Mrs. Toad has a class of $2017$ students, with unhappiness levels $1, 2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class into exactly $15$ groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all $15$ groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into $15$ groups?

Answer: 1121


5. As shown in Figure 2, two circles with equal radii intersect at points $B$ and $C$, with centers $O_{1}$ and $O_{2}$ lying on the other circle, respectively. $AB$ is the diameter of $\odot O_{1}$, and $AO_{2}$ intersects $\odot O_{2}$ at points $M$ and $N$, where point $M$ is between $A$ and $O_{2}$. If the extension of $CM$ intersects the extension of $NB$ at point $P$, then $CP: CN=$ $\qquad$

Answer: \sqrt{3}:1


Nezha has 100 cards, each marked with a natural number from 1 to 100. He gives the even-numbered cards to Ao Bing, and some of the odd-numbered cards to Taiyi Zhenren. The sum of the numbers on the remaining cards is exactly 2021. Nezha has at most $\qquad$ cards left.

Answer: 43


Problem 8.1. The graph of a linear function is parallel to the graph of $y=\frac{5}{4} x+\frac{95}{4}$, passing through $M(-1 ;-25)$, and intersects the coordinate axes $O x$ and $O y$ in $A$ and $B$ correspondingly.

(a) Find the coordinates of $A$ and $B$.

(b) Consider the unity grid in the plane. Find the number of squares containing points of $A B$ (in their interiors).

Answer: 38


335. What is the greatest number of balls of radius 7 that can simultaneously touch, without intersecting, a ball of radius 3?

Transition to space.

Answer: 6


5. (15 points) A light ray falls at an angle $\alpha=60^{\circ}$ on the front surface of a parallel glass plate. The refractive index of the glass is $n=1.6$. By what angle is the ray, reflected from the back surface of the plate and exiting back through the front surface, deflected from the direction of the incident ray?

Answer: 60


4. The time difference between Sydney and Beijing is 3 hours, for example: when it is 12:00 in Sydney, it is 9:00 in Beijing. On a certain day, when it is 9:15 in Sydney, Xiao Ma and Xiao Yang depart from Sydney and Beijing respectively to go to the other's location. Xiao Ma arrives in Beijing at 19:33 Beijing time. The ratio of the travel time for Xiao Ma and Xiao Yang is 7:6. What is the local time in Sydney when Xiao Yang arrives? $\qquad$ —.

Answer: 20:39


Find the smallest integer $n\ge3$ for which there exists an $n$-gon and a point within it such that, if a light bulb is placed at that point, on each side of the polygon there will be a point that is not lightened. Show that for this smallest value of $n$ there always exist two points within the $n$-gon such that the bulbs placed at these points will lighten up the whole perimeter of the $n$-gon.

Answer: n = 6


12. (10 points) There are four colors of light bulbs (with an ample supply of each color), and one light bulb needs to be installed at each vertex of the triangular prism $A B C-A_{1} B_{1} C_{1}$. The requirement is that the light bulbs at the two endpoints of the same edge must be of different colors, and each color of light bulb must be used at least once. How many installation methods are there?

Answer: 216


1. Initially, the numbers 1, 2, and 4 are written on the board. Every minute, Anton erases the numbers written on the board and writes down their pairwise sums instead. After an hour, three enormous numbers are written on the board. What are their last digits? List them in any order, separated by semicolons.

Example of answer format:

$1 ; 2 ; 3$

Answer: 6;7;9


7.5. Petya tells his neighbor Vova: «In our class, there are 30 people, and there is an interesting situation: any two boys have a different number of girlfriends in the class, and any two girls have a different number of friends among the boys in the class. Can you determine how many boys and how many girls we have in the class?» Vova replies: «You must be mistaken, such a situation is impossible!». a) Is Vova right? b) And how would you answer Petya's question?

Answer: 15


## Task $7 / 84$

A man says: "The birth year of my grandson is the product $x y$ of two natural numbers $x$ and $y$. In the year $x^{2}$, he will be $x$ years old."

In which year was the grandson born?

Answer: 1980


8. (10 points) In the inscribed quadrilateral $A B C D$, the degree measures of the angles are in the ratio $\angle A: \angle B: \angle C=2: 3: 4$. Find the length of $A C$, if $C D=12, B C=8 \sqrt{3}-6$.

