). A  move consists of eating all the mana in one pile and dividing the other into two non-empty piles, not necessarily the same size. Players take turns making their moves and the last one wins that I can make a move Show that if the starting number of mana is in at least one of the stacks is even, so the first player can win no matter how his opponent plays (that is, he has a perfect strategy to win).

[p2.](https://artofproblemsolving.com/community/c4h1847975p12452018) If in the  $ \vartriangle ABC $ , two sides are not greater than their corresponding altitudes, how much do the angles of the triangle measure?

p3. Eight players participated in a free-for-all chess tournament. All players obtained different final scores, the one who finished second having obtained as many points as a total were obtained by the four worst-performing players. What was the result between the players who got fourth and fifth place?

p4. Let  $N$  be a number of  $2002$  digits, divisible by  $9$ . Let  $x$  be the sum of the digits of  $N$ . Let  $y$  the sum of the digits of  $x$ . Let  $z$  be the sum of the digits of  $y$ . Determine the value of  $z$ .

[p5.](https://artofproblemsolving.com/community/c4h2916289p26046171) Given a right triangle angle  $T$ , where the coordinates of its vertices are integer numbers, let  $E$  be the number of integer coordinate points that belong to the edge of the triangle  $T$ ,  $I$  the number of integer coordinate points belonging to the interior of the triangle  $T$ . Prove that the area  $A (T)$  of the triangle  $T$  is given by:  $A (T) =\frac{E}{2}+ I$  .

p6. Given a  $4\times 4$  board, determine the value of natural  $k$  with the following properties: $\bullet$  If there are less than  $k$  tiles on the board, then two rows and two columns can be found such that if they are erased, there will be no pieces left on the board. $\bullet$  On the board,  $k$  tiles can be placed in such a way that when erasing any two rows and two any columns, there will always be at least one tile on the board

[p7.](https://artofproblemsolving.com/community/c4h1847977p12452026) Given the segment  $ AB $ , let  $ M $  be one point lying on it. Towards the same side of the plane and with base  $ AM $  and  $ MB $ , the squares  $ AMCD $  and  $ MBEF $  are constructed. Let  $ P $  and  $ Q $  be the respective centers of these squares. Determine how the midpoint of the segment  $ PQ $  moves as the point  $ M $  moves along the segment.

Answer: 9


Annie takes a  $6$  question test, with each question having two parts each worth  $1$  point. On each **part**, she receives one of nine letter grades  $\{\text{A,B,C,D,E,F,G,H,I}\}$  that correspond to a unique numerical score. For each **question**, she receives the sum of her numerical scores on both parts. She knows that  $\text{A}$  corresponds to  $1$ ,  $\text{E}$  corresponds to  $0.5$ , and  $\text{I}$  corresponds to  $0$ .

When she receives her test, she realizes that she got two of each of  $\text{A}$ ,  $\text{E}$ , and  $\text{I}$ , and she is able to determine the numerical score corresponding to all  $9$  markings. If  $n$  is the number of ways she can receive letter grades, what is the exponent of  $2$  in the prime factorization of  $n$ ?

*2020 CCA Math Bonanza Individual Round #10*

Answer: 11


In the rectangular parallelepiped in the figure, the lengths of the segments  $EH$ ,  $HG$ , and  $EG$  are consecutive integers. The height of the parallelepiped is  $12$ . Find the volume of the parallelepiped.
![Image](https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png)

Answer: 144


Let  $ABC$  be a triangle with  $\angle ABC=90^{\circ}$ . The square  $BDEF$  is inscribed in  $\triangle ABC$ , such that  $D,E,F$  are in the sides  $AB,CA,BC$  respectively. The inradius of  $\triangle EFC$  and  $\triangle EDA$  are  $c$  and  $b$ , respectively. Four circles  $\omega_1,\omega_2,\omega_3,\omega_4$  are drawn inside the square  $BDEF$ , such that the radius of  $\omega_1$  and  $\omega_3$  are both equal to  $b$  and the radius of  $\omega_2$  and  $\omega_4$  are both equal to  $a$ . The circle  $\omega_1$  is tangent to  $ED$ , the circle  $\omega_3$  is tangent to  $BF$ ,  $\omega_2$  is tangent to  $EF$  and  $\omega_4$  is tangent to  $BD$ , each one of these circles are tangent to the two closest circles and the circles  $\omega_1$  and  $\omega_3$  are tangents. Determine the ratio  $\frac{c}{a}$ .

Answer: 2


The diagram below shows some small squares each with area  $3$  enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer  $n$  such that the area of the shaded region inside the larger square but outside the smaller squares is  $\sqrt{n}$ .

[asy]
size(150);
real r=1/(2sqrt(2)+1);
path square=(0,1)--(r,1)--(r,1-r)--(0,1-r)--cycle;
path square2=(0,.5)--(r/sqrt(2),.5+r/sqrt(2))--(r*sqrt(2),.5)--(r/sqrt(2),.5-r/sqrt(2))--cycle;
defaultpen(linewidth(0.8));
filldraw(unitsquare,gray);
filldraw(square2,white);
filldraw(shift((0.5-r/sqrt(2),0.5-r/sqrt(2)))*square2,white);
filldraw(shift(1-r*sqrt(2),0)*square2,white);
filldraw(shift((0.5-r/sqrt(2),-0.5+r/sqrt(2)))*square2,white);
filldraw(shift(0.5-r/sqrt(2)-r,-(0.5-r/sqrt(2)-r))*square,white);
filldraw(shift(0.5-r/sqrt(2)-r,-(0.5+r/sqrt(2)))*square,white);
filldraw(shift(0.5+r/sqrt(2),-(0.5+r/sqrt(2)))*square,white);
filldraw(shift(0.5+r/sqrt(2),-(0.5-r/sqrt(2)-r))*square,white);
filldraw(shift(0.5-r/2,-0.5+r/2)*square,white);
[/asy]

Answer: 288


Four copies of an acute scalene triangle  $\mathcal T$ , one of whose sides has length  $3$ , are joined to form a tetrahedron with volume  $4$  and surface area  $24$ .  Compute the largest possible value for the circumradius of  $\mathcal T$ .

Answer: \sqrt{4+\sqrt{3}}


Triangle  $ABC$  has positive integer side lengths with  $AB=AC$ . Let  $I$  be the intersection of the bisectors of  $\angle B$  and  $\angle C$ . Suppose  $BI=8$ . Find the smallest possible perimeter of  $\triangle ABC$ .

Answer: 108


For a positive integer  $n$ ,  $\sigma(n)$  denotes the sum of the positive divisors of  $n$ . Determine   $$ \limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}} $$  **Note:** Given a sequence ( $a_n$ ) of real numbers, we say that  $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$  if ( $a_n$ ) is not upper bounded, and, otherwise,  $\limsup\limits_{n\rightarrow \infty} a_n$  is the smallest constant  $C$  such that, for every real  $K > C$ , there is a positive integer  $N$  with  $a_n < K$  for every  $n > N$ .

Answer: 1


We start with any finite list of distinct positive integers. We may replace any pair  $n, n + 1$  (not necessarily adjacent in the list) by the single integer  $n-2$ , now allowing negatives and repeats in the list. We may also replace any pair  $n, n + 4$  by  $n - 1$ . We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.

Answer: -3


In convex quadrilateral  $KLMN$  side  $\overline{MN}$  is perpendicular to diagonal  $\overline{KM}$ , side  $\overline{KL}$  is perpendicular to diagonal  $\overline{LN}$ ,  $MN = 65$ , and  $KL = 28$ . The line through  $L$  perpendicular to side  $\overline{KN}$  intersects diagonal  $\overline{KM}$  at  $O$  with  $KO = 8$ . Find  $MO$ .

Answer: 90


Two regular hexagons of side length  $2$  are laid on top of each other such that they share the same center point and one hexagon is rotated  $30^\circ$  about the center from the other. Compute the area of the union of the two hexagons.

Answer: 48\sqrt{3}-72


A semicircle has diameter  $\overline{AD}$  with  $AD = 30$ . Points  $B$  and  $C$  lie on  $\overline{AD}$ , and points  $E$  and  $F$  lie on the arc of the semicircle. The two right triangles  $\vartriangle BCF$  and  $\vartriangle CDE$  are congruent. The area of  $\vartriangle  BCF$  is  $m\sqrt{n}$ , where  $m$  and  $n$  are positive integers, and  $n$  is not divisible by the square of any prime. Find  $m + n$ .
![Image](https://cdn.artofproblemsolving.com/attachments/b/c/c10258e2e15cab74abafbac5ff50b1d0fd42e6.png)

Answer: 52


Let  $x,y,z$  be positive real numbers such that  $x^2 + y^2 = 49, y^2 + yz + z^2 = 36$  and  $x^2 + \sqrt{3}xz + z^2 = 25$ . If the value of  $2xy + \sqrt{3}yz + zx$  can be written as  $p \sqrt{q}$  where  $p,q \in \mathbb{Z}$  and  $q$  is squarefree, find  $p+q$ .