Answer: 20


一、(20 points) As
shown in Figure 3, the
shaded part is a
closed figure within
the semicircle $\odot 0$, it is known that chord $M N$
is parallel to chord $P Q$, and
the distance between the parallel lines $M N$ and $P Q$ is $2 \mathrm{~m}, M N=$
$6 \mathrm{~m}, P Q=8 \mathrm{~m}$. Find the length of $\overparen{M P}$ (accurate to $0.001 \mathrm{~m}$ ).

Answer: 2.265


Problem 8.6. For quadrilateral $ABCD$, it is known that $AB=BD, \angle ABD=\angle DBC, \angle BCD=90^{\circ}$. A point $E$ is marked on segment $BC$ such that $AD=DE$. What is the length of segment $BD$, if it is known that $BE=7, EC=5$?

![](https://cdn.mathpix.com/cropped/2024_05_06_d7ec6e60d14a8fdbf750g-30.jpg?height=476&width=517&top_left_y=91&top_left_x=468)

Answer: 17


7. Two people are walking towards each other from points $A$ and $B$ with speeds $v_{1}=6 \mathrm{~m} / \mathrm{s}$ and $v_{2}=4 \mathrm{~m} / \mathrm{s}$. At the moment of their meeting, one of them turned around and walked in the opposite direction, while the other continued in the same direction. The person who turned around, upon returning to their starting point, waited for the second person. They waited for $t_{2}=10 \mathrm{min}$. Determine how much time $t_{1}$ passed from the start of the journey until the meeting. (15 points)

Answer: 20


Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.

Proposed by Mohammad Ahmadi

Answer: n


Task 3. (15 points) At the quality control department of an oil refinery, Engineer Valentina Ivanovna received a research object consisting of about 200 oil samples (a container designed for 200 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or high-sulfur, and density - either light or heavy. The relative frequency (statistical probability) that a randomly selected sample will be a heavy oil sample is $\frac{1}{9}$. The relative frequency that a randomly selected sample will be a light low-sulfur oil sample is $\frac{11}{18}$. How many high-sulfur oil samples does the object contain, if there were no low-sulfur samples among the heavy oil samples?

Answer: 77


The unit-length rods $O A$ and $A B$ are connected by a hinge at $A$. To $A B$, we attach at $B$ a rod $B C$ of equal length, such that $C$ lies in the plane $O A B$ and $\angle A B C = 90^{\circ}$. Keeping $O$ at the origin, we move our structure in the coordinate plane such that $B$ moves along the line through $O$ with direction vector $(1 ; 1-\sqrt{2})$. What path does the point $C$ trace as $A$ moves around the unit circle centered at the origin?

Answer: \dfrac{x^2}{(1 + \sqrt{2})^2} + \dfrac{y^2}{(\sqrt{2} - 1)^2} = 1


2. Given an integer $n \geqslant 2$, find the largest constant $C_{n}$, such that for any real numbers $a_{1}, a_{2}, \cdots, a_{n}, a_{1}+a_{2}+\cdots+a_{n}=0$, we have $\sum_{i=1}^{n} \min \left\{a_{i}, a_{i+1}\right\} \leqslant C_{n} \cdot \min _{1 \leqslant i \leqslant n} a_{i}$ (here we assume $a_{n+1}=a_{1}$ ).

Answer: \dfrac{n}{n-1}


6. Given a convex quadrilateral ABCD. Point $M$ is the midpoint of side BC, and point $N$ is the midpoint of side CD. Segments AM, AN, and $MN$ divide the quadrilateral into four triangles, the areas of which, written in some order, are consecutive natural numbers. What is the maximum possible area of triangle $\mathrm{ABD}$?

Answer: 6


11. (16 points) As shown in Figure 2, given the parabola $y^{2}=2 p x$ $(p>0), A B$ is a chord passing through the focus $F$. If $A B$ makes an angle $\theta(0<\theta \leqslant \frac{\pi}{2})$ with the $x$-axis, find $\angle A O B$.

Answer: \pi - \arctan\left(\dfrac{4}{3 \sin \theta}\right)


Example 2. Form the equation of the line passing through the point $B(-1,2)$ and tangent to the parabola $y=x^{2}+4 x+9$.