Answer: 30


Let  $(a_n)$  be a sequence of integers, with  $a_1 = 1$  and for evert integer  $n \ge 1$ ,  $a_{2n} = a_n + 1$  and  $a_{2n+1} = 10a_n$ . How many times  $111$  appears on this sequence?

Answer: 14


A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by  $\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from  $A$ .

Answer: 384


Suppose $w, x, y, z$ satisfy the following equations:
\[
\begin{align*}
w + x + y + z &= 25, \\
wx + wy + wz + xy + xz + yz &= 2y + 2z + 193
\end{align*}
\]
The largest possible value of $w$ can be expressed in lowest terms as $\frac{w_1}{w_2}$ for some integers $w_1, w_2 > 0$. Find $w_1 + w_2$.

Answer: 27


Given a cube, on each edge, write a number, either $1$ or $-1$. For each face of the cube, multiply the four numbers on the edges of this face and write the product on this face. Finally, add all the eighteen numbers that were written on the edges and faces of the cube.

What is the smallest possible sum that can be obtained?

Answer: -12


Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.

Answer: 25


A triangulation of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Answer: 9089


For a positive integer $n > 1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n) = n - g(n)$. For example, $g(10) = 5$, $f(10) = 5$ and $g(13) = 1$, $f(13) = 12$. Let $N$ be the smallest positive integer such that $f(f(f(N))) = 97$. Find the largest integer not exceeding $\sqrt{N}$.

Answer: 19


Let $p$ be a polynomial with degree less than $4$ such that $p(x)$ attains a maximum at $x = 1$. If $p(1) = p(2) = 5$, find $p(10)$.

Answer: 5


21 Savage has a 12-car garage, with a row of spaces numbered 1, 2, 3, \ldots, 12. How many ways can he choose 6 of them to park his 6 identical cars in, if no 3 spaces with consecutive numbers may be all occupied?

Answer: 357


Let $S = \{1, 2, \ldots, 8\}$. How many ways are there to select two disjoint subsets of $S$?

Answer: 6561


In the diagram below, \( \overline{AB} \) and \( \overline{CD} \) are parallel, \( \angle BXY = 45^\circ \), \( \angle DZY = 25^\circ \), and \( XY = YZ \). What is the degree measure of \( \angle YXZ \)?

Answer: 55


Each cell of a $7 \times 7$ table is painted with one of several colors. It is known that for any two distinct rows, the numbers of colors used to paint them are distinct, and for any two distinct columns, the numbers of colors used to paint them are distinct. What is the maximum possible number of colors in the table?

Answer: 22


Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding
\[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]

Answer: 164


In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.

Answer: 124


Dima has written the numbers $\frac{1}{80!}, \frac{1}{81!}, \ldots, \frac{1}{99!}$ on 20 infinite pieces of paper as decimal fractions. For example, the following is written on the last piece: \( \frac{1}{99!} = 0.\underbrace{00\ldots00}_{155\text{ zeros}}10715\ldots \). Sasha wants to cut a fragment of $N$ consecutive digits from one of these pieces without the comma. What is the maximal $N$ for which Sasha can do this so that Dima cannot guess from which piece Sasha has cut his fragment?

Answer: 155


Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in \{0,1\} such that there exists a homogeneous polynomial $P(x,y,z)$ of degree 2016 with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that
\[P(x,y,z) Q(x,y,z) = P(yz,zx,xy) + 2R(x,y,z)\]
and $P(1,1,1)$ is odd. Determine the size of $S$.

Note: A homogeneous polynomial of degree $d$ consists solely of terms of degree $d$.

Answer: 509545


We will say that a list of positive integers is *admissible* if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that from each admissible list, some numbers can be crossed out such that the sum of the numbers left uncrossed is greater than or equal to $1810-d$ and less than or equal to $1810+d$.

Answer: 10


There are $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visitors decide to take a trip around the empire, visiting a different city each week in some random order. $3$ of these cities are inhabited by mathematicians, who will talk to all visitors about the laws of statistics. A visitor with this knowledge has a probability of $0$ of buying a lottery ticket, else they have a probability of $0.5$ of buying one. What is the expected number of visitors who will play the Martingale Lottery?

Answer: 405


Let $n$ be a positive integer such that $\lfloor\sqrt{n}\rfloor - 2$ divides $n - 4$ and $\lfloor\sqrt{n}\rfloor + 2$ divides $n + 4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x \rfloor$ refers to the greatest integer less than or equal to $x$).

Answer: 956


Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then \( \gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c)) \). Let $s(n)$ be the sum of the distinct prime divisors of $n$. If \( f(n) < \frac{1}{2018} \), compute the smallest possible value of $s(n)$.

Answer: 58


The incircle of triangle $\triangle ABC$ is the unique inscribed circle that is internally tangent to the sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$. How many non-congruent right triangles with integer side lengths have incircles of radius $2015$?

Answer: 8


Let $f : \mathbb{N} \to \mathbb{N}$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in \mathbb{N}$. Find $f(2010)$.

Answer: 3843


4. A person rolls a die, adding up the numbers rolled each time, and stops if the total exceeds 20. Then, when he stops, the number he is most likely to have rolled is $\qquad$

Answer: 21


Problem 5. A messenger was riding a horse to deliver a message to Ilya Muromets. At some point, he noticed that Ilya Muromets had passed by him (and continued walking in the opposite direction). After 10 seconds (when the horse stopped), the messenger dismounted and ran to catch up with Ilya. How many seconds will it take for the messenger to deliver the message if he runs twice as fast as Ilya Muromets, but five times slower than the galloping horse?

Answer: 110


## Task B-4.3.

The sum of all 1002-digit numbers that in their representation have a thousand zeros and two ones is $S$. Determine the remainder obtained when the number $S$ is divided by 3.

Answer: 1


## Task 3 - 040613

A 6th grade class made various paper triangles. The students wanted to store these triangles in a cabinet in their school's mathematics room, which had nine compartments. The students had designated three compartments for the equilateral triangles, three for the isosceles triangles (i.e., those that are not equilateral), and three for the scalene triangles. Within these groups, the figures were to be further divided into acute, right, and obtuse triangles.

Check whether the number of compartments was correctly chosen!

Answer: 7


7.1. Find the number of all integer solutions of the inequality $\sqrt{1-\sin \frac{\pi x}{4}-3 \cos \frac{\pi x}{2}}-\sqrt{6} \cdot \sin \frac{\pi x}{4} \geq 0$, belonging to the interval [1991; 2013].

Answer: 8


I-2 Consider a chessboard $n \times n$ where $n>1$ is a positive integer. We select the centers of $2 n-2$ squares. How many selections are there such that no two selected centers lie on a line parallel to one of the diagonals of the chessboard?

Answer: 2^{n}


18. A taxi company's repair station has 7 taxis to be repaired. If 1 worker repairs these 7 cars, the repair times are $12, 17, 8, 18, 23, 30, 14$ minutes, respectively. Each taxi incurs an economic loss of 2 yuan for every minute it is out of service. Now, 3 workers with the same work efficiency are working separately. To minimize the economic loss, what is the minimum loss in yuan?

Answer: 362


Exercise 12. A circle with diameter $2 n-1$ is drawn at the center of a chessboard $2 n \times 2 n$. How many squares are crossed by an arc of the circle?

Answer: 8n-4


2. In a certain kingdom, there are 32 knights, some of whom are servants to other knights. Each servant can have at most one master, and each master must be richer than any of his servants. If a knight has at least four servants, he is ennobled as a noble. If it is stipulated that a servant of $A$'s servant is not a servant of $A$, then the maximum possible number of nobles is $\qquad$

Answer: 7


## Task 1

The members of a youth brigade helped to prepare apartments for elderly citizens. The value amounted to 1200 Marks. They helped with cable installation. This value amounted to 2100 Marks. In the operation, they saved materials. This value amounted to 1500 Marks.

How many Marks did the youth brigade bill for?

Answer: 4800


19. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. The number of different addition equations is ( $\quad$ ). $(a+b$ and $b+a$ are considered the same equation)

Answer: 1536


19. The airplane robot Roly is leading all the airplane robots to prepare for an air show. If each performance team consists of 12 airplane robots, there are still 5 airplane robots short; if each performance team consists of 15 airplane robots, there are still 8 airplane robots short. It is known that the total number of airplane robots is greater than 15 and less than 100. Therefore, there are $\qquad$ airplane robots.

Answer: 67


## Task B-1.5.

The area of triangle $A B C$ is $32 \mathrm{~cm}^{2}$. Points $M$ and $N$ are chosen on sides $A B$ and $B C$ such that $|A M|:|M B|=|B N|:|N C|=3: 1$. Point $G$ is the intersection of lines $A N$ and $C M$. Determine the ratio of the areas of figures $A G C$ and $M B N G$.