Answer: y = -2x


7. In the Cartesian coordinate system, there are two points $P(-1,1)$ $Q(2,2)$, and the graph of the linear function $y=k x-1$ intersects the extension of line segment $P Q$ (the intersection point does not include $Q$). Then the range of the real number $k$ is $\qquad$.

Answer: (\dfrac{1}{3}, \dfrac{3}{2})


7. Given a polynomial $P(x)$, not identically zero. It is known that for all $x$ the identity $(x-2020) \cdot P(x+1)=(x+2021) \cdot P(x)$ holds. How many roots does the equation $P(x)=0$ have? Answer: 4042

Answer: 4042


24. Use red, pink, yellow, and black 4 colors to paint the 6 sides of a regular hexagon, so that the sides painted yellow and black each have one, and the sides painted red and pink each have 2. Mr. Li has some color blindness. For any two colored regular hexagons $A_{6}$ and $A_{6}^{\prime}$, if $A_{6}^{\prime}$ is obtained by changing the two sides painted red in $A_{6}$ to pink, and the two sides painted pink to red, Mr. Li, due to his color blindness, considers $A_{6}$ and $A_{6}^{\prime}$ to be the same. Find the number of distinct colorings of the hexagon under Mr. Li's identification method.

Answer: 15


489*. Three cyclists set off simultaneously: the first and second from point A, and the third towards them from point B. After 1.5 hours, the first cyclist was at an equal distance from the other two, and after 2 hours from the start, the third cyclist was at an equal distance from the first and second. How many hours after departure was the second cyclist at an equal distance from the first and third?

Answer: 3


On the plane, there is an angle of $60^{\circ}$. A circle touches one side of this angle, intersects the other side at points $A$ and $B$, and intersects the angle bisector at points $C$ and $D$. $A B = C D = \sqrt{6}$. Find the area of the circle bounded by this circle.

Answer: \sqrt{3}\pi


Example 13. There are $n$ parabolas on a plane, where each two intersect at only two points, and no three intersect at the same point. Find how many parts these $n$ parabolas divide the plane into.

Answer: n^2 + 1


4. Every 24 hours, the clock runs ahead by 3 minutes. They were set accurately. After what shortest time will the clock hands show the exact time again?

Answer: 240


Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are positive integers. What is $a+b$?

[i]2021 CCA Math Bonanza Individual Round #12[/i]

Answer: 112


Let \( f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0} \) and \( g(x) = c_{n+1} x^{n+1} + c_{n} x^{n} + \cdots + c_{0} \) be two non-zero polynomials with real coefficients such that \( g(x) = (x+r) f(x) \), where \( r \) is a real number. If \( a = \max \left( \left| a_{n} \right|, \ldots, \left| a_{0} \right| \right) \) and \( c = \max \left( \left| c_{n+1} \right|, \ldots, \left| c_{0} \right| \right) \), prove that \( \frac{a}{c} \leqslant n+1 \).

Answer: \frac{a}{c} \leqslant n+1


6. (10 points) April has 30 days. If there are 5 Saturdays and Sundays, then April 1st is a

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

Note: The note above is not part of the translation but an instruction that was given. Here is the translation as requested:

6. (10 points) April has 30 days. If there are 5 Saturdays and Sundays, then April 1st is a

Answer: Saturday


165. The Decisive Vote. The Congress of the United Society of Wandering Beggars (better known as the Union of Vagrants) convened to decide whether to declare a strike, demanding a shorter working day and increased alms. It was decided that during the vote, those members of the society who would vote in favor of the strike would remain standing, while those against would sit down.

- Gentlemen,-- said the chairman of the meeting after counting the votes,--I have the pleasure to inform you that the strike has been approved by a majority constituting one quarter of the opposition. (Loud cries of approval.)
- Mr. Chairman,-- someone shouted from the back,--some of us couldn't sit down.
- Why not?
- Because there are no chairs here.
- Then, perhaps, those who wanted to sit but couldn't, would not mind raising their hands... I see there are twelve of you, so the strike is canceled by a majority of one vote. (Whistles and disorder in the hall.)

How many members of the Society of Beggars participated in the vote?

Answer: 207


1. A motorcyclist set out from point $A$ with an initial speed of 90 km/h, uniformly increasing his speed (that is, over equal time intervals, his speed increases by the same amount). After three hours, the motorcyclist arrived at point $B$, passing through $C$ along the way. After that, he turned around and, still uniformly increasing his speed, headed back. Two hours later, he passed point $C$ again at a speed of 110 km/h and continued on to $A$. Find the distance between points $A$ and $C$.