Answer: 1:1


[

Tangent Circles

[ The perpendicular is shorter than the inclined. Inequalities for right triangles ]

Point $E$ on side $B C$ and point $F$ on side $A D$ of the convex quadrilateral $A B C D$ are positioned such that $B E=2 E C$, $A F=2 F D$. On segment $A E$ is the center of a circle of radius $r$, tangent to sides $A B, B C$ and $C D$. On segment $B F$ is the center of a circle of the same radius $r$, tangent to sides $A B, A D$ and $C D$. Find the area of quadrilateral $A B C D$, given that the specified circles are externally tangent to each other.

#

Answer: 8r^{2}


1. Joey is standing at a corner of the rectangular field shown. He walks the perimeter of the field 5 times. How many metres does Joey walk?

Answer: 1500


4. (10 points) A sequence of numbers: $8,3,1,4, . \cdots$, starting from the third number, each number is the unit digit of the sum of the two closest preceding numbers. What is the 2011th number? $\qquad$ .

Answer: 2


Let $n \geqslant 1$ be a positive integer. Consider a line of $n$ lamps. Initially, some of the lamps are on, and each minute, the lamps that have exactly one neighboring lamp on turn on, and the others turn off (the lamps at the ends have only one neighboring lamp).

For which integers $n$ are we certain that, regardless of the initial configuration of the lamps, they will all be off after a certain time?

Answer: 2^k-1


9. If positive integers $m, n$ satisfy $\frac{(m+n)!}{n!}=5040$, then the value of $m!n$ is $\qquad$

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

Answer: 144


6. Petya saw a bank advertisement: "Deposit 'Super-Income' - up to $10 \%$ annually!". He became interested in this offer and found it on the bank's website. The conditions turned out to be slightly different, as reflected in the table.

| Interest period | from 1 to 3 months | from 4 to 6 months | from 7 to 9 months | from 10 to 12 months |
| :---: | :---: | :---: | :---: | :---: |
| Rate | $10 \%$ | $8 \%$ | $8 \%$ | $8 \%$ |

*     - interest is credited at the end of the period.

What is the actual effective annual interest rate of this deposit? Round your answer to the tenths.

Answer: 8.8


Problem 10.5. One winter day, 43 children were throwing snowballs at each other. Each of them threw exactly one snowball at someone else. It is known that:

- the first threw a snowball at the one who threw a snowball at the second,
- the second threw a snowball at the one who threw a snowball at the third,
- the forty-third threw a snowball at the one who threw a snowball at the first.

What is the number of the one who threw a snowball at the third?

Answer: 24


Six, there are 6000 points inside a circle, and no three points are collinear:
(1) Can this circle be divided into 2000 parts, each containing exactly three points? How to divide it?
(2) If the three points in each part satisfy: the distance between any two points is an integer and does not exceed 9, then using the three points in each part as vertices to form triangles, how many of these triangles are of the same size at least?

Answer: 22


6. If 5 consecutive natural numbers are all composite, then this group of numbers is called a "twin 5 composite". So, among the natural numbers not exceeding 100, there are $\qquad$ groups of twin 5 composite.

Answer: 10


3. The fox builds "pyramids" from 5 cubes in the following way: each "pyramid" consists of one or several levels; on each level, the number of cubes is strictly less than on the previous one; each new level consists of one or several consecutive cubes. You can see an example of a "pyramid" made of ten cubes in the picture:
![](https://cdn.mathpix.com/cropped/2024_05_06_d59521f9eee01ea62528g-3.jpg?height=214&width=870&top_left_y=244&top_left_x=206)

How many different "pyramids" can the fox build?

(It is important where the right edge and the left edge of the "pyramid" are — the "pyramids" in the picture are different).

Answer: 7


14 Labeled as $1,2, \cdots, 100$, there are some matches in the matchboxes. If each question allows asking about the parity of the sum of matches in any 15 boxes, then to determine the parity of the number of matches in box 1, at least how many questions are needed?

Answer: 3


Exercise 6. We have 102 distinct gifts. We want to distribute them to 100 winners of a contest, so that each winner receives at least one gift. Let $N$ be the number of ways to do this. Calculate

$$
\frac{N \times 48}{1 \times 2 \times 3 \times \cdots \times 102}
$$

Answer: 60200


4. (8 points) Two identical equilateral triangles can form a rhombus. If the side length of the equilateral triangle is 10, then the area of the largest square that can fit inside this rhombus is $\qquad$ .

Answer: 50


5. For what largest $p$ does there exist a convex $n$-gon in which the lengths of the diagonals take on no more than two distinct values? (I. Rubanov)

Answer: 7


6. There are 900 three-digit numbers (100, $101, \cdots, 999$). If these three-digit numbers are printed on cards, with one number per card, some cards, when flipped, still show a three-digit number, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, which when flipped shows $[\varepsilon \subseteq$, so some cards can be used in two ways. Therefore, the maximum number of cards that can be saved is $\qquad$.

Answer: 34


A sequence of positive integers is constructed by listing the first 4, then skipping one, listing the next 5, skipping 2, listing 6, skipping 3, and, on the $n$th iteration, listing $n + 3$ and skipping $n$. The sequence begins 1, 2, 3, 4, 6, 7, 8, 9, 10, 13. What is the 500,000th number in the sequence?

Answer: 996506


Suppose that $$\begin{array}{@{}c@{\;}c@{}c@{}c} && &P_b \\ & \times & & P_b \\ \cline{2-4} & & Q & P_b, \\ \end{array} $$where $P$, $Q$, and $b$ represent three distinct digits 1-9. If $Q=\frac{P}{2}$, and $P$ is two less than $b$, what is the value of the digit $P$?

Answer: 4


Consider the following two strings of digits: $11001010100101011$ and $110100011000100$. First consider them to be in base $10$ and sum them to get $n$. Then consider them to be in binary, sum them, write the answer in binary, then interpret the digits of the sum as if they were in base $10$ to get $m$. What is $n-m$?

Answer: 0


In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$.

[asy]
pair A, K, O, C, M, B, X, Y, Z, J, T;
O=(0,0);
C=(32,0);
M=(50,0);
B=(68,0);
A=(-68,0);
K=(A+C)/2;
X=(0,68);
Y=(-18,50);
Z=(50,18);
J=(7,43.3);
T=(59,15.6);
path nom, bigc, middlec, smallc;
nom=A--B--(100,100)--(-100,100)--cycle;
bigc=A..X..B--cycle;
middlec=A..Y..C--cycle;
smallc=C..Z..B--cycle;
fill(bigc, gray(.5));
fill(middlec, white);
fill(smallc, white);
draw(smallc);
draw(middlec);
draw(bigc);
draw(A--B);
label("A", A, S);
label("K", K, S);
label("O", O, S);
label("M", M, S);
label("C", C, S);
label("B", B, S);
label("S", J, SW);
label("E", T, SW);
label("$l$", (.9(J-T)+J), NW);
draw((.9(J-T)+J)--(.5(T-J)+T));
dot(K);
dot(O);
dot(M);
dot(J);
dot(T);
[/asy] Line $l$ is drawn to touch the smaller semi-circles at points $S$ and $E$ so that $KS$ and $ME$ are both perpendicular to $l$.  Determine the area of quadrilateral $KSEM$.

Answer: 2040


 Players $A$ and $B$ play a game with $N\geq2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\cdots$ by the following rules:
$\bullet$ On every move of his $B$ passes $1$ coin from every box to an adjacent box.
$\bullet$ On every move of hers $A$ chooses several coins that were not involved in $B$'s previous
move and are in different boxes. She passes every chosen coin to an adjacent box.
Player $A’$s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of
how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

Answer: 4022


Let $\mathcal{A}$ denote the set of all polynomials in three variables $x,y,z$ with integer coefficients. Let $B$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as

$$(x+y+z)P(x,y,z)+(xy+yz+zx)Q(x,y,z)+xyzR(x,y,z)$$

with $P,Q,R\in\mathcal{A}$ .Find the smallest non-negative integer $n$ such that $x^{i}y^{j}z^{k}\in\mathcal{B}$ for all non negative integers $i,j,k$ satisfying $i+j+k\geqslant n$.

Answer: 4


A1. Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.

Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.

Answer: 50


Ninth and tenth graders participated in a chess tournament. There were ten times as many tenth graders as ninth graders. The total score of tenth graders was 4.5 times that of the ninth graders. What was the ninth graders score?

Answer: 10


In a convex 1950-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?

Answer: 1949


Dividing $x^{1951}-1$ by $P(x)=x^4+x^3+2x^2+x+1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.

Answer: -1


Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.

Answer: 5


The distance between towns A and B is 999 km. At every kilometer of the road that connects A and B a sign shows the distances to A and B as follows:
$$\boxed{0|999}\quad\boxed{1|998}\quad\boxed{2|997}\quad\ldots\quad\boxed{998|1}\quad\boxed{999|0}$$
How many signs are there, with both distances written with the help of only two distinct digits?

Answer: 40


A factory produces rattles shaped in the form of a ring with 3 red and 7 blue spherical beads on it. Two rattles are said to be of the same type if one can be obtained from the other one by moving a bead along the ring or by flipping the ring over in space. How many different types of rattles can be manufactured?

Answer: 8


We drew 11 horizontal and 11 vertical intersecting straight lines on a sheet of paper. We call a segment of one of the straight line drawn that connects two neighboring intersections a “link”. What least number of links must we erase in order for each intersection to be a junction of not more than 3 links?

Answer: 41


A total of $2010$ coins are distributed in $5$ boxes. Initially, the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that, she can perform the following transformation any number of times: from a box with at least $4$ coins, she can transfer one coin to each of the other boxes.

What is the maximum number of coins that Martha can take away?

Answer: 2004


In triangle $ABC$, where $\angle ABC$ is obtuse, point $D$ is located on side $AC$ such that $\angle ABD$ is a right angle. Point $E$ is on side $AC$ between points $A$ and $D$, such that line $BD$ bisects $\angle EBC$. Given that $AC = 35$, $BC = 7$, and $BE = 5$, find the length of $CE$.

Answer: 10


Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions:

1. Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime.
2. All other pairs of the five integers are relatively prime.

Answer: 47


Determine all $n \geq 3$ for which there exist $n$ positive integers $a_1, a_2, \ldots, a_n$ such that any two of them have a common divisor greater than 1, but any three of them are coprime. Furthermore, assuming that the numbers $a_i$ are less than 5000, find the greatest possible $n$.

Answer: 4


The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.

Find the smallest possible number of independent deputies after the vacation.

Answer: 50


Navi and Ozna are playing a game where Ozna starts first and the two take turns making moves. A positive integer is written on the board. A move consists of either:

1. Subtracting any positive integer at most $2015$ from it, or
2. If the integer on the board is divisible by $2014$, dividing by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \geq 2015$, ensuring that all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.

Answer: 2013


Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$.

Answer: 630


Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$, each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\frac{a\sqrt{b}-c}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$.

Answer: 46


In a party of $2020$ people, some pairs of people are friends. We define a person's popularity as the size of the largest group of people in the party containing them such that every pair of people in that group are friends. A person has a popularity number of $1$ if they have no friends. What is the largest possible number of distinct popularities in the party?

Answer: 1010


Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times does $111$ appear in this sequence?

Answer: 14


In the diagram, congruent rectangles $ABCD$ and $DEFG$ share a common vertex $D$. The sides $BC$ and $EF$ intersect at point $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$, find the area of the region $ABHED$.

Answer: 54


Five standard six-sided dice are rolled.  We are told there is no three-of-a-kind, but there is a pair of dice that show the same number.  These two dice are set aside, and the other three dice are re-rolled.  What is the probability that after re-rolling these three dice, at least three of the five dice show the same value?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 13


A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?

[asy]
path a=(0,0)--(10,0)--(10,10)--(0,10)--cycle;
path b = (0,10)--(6,16)--(16,16)--(16,6)--(10,0);
path c= (10,10)--(16,16);
path d= (0,0)--(3,13)--(13,13)--(10,0);
path e= (13,13)--(16,6);
draw(a,linewidth(0.7));
draw(b,linewidth(0.7));
draw(c,linewidth(0.7));
draw(d,linewidth(0.7));
draw(e,linewidth(0.7));
draw(shift((20,0))*a,linewidth(0.7));
draw(shift((20,0))*b,linewidth(0.7));
draw(shift((20,0))*c,linewidth(0.7));
draw(shift((20,0))*d,linewidth(0.7));
draw(shift((20,0))*e,linewidth(0.7));
draw((20,0)--(25,10)--(30,0),dashed);
draw((25,10)--(31,16)--(36,6),dashed);
draw((15,0)--(10,10),Arrow);
draw((15.5,0)--(30,10),Arrow);
label("$W$",(15.2,0),S);
label("Figure 1",(5,0),S);
label("Figure 2",(25,0),S);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 13


Shestakov C.A.
In a convex quadrilateral $ABCD$, points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ respectively.
Segments $AE$, $AF$, and $EF$ divide the quadrilateral into four triangles, the areas of which are (in some order) consecutive natural numbers. What is the greatest possible value of the area of triangle $ABD$?

Answer: 6


4. The graphs of the functions $y=x^{2}-x$ and $y=\cos 10 \pi x(x \geqslant 0)$ intersect at $\qquad$ points.

Answer: 17


In triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $OA = ON = 11$ and $OH = 7.$ Compute $BC^2$.

Answer: 288


Let's share the thought of dividing every three-digit (natural) number by the sum of its digits. What is the smallest quotient that results?

---

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

Answer: \frac{199}{19}


23. Using the digits $1,2,3,4$ (digits can be repeated), how many $n(n \geqslant 2)$-digit numbers can be formed that contain an odd number of 1s, an even number of 2s, and at least one 3?

Answer: \dfrac{4^n - 3^n + (-1)^n}{4}


1.3. At the Faculty of Mechanics and Mathematics of Moscow State University, there are $n$ female students and $2 n-1$ male students (both female and male students are numbered). It is known that the $k$-th female student is acquainted with male students numbered from 1 to $2 k-1$. How many ways are there to form $n$ pairs, each consisting of an acquainted male and female student?

## Second Round

Answer: n!


6. Let $a, b \in \mathbf{N}_{+}$, and $a+b \sqrt{2}=(1+\sqrt{2})^{100}$. Then the last two digits of $a b$ are $\qquad$ .

Answer: 64


In the figure, there is a square $A B C D$, in which four identical isosceles triangles $A B E, B C F, C D G$, and $D A H$ are placed, all shaded gray. The sides of the square $A B C D$ are the bases of these isosceles triangles. We know that the gray areas of the square $A B C D$ together have the same area as its white area. Furthermore, we know that $|H F|=12 \mathrm{~cm}$. Determine the length of the side of the square $A B C D$.

(L. Šimünek)

![](https://cdn.mathpix.com/cropped/2024_04_17_4274193d47a1b7e4f9cbg-2.jpg?height=526&width=594&top_left_y=588&top_left_x=731)

Idea. Appropriately divide the figure.

Answer: 24


14 、Using "10, 1, $ \times , \div$ and adding () to calculate $6, 8, 8, 9$, to make the result 24, for example $9 \times 8 - 8 \times 6 = 24$, please write another expression _.

Please write another expression _.

Answer: (8 \times 6) \div ((9 - 8) + (8 \div 8)) = 24


Let \( n \in \mathbf{Z}_{+} \). Arrange \( n+1 \) bowls in a row, numbered from left to right as \( 0, 1, \cdots, n \). Initially, there are \( n \) stones in bowl 0, and no stones in the other bowls. Person A performs the following operations until all the stones are moved to bowl \( n \): they choose a bowl with \( k \) stones (\( k \geq 1 \)), and move exactly one stone from the chosen bowl to another bowl to the right, where the number of the latter bowl differs from the former by at most \( k \). Let \( T \) be the minimum number of operations required for person A to achieve the goal. Prove:
$$
T \geq \left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \cdots + \left\lceil \frac{n}{n} \right\rceil .
$$

Answer: T \geq \left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \cdots + \left\lceil \frac{n}{n} \right\rceil


5. (1986, Shanghai) The three sides of a triangle are all positive integers, one of which has a length of 4, but it is not the shortest side. How many different triangles are there? ...

The translation is provided while retaining the original text's line breaks and format.

Answer: 8


Professor Greenfield, a renowned expert in agricultural economics, is studying the impact of globalization on food systems. She discovers that a local farm produces 1,500 kilograms of corn per month. Due to globalization efforts, the farm plans to expand its market internationally, which is expected to increase their production by 20% each month for the next three months. How many kilograms of corn will the farm produce in total over these three months if the planned increase is achieved each month?

Answer: 5460


Grandpa George, who lived during the colonial era, often shares stories about how the city has changed over the years. He remembers that back then, the city's main square had just 12 old colonial buildings. Over time, as the city modernized, 3 of those buildings were replaced with new skyscrapers. Recently, the city council decided to preserve its history by restoring 5 of the remaining old buildings. However, 2 more colonial buildings were demolished to make way for a new park. How many of the original colonial buildings are still standing in the city's main square today?