Answer: 76


Find the smallest positive integer \( n \) such that there exist \( n \) distinct positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfying the following three conditions:
1. The least common multiple of \( a_{1}, a_{2}, \cdots, a_{n} \) is 2002.
2. For any \( 1 \leq i < j \leq n \), the greatest common divisor of \( a_{i} \) and \( a_{j} \) is greater than 1.
3. The product \( a_{1} a_{2} \cdots a_{n} \) is a perfect square and a multiple of 32.

Additionally, for this smallest positive integer \( n \), find all \( n \)-tuples \( \left\{a_{1}, a_{2}, \cdots, a_{n}\right\} \) that satisfy conditions (1), (2), and (3).

Answer: 7


7. From the sides and diagonals of a regular 12-sided polygon, three different segments are randomly selected. The probability that their lengths are the side lengths of a triangle is $\qquad$

Answer: \dfrac{223}{286}


7. As shown in Figure 1, given a regular tetrahedron $P-A B C$ with all edge lengths equal to 4, points $D, E, F$ are on edges $P A, P B, P C$ respectively. Then the number of $\triangle D E F$ that satisfy $D E = E F = 3, D F = 2$ is $\qquad$.

Answer: 3


5. Let the polynomial $p(x)$ have a degree not exceeding 3, and $p(0)=$ $1, p(3)=0,|p(2+x)|=|p(2-x)|$. If the leading coefficient of $p(x)$ is negative, then $p(x)=$ $\qquad$ $ـ$

Answer: -\dfrac{1}{6}x^3 + x^2 - \dfrac{11}{6}x + 1


There is a certain number of identical plastic bags that can be nested within each other. If all the bags are nested inside one of the bags, we call this situation a "bag of bags." Count the number of ways to create a "bag of bags" using 10 bags.

Answer: 719


# Task № 2

The captain's assistant, observing the loading of the ship, smoked one pipe after another from the start of the loading. When $2 / 3$ of the number of loaded containers became equal to $4/9$ of the number of unloaded containers, and the ship's bells struck noon, the old seafarer began to light another pipe. When he finished smoking it, the ratio of the number of loaded containers to the number of unloaded containers became the inverse of the ratio that existed before he started smoking that pipe. How many pipes did the second assistant smoke during the loading (assuming that the loading speed, as well as the smoking speed, remained constant throughout the process.)

Answer: 5


Four boys bought a boat for $\textdollar 60$ . The first boy paid one half of the sum of the amounts paid by the other boys;
the second boy paid one third of the sum of the amounts paid by the other boys;
and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay?
$\textbf{(A) }\textdollar 10\qquad \textbf{(B) }\textdollar 12\qquad \textbf{(C) }\textdollar 13\qquad \textbf{(D) }\textdollar 14\qquad \textbf{(E) }\textdollar 15$

Answer: 13


7.

In triangle $A B C$, the bisector $A L$ ( $L \in B C$ ) is drawn, and $M$ and $N$ are points on the other two bisectors (or their extensions) such that $M A=M L$ and $N A=N L, \angle B A C=50^{\circ}$.

Find the measure of $\angle M A N$ in degrees.

Answer: 65


Three distinct positive numbers are three consecutive terms of an arithmetic progression. Can these same three numbers be three (not necessarily consecutive) terms of a geometric progression?

Answer: \text{Yes}


3. In a row, 100 knights and 100 liars are standing (in some order). The first person was asked: "Are you a knight?", and the rest were asked in turn: "Is it true that the previous person answered 'Yes'?" What is the maximum number of people who could have said "Yes"? Knights always tell the truth, liars always lie.

Answer: 150


7. Builders need to tile the floor in a small house with small square tiles. The client says that the variant with more red tiles is preferable. For variants with the same number of red tiles, the one with more orange tiles is preferable; and generally, the client's preferences for colors are: red, orange, yellow, green, blue, indigo. His wife, however, wants the opposite:

- the number of indigo tiles should be at least as many as the total number of red, orange, yellow, green, and blue tiles;
- the number of blue tiles should be at least as many as the total number of red, orange, yellow, and green tiles;
- the number of green tiles should be at least as many as the total number of red, orange, and yellow tiles;
- the number of yellow tiles should be at least as many as the total number of red and orange tiles;
- the number of orange tiles should be at least as many as the number of red tiles.