Answer: 7


Judge Thompson, a retired judge with extensive knowledge of estate tax case laws, has been asked to help his neighbor, Mrs. Wilson, understand how much estate tax she might owe. Mrs. Wilson's estate is valued at $2 million, and the tax rate for estates is 20% on amounts over $1 million. However, there is an exemption for the first $500,000 of the estate. How much estate tax will Mrs. Wilson owe after accounting for the exemption?

Answer: 100000


A rock guitarist is planning a series of live performances to connect with their audience. Each show brings in 200 fans. The guitarist wants to perform 3 concerts in one week. During each concert, they plan to play 15 songs. After each song, they take a 2-minute break to interact with the audience. If each song lasts 4 minutes, how many minutes in total will the guitarist spend playing and interacting with the audience during the entire week?

Answer: 270


A local Cheltenham resident who uses hearing aids visits a store to buy new batteries for their hearing aids. Each pack contains 6 batteries, and the resident's hearing aids require 2 batteries each. If the resident needs enough batteries to last for 3 months, and they change the batteries once every 2 weeks, how many packs of batteries should they buy?

Answer: 4


A travel blogger named Alex is planning a trip to explore unique locations in France. Alex wants to visit 5 hidden villages, each in a different region. The distances between these villages are as follows: from the first village to the second is 25 kilometers, the second to the third is 40 kilometers, the third to the fourth is 35 kilometers, and the fourth to the fifth is 30 kilometers. However, Alex wants to make a detour to visit a scenic viewpoint that is 10 kilometers off the route between the third and fourth villages. How many kilometers will Alex travel in total during this trip?

Answer: 150


Mr. Thompson, a traditionalist, enjoys spending his afternoons reading classic books in his quiet living room. He has a collection of 72 books on his wooden bookshelf. Every month, he decides to read 3 new books that he carefully selects from his local library, avoiding any digital versions. However, he also gives away 1 book each month to his neighbor, who shares his love for timeless stories. If Mr. Thompson continues this pattern for 6 months, how many books will he have on his shelf at the end of that period?

Answer: 66


Consider a shaded region consisting of 13 unit squares. This region is located in the first quadrant, beginning along the y-axis and stretching rightwards with the following dimensions: 1 unit wide and 5 units high from the origin and an attached section 3 units wide and 2 units high. This shape is rotated about the x-axis. Calculate the volume of the solid formed.

Answer: 37\pi


Two cylindrical cans must have the same volume. The height of one can is five times the height of the other. If the radius of the narrower can is 10 units, what is the radius of the wider can? Express your answer in simplest radical form.

Answer: 10\sqrt{5}


Right triangle $T$ is inscribed in circle $A$, which has radius $13$. Circle $B$ with radius $4$ is internally tangent to circle $A$ at one vertex of $T$. Circle $C$, with radius $3$, is internally tangent to circle $A$ at another vertex of $T$. The circles $B$ and $C$ are externally tangent to another circle $E$, which has radius $r$. If the angle between the radii of circle $A$ touching vertices related to circles $B$ and $C$ is $90^\circ$, find $r$.

Answer: \frac{\sqrt{181} - 7}{2}


Tracy had a bag of candies. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to her friend Rachel. Tracy and her mom then each ate 20 candies from what Tracy had left. Finally, Tracy's sister took between two and six candies, leaving Tracy with ten candies. How many candies did Tracy have at the start?

Answer: 108


Let $t$ represent all real values. Consider the curve formed by the set of points $(x, y)$ defined by the equations:
\[
x = t^3 - 3t + 1,
\]
\[
y = t^4 - 4t^2 + 4.
\]
Determine the ordered pair $(x,y)$ where the curve crosses itself.

Answer: (1,1)


In right triangle $ABC$, where $AB = BC$, point $D$ is the midpoint of $\overline{BC}$. Point $E$ is on the extension of $\overline{AC}$ such that $CE = 15$ units. If $D$ is also the midpoint of $\overline{AE}$, what is the length of $\overline{BD}$? Express your answer as a decimal to the nearest tenth.

Answer: 7.5


A rectangle $EFGH$ has dimensions $12$ by $15$. An equilateral triangle is inscribed such that no point of the triangle lies outside $EFGH$. Calculate the maximum possible area of such a triangle.

Answer: 48\sqrt{3}


Initially, there were twenty states that could opt to change their primary energy source to solar. If fifteen states decide to go ahead, and two later revert to their previous energy sources, what will be the new ratio rounded to the nearest tenth?

Answer: 0.7


On a one-way single-lane road with a speed limit of 80 kilometers per hour, vehicles must maintain a safe following distance calculated as one vehicle length for each 10 kilometers per hour of speed or fraction thereof. There's a monitoring device alongside the road that counts the number of vehicles passing within one hour. Assuming each vehicle is 5 meters long, determine the maximum number of complete vehicles passing the monitoring device in an hour if no vehicles exceed the speed limit. Calculate the quotient when this number is divided by 10.

Answer: 177


Emily reorganized her playing cards with a new pattern: the numbers repeat in the sequence $A, 2, 3, \ldots, K$, but the suits change every four cards: Hearts, Diamonds, Clubs, and Spades in that order. She wonders what the suit and number of the 101st card would be in this new arrangement.

Answer: 10 \text{ of Hearts}


The base of a triangle is of length $b$, and the altitude is of length $h$. Two rectangles are inscribed in the triangle. The first rectangle has a height of $x$, and the second rectangle, placed directly above the first, has a height of $2x$. Both rectangles share the same base, which lies along the base of the triangle. Find the area of the second rectangle.

Answer: \frac{2bx(h-3x)}{h}


In the given diagram, there are two polygons: a regular octagon and a regular pentagon, sharing a common vertex. What is the sum, in degrees, of the measures of angles $PQR$ and $PQS$ located at the shared vertex?

Answer: 243^\circ


Let \(a\), \(b\), and \(c\) be real numbers such that \(a+b+c = t\). Additionally, let \(k\) be a positive real constant. Find, in terms of \(t\) and \(k\), the minimum value of \(ka^2 + b^2 + kc^2\).

Answer: \frac{kt^2}{k+2}


$\triangle XYZ$ is inscribed inside $\triangle PQR$ such that points $X, Y, Z$ lie on $RQ, QP, PR$, respectively. The circumcircles of $\triangle PXZ, \triangle QYX, \triangle RZY$ have centers $O_4, O_5, O_6$, respectively. $PQ = 26, QR = 29, RP = 25$, and $\stackrel{\frown}{PY} = \stackrel{\frown}{RZ},\ \stackrel{\frown}{QZ} = \stackrel{\frown}{PX},\ \stackrel{\frown}{QY} = \stackrel{\frown}{RX}$. The length of $RQ$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

Answer: 30


Rational Woman and Irrational Woman decide to drive around two racetracks. Rational Woman drives on a path parameterized by
\[
x = 2 + \cos t, \quad y = \sin t,
\]
and Irrational Woman drives on a path parameterized by
\[
x = 3 + 3 \cos \frac{t}{2}, \quad y = \sin \frac{t}{2}.
\]
Find the smallest possible distance between any two points, each belonging to one of the tracks.

Answer: 0


Let $\overline{AD}, \overline{BE}, \overline{CF}$ be the altitudes of acute triangle $ABC$. If
\[7 \overrightarrow{AD} + 3 \overrightarrow{BE} + 8 \overrightarrow{CF} = \mathbf{0},\]
compute $\angle BAC$ in degrees.

Answer: 60^\circ


Construct a triangular array of numbers starting with initial values 1 on the sides and interior numbers obtained by adding the two adjacent numbers from the previous row. Rows 1 through 4 are shown:

\[
\begin{array}{ccccccccc}
& & & 1 & & 1 & & & \\
& & 1 & & 2 & & 1 & & \\
& 1 & & 3 & & 3 & & 1 & \\
1 & & 4 & & 6 & & 4 & & 1 \\
\end{array}
\]

Find the sum of the numbers in the 50th row.

Answer: 2^{50}


The graph of the function $y = g(x)$ is displayed. For all $x > 3$, $g(x) > 0.5$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A$, $B$, and $C$ are integers, determine $A+B+C$. The vertical asymptotes of $g$ are at $x = -3$ and $x = 2$, and the horizontal asymptote is at $y = 1/A$ with $1/A < 1$.

Answer: -8


In an equilateral triangle $ABC$, lines $DE$, $FG$, and $HI$ are drawn parallel to side $BC$, with $AD = DE = EG = GI$. Find the ratio of the area of trapezoid $HIAB$ to the area of triangle $ABC$.

Answer: \frac{7}{16}


**Martin's Coin Collection**

Martin organizes his coin collection from different regions and by the decade in which they were minted. The values of coins from different regions he collected are: USA and Mexico, $8$ cents each; Canada $5$ cents each; and Brazil $7$ cents each. (USA and Canada are North American countries and Mexico and Brazil are in Central and South America, respectively.) He has the following number of coins from different decades:

- **1950s**: USA: 5, Canada: 9, Mexico: 7, Brazil: 3
- **1960s**: USA: 11, Canada: 6, Mexico: 4, Brazil: 8
- **1970s**: USA: 9, Canada: 10, Mexico: 14, Brazil: 11
- **1980s**: USA: 8, Canada: 12, Mexico: 20, Brazil: 14

In dollars and cents, how much did his Central and South American coins minted before the $80\text{'s}$ cost him?

Answer: \$3.54


Kim earned scores of 89, 85, 91, and 87 on her first four mathematics examinations. If Kim receives a score of 82 on the fifth exam, what will be the change in her average score?

Answer: -1.2


Define $f(n)$ to be the sum of all divisors of a positive integer $n$. If $f(f(n)) = n + 3$, then call $n$ superdeficient. Determine how many superdeficient positive integers exist.

Answer: 0


Liam was given some coins by his grandparents. He lost one-third of them during a bike ride. After searching, he found five-sixths of the coins he had lost. Before finding these coins, however, he accidentally spent one-fourth of the total coins he initially had. How many fractions of the original total coins is Liam still missing after all these events?

Answer: \frac{11}{36}


Six different numbers are 1873, 1997, 2021, 2028, 2117, and 2127. Find the mean of two numbers when it's known that the mean of four of these numbers is 2012.

Answer: 2057.5


Sandy's daughter has a playhouse in the back yard. She plans to cover the two shaded side walls and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 10-foot by 15-foot sections that cost $\$35$ each. If Sandy can cut the siding when she gets home, how many dollars will be the cost of the siding Sandy must purchase?

Answer: \$70


Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in $4n^2$ squares with dimensions $1\times1$. What is the highest possible number of diagonals that can be drawn in $1\times1$ squares, such that each two diagonals have no common points.

Answer: 2n^2


Let \( S \) be the unit circle on the complex plane (i.e., the set of complex numbers with modulus equal to 1), and \( f \) be a mapping from \( S \) to \( S \). For any \( z \in S \), define

\[
f^{(1)}(z) = f(z),\quad f^{(2)}(z) = f(f(z)),\quad \cdots,\quad f^{(k)}(z) = \underbrace{f(f(\cdots(f}_{k \text{ times}}(z) \cdots))),
\]

If \( c \in S \) satisfies

\[
f^{(1)}(c) \neq c,\quad f^{(2)}(c) \neq c,\quad \cdots,\quad f^{(n-1)}(c) \neq c,\quad f^{(n)}(c) = c,
\]

we call \( c \) an \( n \)-periodic point of \( f \).

Let \( m \) be a natural number greater than 1 and define \( f \) as follows:

\[
f(z) = z^{m} \quad (z \in S).
\]

Find the total number of 1989-periodic points of \( f \).

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


2. As shown in Figure 2, in parallelogram $A B C D$, $\angle A B C=72^{\circ}, A F \perp B C$ at point $F, A F$ intersects $B D$ at point $E$. If $D E=2 A B$, then $\angle A E D=$ $\qquad$ .

Answer: 66^\circ


Problem 5. (5 points) In a tournament, 55 boxers participated, and after each match, the winner left the tournament. The matches were held consecutively. It is known that in each match, boxers whose difference in the number of victories (at that moment) was no more than 1, met. How many matches could the winner of the tournament have played at most?

Answer: 8


8.1. Find ten natural numbers whose sum and product are equal to 20.

Answer: 1


Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance between points $A$ and $C$ on the lateral surface of the frustum.

Answer: 4 \sqrt{13 - 6 \sqrt{2}}


3. The original equation is equivalent to

$$
\begin{aligned}
\left(\frac{1-\cos 2 x}{2}\right)^{5}+\left(\frac{1+\cos 2 x}{2}\right)^{5}=\frac{29}{16} \cos ^{4} 2 x & \Longleftrightarrow \\
& \Longleftrightarrow \frac{2+20 \cos ^{2} 2 x+10 \cos ^{4} 2 x}{32}=\frac{29}{16} \cos ^{4} 2 x
\end{aligned}
$$

From which $24 \cos ^{4} 2 x-10 \cos ^{2} 2 x-1=0, \cos ^{2} 2 x=\frac{1}{2}, \cos 4 x=0$.

Answer: \frac{\pi}{8} + \frac{k\pi}{4}


Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

Answer: 13


182. Quarrelsome Children. A man married a widow, and each of them had children from their previous marriages. After 10 years, a battle broke out in which all the children (by then there were 12) participated. The mother ran to the father, shouting:

- Come quickly. Your children and my children are beating our children!

Each now had 9 children of their own.

How many children were born in these 10 years?

Answer: 6


3. At the high school reunion, former students of one class arrived. All the men shook hands with each other, the women kissed each other (on the cheek), and each man kissed each woman once on the hand. How many people attended the dinner if there were a total of 288 kisses and 78 handshakes?

Answer: 25


4. Variant 1.

The numbers $96, 28, 6, 20$ were written on the board. One of them was multiplied, another was divided, a third was increased, and a fourth was decreased by the same number. As a result, all the numbers became equal to one number. Which one?

Answer: 24


[ Isosceles, inscribed, and circumscribed trapezoids ] [ Projections of the bases, sides, or vertices of a trapezoid ]

The height of an isosceles trapezoid, dropped from the vertex of the smaller base to the larger base, divides the larger base into segments that are in the ratio 2:3. How does the larger base relate to the smaller one?

Answer: 5:1


Example 1 Write two integers on the blackboard, then erase one of them and replace it with the sum or difference of the other two numbers. If this process is repeated several times, and the final result is (17.1999, 2015), then - what could the initial numbers on the blackboard be: (1)(2,2,2); (2)(3,3,3).

Answer: (3,3,3)


As shown in the figure, it is known that $\mathrm{PA}$ and $\mathrm{PB}$ are tangent to circle $\odot O$ at points $\mathrm{A}$ and $\mathrm{B}$ respectively, and $\mathrm{PCD}$ is a secant line of circle $\odot O$. $\mathrm{E}$ is the midpoint of $\mathrm{AB}$. Prove that $\angle \mathrm{ACD} = \angle \mathrm{BCE}$.

Answer: \angle \mathrm{ACD} = \angle \mathrm{BCE}


Let $k>0$ be a fixed integer. Compute the minimum integer $N$ (in terms of $k$ ) for which there exists a set of $2 k+1$ distinct positive integers that has sum greater than $N$, but for which every subset of size $k$ has sum at most $N / 2$.

Answer: 2k^3 + 3k^2 + 3k


We erect a straight cone over the base of a hemisphere with radius $r$, the height of which is 2r. Calculate the volume of the common solid formed by the two bodies.

Answer: \dfrac{14}{25} \pi r^3


11、Given the ellipse $C: \frac{x^{2}}{3}+y^{2}=1$ with its upper vertex $A$, a line $l$ that does not pass through $A$ intersects the ellipse $C$ at points $P$ and $Q$, and $A P \perp A Q$. Find the maximum value of the area of $\triangle A P Q$.

---

Note: The format and line breaks have been preserved as requested.

Answer: \dfrac{9}{4}


4. Variant 1.

A rectangle was cut into three rectangles, two of which have dimensions 9 m $\times$ 12 m and 10 m $\times$ 15 m. What is the maximum area the original rectangle could have had? Express your answer in square meters.

Answer: 330


Alex, a professional wrestler who advocates for LGBTQ+ representation, is planning a special wrestling event to celebrate diversity in sports. They have decided to donate $5 for every ticket sold to a local LGBTQ+ charity. The event is expected to attract 300 fans. Additionally, the arena where the event will be held can accommodate 500 fans, and Alex has reserved 10% of the tickets exclusively for members of the LGBTQ+ community at a discounted rate of $10 per ticket. The remaining tickets are sold at the regular price of $20 each.

Calculate the total amount of money Alex will donate to the charity if all the tickets are sold out, considering both the discounted and regular ticket sales.

Answer: 2500


Emily is a stay-at-home mom who manages the household while her partner works. She has 5 parenting tips that she wants to share with a group of friends. During a gathering, she shares 3 tips with her friend Sarah and 2 tips with her friend Julie. Later, she writes down another 4 new tips that she plans to share during their next meeting. How many parenting tips does Emily have in total now?

Answer: 9


Lars is an underground music enthusiast from Germany who exchanges mixtapes and music recommendations with his fans. He decides to create mixtapes that feature 12 tracks each, carefully chosen from his extensive collection of 144 songs. He plans to send one mixtape to each of his 9 fans.

To make the mixtapes unique, he ensures that no two mixtapes have more than 3 tracks in common. After selecting the tracks for the mixtapes, he realizes he has 18 extra songs that he didn't use in any mixtape.

How many different tracks did Lars use in total across all 9 mixtapes?