The floor requires 100 tiles.

How many green tiles will be needed if the most preferable variant for the client is chosen, which also meets the conditions set by his wife?

Answer: 13


3. A $4 \times 4 \times h$ rectangular prism can hold 8 small spheres with a radius of 1 and 1 large sphere with a radius of 2. Then the minimum value of $h$ is $\qquad$ .

Answer: 2 + 2\sqrt{7}


1. Solve the equation $1+2^{n}+3^{n}+5^{n}=2 k$ in integers.

Answer: n = 0


5. Take any three points $A, B, C$ on the hyperbola $x^{2}-y^{2}=2$, then the locus equation of the orthocenter $H$ of $\triangle A B C$ is $\qquad$.

Answer: x^2 - y^2 = 2


11. (20 points) As shown in Figure 1, the ellipse $E: \frac{x^{2}}{4}+y^{2}=1$ has its right focus at $F$, and the circle $\odot O: x^{2}+y^{2}=1$ has a tangent line $AB$ that intersects the ellipse $E$ at points $A$ and $B$ ($A$ and $B$ are both on the right side of the $y$-axis). Find the maximum value of the inradius of $\triangle ABF$.

Answer: \dfrac{3 - \sqrt{3}}{4}


Inside circle \( O \), chords \( CD \) and \( EF \) are parallel and intersect the diameter \( AB \) at a \( 45^\circ \) angle. \( CD \) and \( EF \) intersect the diameter \( AB \) at points \( P \) and \( Q \) respectively. Given that the radius of circle \( O \) is 1, prove that \( P C \cdot Q E + P D \cdot Q F < 2 \).

Answer: PC \cdot QE + PD \cdot QF < 2


Example 17 As shown in Figure 1.4.19, a semicircle is constructed inside trapezoid $ABCD$, tangent to the upper base and the two non-parallel sides, with its diameter lying on the lower base. If $AB=2, CD=3$, find the length of the lower base $BC$.

 untranslated text remains the same as the original, only the content within the example has been translated.

Answer: 5


12. (10 points) Given the equation $a b c d=a a d \times e$, where different letters represent different digits, what is the maximum value of the four-digit number $a b c d$?

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

Answer: 3015


6. As shown in Figure 1, in a cylindrical ping-pong ball container with a height of 10 and a base radius of 1, a ping-pong ball is attached to both the top and bottom walls of the container. The ping-pong ball on the bottom wall is tangent to the bottom and the side of the container, and the ping-pong ball on the top wall is tangent to the top and the side of the container (the thickness of the container and the ping-pong balls is negligible). A plane is tangent to both ping-pong balls. It is known that the shape formed by the plane intersecting the edge of the container is an ellipse. Then the eccentricity of this ellipse is $\qquad$

Answer: \dfrac{\sqrt{15}}{4}


One, (Full marks 50 points) Given a right trapezoid $A B C D$ with a point $E$ on the right-angle leg $A B$, such that $\triangle C D E$ is an equilateral triangle, and $\frac{A E}{C D}=\frac{\sqrt{6}}{4}$. Find the value of $\frac{A E}{A B}$.

Answer: \dfrac{\sqrt{5}-1}{2}


4. As shown in the figure, there is a circle and three squares. The vertices of the middle square are all on the circle. The intersection points of the circle with the largest square and the vertices of the smallest square are the midpoints of their respective line segments. The area of the largest square is 12 square centimeters. What is the area of the smallest square in square centimeters?

Answer: 3


5. Qiqi arranged a pattern with matchsticks on the table, and the pattern seen in the mirror on the table is 2022. If Qiqi wants to see the pattern of matchsticks on the table as 2022 in the same position, she needs to move at least $\qquad$ matchsticks.

Answer: 6


From point \( A \) on a circle, two bodies start moving simultaneously in opposite directions: the first one with a constant velocity \( v \), and the second one with uniform acceleration \( a \) and an initial velocity of 0. After what time did the bodies meet for the first time, if the second time they met again at point \( A \)?

Answer: \dfrac{v(\sqrt{5} - 1)}{a}


2B. Find $x$ for which the function $y=\left(x-x_{1}\right)^{2}+\left(x-x_{2}\right)^{2}+\ldots+\left(x-x_{1996}\right)^{2}$ takes the smallest value.

Answer: \dfrac{x_1 + x_2 + \ldots + x_{1996}}{1996}


16. (15 points) Let the function
$$
f(x)=x^{2}-\left(k^{2}-5 a k+3\right) x+7(a, k \in \mathbf{R}) \text {. }
$$

For any $k \in[0,2]$, if $x_{1}, x_{2}$ satisfy
$$
x_{1} \in[k, k+a], x_{2} \in[k+2 a, k+4 a] \text {, }
$$

then $f\left(x_{1}\right) \geqslant f\left(x_{2}\right)$. Find the maximum value of the positive real number $a$.

Answer: \dfrac{2\sqrt{6} - 4}{5}


Four. (20 points) As shown in Figure 3, let $F$ be one of the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, and $A$ be the vertex of the ellipse that is farthest from $F$. Take 2008 distinct points $P_{i}(i=1,2, \cdots, 2008)$ on the minor axis $B C$ of the ellipse. Let $F P_{i}$ intersect $A B$ or $A C$ at point $M_{i}$, and $A P_{i}$ intersect $F B$ or $F C$ at point $N_{i}$. How many pieces do the lines $M_{i} N_{i}(i=1,2, \cdots, 2008)$ divide the ellipse into?

Answer: 2009


3. There is a $10 \times 2$ unit square grid. If two cells in the grid share a common edge, they are called "adjacent". Then, there are $\qquad$ ways to mark nine cells such that no two of them are adjacent.

Answer: 36


11. (1973 USA Mathematical Olympiad) In a given regular $(2n+1)$-sided polygon, three different vertices are chosen at random. If all such choices are equally likely, find the probability that the center of the polygon lies in the interior of the triangle formed by the randomly chosen points.

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


Example 9. Simplify the expression $(\bar{A}+B)(\bar{A}+\bar{B})$.

Answer: \bar{A}


17. As shown in the figure, $\triangle \mathrm{AED}$ is a right-angled triangle, with the lengths of the two legs being $\left\{\begin{array}{l}A E=15 \\ D E=20\end{array}\right.$ . A square $\mathrm{ABCD}$ is constructed with $\mathrm{AD}$ as one side, and a parallelogram $\mathrm{ABFE}$ is constructed with $\mathrm{AB}$ and $\mathrm{AE}$ as sides. $\mathrm{EF}$ intersects $\mathrm{AD}$ at point $\mathrm{G}$, and $\mathrm{AC}$ intersects $\mathrm{FG}$ at point $\mathrm{H}$. The difference in area (larger area minus smaller area) between $\triangle A G H$ and $\triangle C F H$ is $\qquad$ -

Answer: \dfrac{17}{2}


13.5.3 * Given: The hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>b>0)$ has an eccentricity $e=2+\sqrt{6}-\sqrt{3}-\sqrt{2}$, and a line $e$ passing through its right focus $F_{2}$ and perpendicular to the $x$-axis intersects the hyperbola at points $A$ and $B$. Find the value of $\angle A F_{1} F_{2}$.

Answer: \dfrac{\pi}{12}


Question As shown in Figure 1, an ant crawls up from the bottom of the tree along the trunk. At each fork, the ant randomly chooses a path. What is the probability that it crawls to point $D$?

Answer: \dfrac{1}{4}


5. If there exists a complex number $z$ with a modulus of 1, such that
$$
z^{2}+a z+a^{2}-1=0,
$$

then the real number $a$ has $\quad$ possible values.

Answer: 5


With 5 non-zero digits, we can form 120 numbers without repeating any digit in a single number. Let $S$ be the sum of all these numbers. Determine the sum of the digits of $S$, being:

a) 1, 3, 5, 7, and 9 the 5 digits;

b) 0, 2, 4, 6, and 8 the 5 digits, remembering that 02468 is a number with 4 digits and, therefore, we will not have 120 numbers in this case.

Answer: 12


Divide the sequence $\{2n+1\}$ cyclically into one-term, two-term, three-term, four-term groups as follows: $(3), (5,7), (9,11,13), (15,17,19,21), (23), (25,27), (29,31,33), (35,37,39,41), (43), \cdots$. What is the sum of the numbers in the 100th group?