Answer: 126


Alex is a software engineer who is developing an algorithm to optimize resource allocation for a bike-sharing system in a city. The city has 5 different bike stations. Each station initially has a certain number of bikes: Station A has 12 bikes, Station B has 8 bikes, Station C has 15 bikes, Station D has 10 bikes, and Station E has 5 bikes. On a busy Monday morning, the demand for bikes is as follows:

- Station A needs 10 additional bikes.
- Station B needs 3 additional bikes.
- Station C has 4 too many bikes.
- Station D needs 6 additional bikes.
- Station E has 2 too many bikes.

Alex wants to redistribute the bikes to meet the demand without moving any bikes between stations more than necessary. How many bikes in total does Alex need to move between the stations to optimize the allocation?

Answer: 6


Professor Artie, a professor of art history, is organizing an art exhibition to demonstrate the concept of artistic subjectivity and interpretation. She has selected 5 different pieces of art for the exhibition, each representing a unique interpretation of a historical event. To highlight the diversity of interpretations, she decides to display each piece in 3 different rooms, each room offering a different lighting setup that influences how the art is perceived.

Professor Artie wants to ensure that each piece is viewed 7 times over the course of the exhibition by different groups of students. If she plans to have 2 tours per room each day, how many days will the exhibition last if she wants each piece to be viewed the desired number of times?

Answer: 6


Coach Li is the caring and dedicated coach of a girls' soccer team. She wants to make sure her players are well-prepared for their upcoming tournament. To do this, she plans to divide her 18 players into 3 equal groups for specialized training sessions. Each group will have a different focus: passing, shooting, and defense. Coach Li has scheduled 45 minutes for each training session and wants to ensure that each group spends an equal amount of time on each focus area. If Coach Li wants to repeat this cycle of training sessions twice before the tournament, how many total minutes will each player spend on training for passing, shooting, and defense together?

Answer: 270


Every Sunday, a Catholic parishioner named Maria travels from her home near Easton to attend the 10:00 AM Mass at Corpus Christi Parish in Lawrence. The distance from her home to the parish is 15 miles. She also volunteers at the parish every Sunday, spending 2 hours helping with community activities. After volunteering, Maria visits a local bakery 3 miles from the parish to buy bread for her family. If Maria then returns home directly from the bakery, calculate the total distance, in miles, that Maria travels on Sundays.

Answer: 36


Alex is a successful real estate developer who is working on a new construction project in a city that has recently implemented economic policies to encourage development. These policies include a 10% tax rebate on construction costs and a 5% increase in property value once the project is completed. Alex's current project involves constructing a new apartment complex with a base construction cost of $5,000,000.

Calculate the total effective cost of the project after applying the tax rebate, and then determine the expected property value after the 5% increase. What is the difference between the expected property value and the total effective cost of the project?

Answer: 750000


A language policy advisor is working on a program to support linguistic diversity in a city. She plans to set up language workshops for 5 different languages. Each workshop can host 20 participants. To ensure diverse participation, she wants to invite 3 different age groups: children, teenagers, and adults. She decides that each age group should have an equal number of participants in each workshop.

If the advisor plans to run a total of 4 workshops for each language, how many participants from each age group should she invite for all the workshops combined?

Answer: 133


Alex, a timid office worker, decides to overcome their fear of heights by taking up rock climbing. To start, Alex chooses a beginner climbing wall at the local gym, which is 12 feet tall. Every week, Alex plans to climb a wall that is 3 feet taller than the previous one to gradually build confidence and skills. After 4 weeks, what will be the height of the climbing wall that Alex will attempt to climb?

Answer: 21


A social worker is helping a single parent and their teenager with budgeting for the month. The single parent earns $2,400 a month. Their monthly expenses are as follows: $800 for rent, $300 for groceries, $150 for utilities, $100 for transportation, and $50 for internet. The teenager has a part-time job and earns $200 a month, which they contribute towards family expenses. After covering all these expenses, the social worker wants to help them create a savings plan. How much money will the family have left to save at the end of the month?

Answer: 1200


Alex has a rare phobia of balloons popping, called globophobia. To overcome their fear, Alex decides to take small steps by gradually exposing themselves to balloons in a controlled setting. On the first day, Alex bravely handles 3 balloons. Each day, they increase the number of balloons by 2 more than the previous day. If Alex continues this pattern for a total of 7 days, how many balloons does Alex handle on the seventh day?

Answer: 45


Maria is a devoted member of her local church and attends services regularly. She hopes her son, Alex, will join her more often. During a special religious season lasting 40 days, Maria attends church 5 times a week. Alex decides to accompany her one Sunday, and Maria is thrilled. For every 5 services Maria attends, Alex agrees to attend 1 additional service. How many more services will Alex attend with Maria during this 40-day period?

Answer: 6


Professor Garcia, a faculty advisor, is helping the student committee plan a social justice event on campus. They want to distribute flyers to promote the event. The committee has decided to print 120 flyers. Each flyer costs $0.15 to print. Additionally, they need to buy snacks for the volunteers, which will cost $25 in total. Professor Garcia also suggests renting a small sound system for $50 for the event. If the committee has a budget of $100, how much more money do they need to cover all the expenses?

Answer: 0


2. Let $\alpha, \beta$ be acute angles. When
$$
-\frac{1}{\cos ^{2} \alpha}+\frac{1}{\sin ^{2} \alpha \sin ^{2} \beta \cos ^{2} \beta}
$$

takes the minimum value, the value of $\operatorname{tg}^{2} \alpha+\operatorname{tg}^{2} \beta$ is

Answer: 3


The unit-sided square $ABCD$ is rotated $90^{\circ}$ around vertex $C$. What area does side $AB$ sweep?

Answer: \dfrac{\pi}{4}


2. For a positive integer $n$, let $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are coprime to $n$, and let $f(n)$ denote the smallest positive integer greater than $n$ that is not coprime to $n$. If $f(n)=m$ and $\varphi(m)=n$, then the pair of positive integers $(n, m)$ is called a friendly pair. Find all friendly pairs.
(Liu Kaifeng, problem contributor)

Answer: (2, 4)


Problem 6. Master Li Si Qing makes fans. Each fan consists of 6 sectors, painted on both sides in red and blue (see fig.). Moreover, if one side of a sector is painted red, the opposite side is painted blue and vice versa. Any two fans made by the master differ in coloring (if one coloring can be transformed into another by flipping the fan, they are considered the same). What is the maximum number of fans the master can make?

Answer: 36


Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called [i]parallelogram-free[/i] if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?

Answer: 2n - 1


3. (8 points) The teacher has a calculator, and each student comes up in turn to input a natural number. The first student inputs 1, the second student inputs 2, the third student inputs 3. Since the buttons 4 and 8 on the calculator are broken, the next student can only input 5, and the following students input 6, 7, 9, 10, 11, 12, 13, 15, and so on. According to this input method, Xiao Ming is the 100th student. The natural number Xiao Ming inputs is $\qquad$.

Answer: 155


4. An observer views the segment $A B$ from two points $C$ and $D$ which are $300 \mathrm{~m}$ apart, at angles of $30^{\circ}$. The lines $A D$ and $B C$ are mutually perpendicular. Determine the length of the segment $A B$.

Answer: 100\sqrt{3}


8. If the remainder of $\underbrace{11 \cdots 1}_{n+1 \uparrow} 1$ divided by 3102 is 1, then the smallest positive integer $n$ is $\qquad$ .

Answer: 138


Four, (15 points) On a plane, $n$ points are called a "standard $n$-point set" if among any three of these points, there are always two points whose distance is no more than 1. To ensure that a circular paper with a radius of 1 can cover at least 25 points of any standard $n$-point set, find the minimum value of $n$.

Answer: 49


14. In an equilateral triangle $\mathrm{ABC}$, take any point $\mathbf{P}$ inside it, and draw perpendiculars $\mathrm{PD}, \mathrm{PE}$, and $\mathrm{PF}$ from this point to $\mathrm{BC}, \mathrm{CA}, \mathrm{AB}$ respectively, with $\mathrm{D}$, $\mathrm{E}$, and $\mathrm{F}$ being the feet of the perpendiculars. Calculate $\frac{P D+P E+P F}{B D+C E+A F}$.

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


Let $n,k$ be arbitrary positive integers. We fill the entries of an $n\times k$ array with integers such that all the $n$ rows contain the integers $1,2,\dots,k$ in some order. Add up the numbers in all $k$ columns – let $S$ be the largest of these sums. What is the minimal value of $S$?

Answer: \left\lceil \frac{n(k+1)}{2} \right\rceil


Example 12 (1995 National High School League Question) Given the family of curves $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+$ $\cos \theta+1) y=0$ ( $\theta$ is a parameter). Find the maximum value of the length of the chord intercepted by the family of curves on the line $y=2 x$.