A. 1992

B. 1990

C. 1873

D. 1891

Answer: 1992


Two people, A and B, take turns choosing numbers from the set of integers {1, 2, ..., 17}. The rules are:
1. A number cannot be chosen if it has already been picked by either player.
2. A number cannot be chosen if it is double or half of a number already picked by either player.
3. The player who cannot choose a number loses.

Player A has already chosen the number 8. To ensure a win, which number should player B choose next?

Answer: 6


After watching a movie, viewers rated the film with an integer score from 0 to 10. At each moment in time, the movie rating was calculated as the sum of all given ratings divided by their count. At a certain moment in time \( T \), the rating was an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers that could have voted after moment \( T \)?

Answer: 5


A polygon has a perimeter of $800 \mathrm{~cm}$. What is the side length of a square whose area is equal to the area of the polygon, if each side of the polygon is a tangent to a circle with a radius of $100 \mathrm{~cm}$?

Answer: 200\,


A problem-solving robot with an initial IQ balance of 25 solves a set of questions with point values ranging from $1$ to $10$. Each time it correctly solves a problem, the robot’s IQ balance decreases by the point value of the problem and then increases by 1. If the robot's IQ balance becomes less than the point value of a problem, it fails to solve that problem. What is the maximum score the robot can achieve?

Answer: 31


Find the largest number \( n \) having the following properties:

(a) No two digits of \( n \) are equal.

(b) The number formed by reversing the digits of \( n \) is divisible by 8.

Remark: \( n \) cannot start with 0, but it can end with 0.

Answer: 8697543210


A rectangular strip of length 16 was cut into two strips of lengths 9 and 7. These two strips were placed on a table as shown in the picture.

It is known that the area of the part of the table covered only by the left strip is 27, and the area of the part of the table covered only by the right strip is 18. Find the area of the part of the table covered by both strips.

Answer: 13.5


In the city where the Absent-Minded Scientist lives, phone numbers consist of 7 digits. The Scientist easily remembers a phone number if it is a palindrome, meaning it reads the same forwards and backwards. For example, the number 4435344 is easy for the Scientist to remember because it is a palindrome. However, the number 3723627 is not a palindrome, making it difficult for the Scientist to remember. Find the probability that the phone number of a new random acquaintance is easy for the Scientist to remember.

Answer: 0.001


Mr. Guo has a cake to divide among 4 or 5 children. He cuts the cake into several pieces, each of varying sizes. Regardless of whether there are 4 or 5 children, they can each take some pieces so that everyone gets an equal amount. How many pieces does Mr. Guo need to cut the cake into at least?

Answer: 8


Mark on the line with red all points of the form $81x + 100y$, where $x, y$ are natural numbers, and paint the remaining integer points blue. Find a point on the line such that any symmetrical integer points relative to it are painted in different colors.

Answer: 4140.5


There are 60 people sitting at a large round table; each person is either a knight or a liar. Each person made the statement: "Of the five people sitting consecutively to my right, at least two are liars." How many knights can be sitting at this table?

Answer: 40


In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player had the same number of points. Determine the minimal total number of draws.

Answer: 4


A *form* is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, \ldots, m_n$ satisfy $m_1 \geq \ldots \geq m_n$. An *angle* in a *form* consists of a box $v$ and all the boxes to the right of $v$ and all the boxes above $v$. The size of a *form* of an *angle* is the number of boxes it contains. Find the maximum number of *angles* of size $11$ in a form of size $400$.

Answer: 36


Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9) > f(16)$. The probability that $f(1) > f(16) > f(25)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1, 2, \ldots, 100\}$.

Answer: 730


Siva has the following expression, which is missing operations:
$$\frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32}.$$

For each blank, he flips a fair coin:
- If it comes up heads, he fills it with a plus.
- If it comes up tails, he fills it with a minus.

Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coin flips and operations.

If the probability that the positive difference between his computed values is greater than $\frac{1}{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.

Answer: 39


Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$.

Answer: 1923


We say that an integer $x \in \{1, \dots, 102\}$ is \textit{square-ish} if there exists some integer $n$ such that $x \equiv n^2 + n \pmod{103}$. Compute the product of all \textit{square-ish} integers modulo $103$.