Answer: 8\sqrt{5}


A positive integer is called a "random number" if and only if:
(1) All digits are non-zero.
(2) The number is divisible by 11.
(3) The number is divisible by 12, and if the digits of the number are permuted in any way, the resulting number is still divisible by 12.
How many such ten-digit random numbers are there?

Answer: 50


Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order.  Compute the number $y$.

Answer: 132


During an early morning drive to work, Simon encountered \( n \) sets of traffic lights, each set being red, amber, or green as he approached it. He noticed that consecutive lights were never the same colour. Given that he saw at least two red lights, find a simplified expression, in terms of \( n \), for the number of possible sequences of colours Simon could have seen.

Answer: 3 \cdot 2^{n-1} - 4n + 2


Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$  and show that the tangents at $A,D$ to the circle meet on the line $BC.$

Answer: \dfrac{3}{4}


Problem 5. (20 points) A point is randomly thrown onto the segment $[6 ; 10]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-3 k-10\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.

Answer: \dfrac{1}{3}


15. As shown in the figure, the area of square $\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter. Therefore, the area of the shaded part is $\qquad$ square centimeters.

Answer: 72


4.1. (13 points) A guard has detained a stranger and wants to drive him away. But the person caught said that he had made a bet with his friends for 100 coins that the guard would not drive him away (if the guard does, he has to pay his friends 100 coins, otherwise they pay him), and, deciding to bribe the guard, offered him to name a sum. What is the largest number of coins the guard can ask for so that the stranger, guided solely by his own profit, would definitely pay the guard?

Answer: 199


B2. The quadratic function $h$ has its vertex at the intersection of the graphs of the functions $f(x)=\log _{3} x$ and $g(x)=-x+4$ and has an initial value of -17. Write the formula of the quadratic function $h$ in standard and vertex form. Draw the graphs of both functions $f$ and $g$ in the same Cartesian coordinate system.

Naloge za 3. letnik

Answer: -2(x - 3)^2 + 1


4. Find the number of distinct quadratic trinomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two distinct roots, which are powers of the number 7 with integer non-negative exponents, and at the same time, their coefficients in absolute value do not exceed $343^{36}$.

Answer: 2969


33. Find the smallest number \( n > 1980 \) such that the number

\[
\frac{x_{1}+x_{2}+x_{3}+\ldots+x_{n}}{5}
\]

is an integer for any assignment of integers \( x_{1}, x_{2}, x_{3}, \ldots, x_{n} \), none of which are divisible by 5.

Answer: 1985


Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Answer: 13


Let $S = 2^3 + 3^4 + 5^4 + 7^4 + \cdots + 17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.

Answer: 93


Let $w, x, y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold:

- $\frac{1}{w} + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3,$
- $wx + wy + wz + xy + xz + yz = 14,$
- $(w + x)^3 + (w + y)^3 + (w + z)^3 + (x + y)^3 + (x + z)^3 + (y + z)^3 = 2160,$
- $w + x + y + z + i\sqrt{n} \in \mathbb{R}.$

Compute $n$.

Answer: 47


Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x \neq y$, we have $xy \notin A$. Determine the maximum possible size of $A$.

Answer: 999001


Let $X$ be the set consisting of twenty positive integers $n, n+2, \ldots, n+38$. Determine the smallest value of $n$ such that any three numbers $a, b, c \in X$, not necessarily distinct, can form the sides of an acute-angled triangle.

Answer: 92


For each integer $k \ge 2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. For example, $X_2 = 10241048576$. If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n \ge 2$ is an integer.

Answer: 5


In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$.

If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain?

Answer: 6


The continent of Trianglandia is an equilateral triangle with a side length of $9$. It is divided into $81$ smaller equilateral triangles, each with a side length of $1$. Each smaller triangle (country) can choose to build a wall along at most one of its $3$ sides. Due to ongoing conflicts, no two adjacent countries can have their walls touch, even at a corner. What is the maximum number of walls that can be constructed in Trianglandia?

Answer: 45


Navi and Ozna are playing a game where Ozna starts first, and the two take turns making moves. A positive integer is written on the board. A move consists of either:

1. Subtracting any positive integer at most 2015 from it, or
2. If the integer on the board is divisible by $2014$, dividing it by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \ge 2015$ and all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.

Answer: 2013


Pete chooses $1004$ monic quadratic polynomials $f_1, \cdots, f_{1004}$ such that each integer from $0$ to $2007$ is a root of at least one of them. Vasya considers all equations of the form $f_i = f_j$ for $i \neq j$ and computes their roots. For each such root, Pete has to pay Vasya $1$ ruble. Find the least possible value of Vasya's income.

Answer: 0


Initially, five variables are defined: $a_1 = 1, a_2 = 0, a_3 = 0, a_4 = 0, a_5 = 0$. On a turn, Evan can choose an integer $2 \le i \le 5$. Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2$, then now $a_1 = 1, a_2 = 0 + 1 = 1, a_3 = 0, a_4 = 0, a_5 = 0$. Find the minimum number of turns Evan needs to make $a_5$ exceed 1,000,000.

Answer: 127


Let $P(A)$ be the arithmetic mean of all elements of the set $A = \{ a_1, a_2, \ldots, a_n \}$, defined as $P(A) = \frac{1}{n} \sum^{n}_{i=1} a_i$. A subset $B$ of $A$ is called a "balanced subset" if $B$ is non-empty and $P(B) = P(A)$. Given the set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, find the number of all "balanced subsets" of $M$.

Answer: 51


The following diagram shows an equilateral triangle $\triangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\triangle ABC$ a distance $\frac{1}{8} AB$ along a side of $\triangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has an area of $300$. Find the area of $\triangle ABC$.

Answer: 768


Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:
- All four rectangles share a common vertex $P$.
- For each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$.
- For each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$.

Compute the total area covered by the union of the four rectangles.

Answer: 30


A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$.

Answer: 2083


Equilateral  $\triangle ABC$  has side length  $600$ . Points  $P$  and  $Q$  lie outside of the plane of  $\triangle ABC$  and are on the opposite sides of the plane. Furthermore,  $PA=PB=PC$ , and  $QA=QB=QC$ , and the planes of  $\triangle PAB$  and  $\triangle QAB$  form a  $120^{\circ}$  dihedral angle (The angle between the two planes). There is a point  $O$  whose distance from each of  $A,B,C,P$  and  $Q$  is  $d$ . Find  $d$ .

Answer: 450


Denote by  $S(n)$  the sum of digits of  $n$ . Given a positive integer  $N$ , we consider the following process: We take the sum of digits  $S(N)$ , then take its sum of digits  $S(S(N))$ , then its sum of digits  $S(S(S(N)))$ ... We continue this until we are left with a one-digit number.

We call the number of times we had to activate  $S(\cdot)$  the **depth** of  $N$ .

For example, the depth of 49 is 2, since  $S(49)=13\rightarrow S(13)=4$ , and the depth of 45 is 1, since  $S(45)=9$ .

[list=a]
[*] Prove that every positive integer  $N$  has a finite depth, that is, at some point of the process we get a one-digit number.
[*] Define  $x(n)$  to be the <u>minimal</u> positive integer with depth  $n$ . Find the residue of  $x(5776)\mod 6$ .
[*] Find the residue of  $x(5776)-x(5708)\mod 2016$ .
[/list]

Answer: 0


A basket is called "*Stuff Basket*" if it includes  $10$  kilograms of rice and  $30$  number of eggs. A market is to distribute  $100$  Stuff Baskets. We know that there is totally  $1000$  kilograms of rice and  $3000$  number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide  $100$  Stuff Baskets?

Answer: 99


Let  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.

Answer: 84


Suppose that points  $A$  and  $B$  lie on circle  $\Omega$ , and suppose that points  $C$  and  $D$  are the trisection points of major arc  $AB$ , with  $C$  closer to  $B$  than  $A$ . Let  $E$  be the intersection of line  $AB$  with the line tangent to  $\Omega$  at  $C$ . Suppose that  $DC = 8$  and  $DB = 11$ . If  $DE = a\sqrt{b}$  for integers  $a$  and  $b$  with  $b$  squarefree, find  $a + b$ .

Answer: 37


Circles  $\mathcal{C}_1$ ,  $\mathcal{C}_2$ , and  $\mathcal{C}_3$  have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line  $t_1$  is a common internal tangent to  $\mathcal{C}_1$  and  $\mathcal{C}_2$  and has a positive slope, and line  $t_2$  is a common internal tangent to  $\mathcal{C}_2$  and  $\mathcal{C}_3$  and has a negative slope. Given that lines  $t_1$  and  $t_2$  intersect at  $(x,y)$ , and that  $x=p-q\sqrt{r}$ , where  $p$ ,  $q$ , and  $r$  are positive integers and  $r$  is not divisible by the square of any prime, find  $p+q+r$ .

Answer: 27