Answer: 52


The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120$. What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?

Answer: 142


Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only condition is that if the union of two different subsets $A$ and $B$ equals $M$, then $A$ and $B$ must be given different colors. What is the least possible value of $k$?

Answer: 2014


Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?

Answer: 5052


Eva draws an equilateral triangle and its altitudes. In the first step, she draws the center triangle of the equilateral triangle. In the second step, she draws the center triangle of this center triangle, and so on. After each step, Eva counts all triangles whose sides lie completely on the drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

Answer: 7


There are $24$ participants who attended a meeting. Each pair of them either shook hands once or did not. A total of $216$ handshakes occurred during the meeting. For any two participants who have shaken hands, at most $10$ among the remaining $22$ participants have shaken hands with exactly one of these two persons. Define a *friend circle* as a group of $3$ participants where each person has shaken hands with the other two. Find the minimum possible number of friend circles.

Answer: 864


A sequence with the first term $a_0$ is defined such that $a_{n+1} = 2a_n^2 - 1$ for $n \geq 0$. Let $N$ denote the number of possible values of $a_0$ such that $a_0 = a_{2020}$. Find the number of factors of $N$.

Answer: 4


A nonempty word is called pronounceable if it alternates between vowels (A, E, I, O, U) and consonants (all other letters) and contains at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.

Answer: 68


Eight 6-sided dice are rolled. What is the probability that exactly four of these dice show the number 3?

Answer: \frac{43750}{1679616}


Nalia needs to travel from point $X$ to $Y$, then from $Y$ to $Z$, and then from $Z$ to $X$. Each of these segments is either by road or rail. The cities form a right-angled triangle with $Z$ located 4000 km from $X$ and $Y$ located 5000 km from $X$. Compute the total distance Nalia travels on her journey.

Answer: 12000


A sphere intersects the xy-plane in a circle centered at $(1,3,0)$ with a radius of 2. The sphere also intersects the yz-plane in a circle centered at $(0,3,-8),$ with radius $r.$ Find $r$ and verify the radius of the sphere.

Answer: 2\sqrt{17}


A hurricane in Melbourne, Australia, caused $\$$45 million in damage. Initially, the exchange rate was 1.8 Australian dollars for 1 British pound. Due to economic changes during the storm, the exchange rate shifted to 1.7 Australian dollars for 1 British pound. Calculate the damage in British pounds considering an average exchange rate for the period.

Answer: 25,714,286


Let $G$ and $H$ denote the centroid and orthocenter of triangle $ABC$, respectively. If $AG = AH$, find all possible values of angle $A$ (in degrees).

Answer: 60^\circ


Alex has 12 different kinds of lunch meat and 11 different kinds of cheese. He can make a sandwich with one kind of meat and up to two kinds of cheese (it does not matter in which order he chooses the cheese). Additionally, there are 3 different kinds of bread he can choose from. How many different sandwiches could Alex make?

Answer: 2412


In $\triangle ABC$, point D is on side BC such that BD = 4 and DC = 14. Calculate the ratio of the area of $\triangle ABD$ to the area of $\triangle ADC$. Additionally, find what fraction of side BC does point D divide the side into.

Answer: \frac{2}{9}


A square has a side length of 8 cm, and each vertex of the square is the center of a circle. Each circle has a radius of 3 cm. Calculate the area of the shaded region formed in the square, assuming that the circles intersect similar to the original problem.

Answer: 64 - 9\pi \text{ cm}^2


David works at a widget factory. On Monday, David produces $w$ widgets every hour for $t$ hours, knowing that $w = 2t$. Feeling more motivated on Tuesday, he works for one fewer hour and increases his production rate by $5$ widgets per hour. Determine how many more widgets David produces on Monday than he does on Tuesday.

Answer: -3t + 5


Jane has received the following scores on her quizzes: 98, 97, 92, 85, 93, 88, and 82. What is her mean score now?

Answer: 90.71


A bridge is constructed by linking a wooden plank between two triangular supports, as shown in the diagram below:

```
[Drawing not to scale]
       A         D
       /\        /\
      /  \      /  \
     /    \    /    \
    B------C  E------F
```
Both triangles are isosceles with equal heights, such that AB = AC and DE = DF. The apex angles of the triangles are $\angle BAC = 25^\circ$ and $\angle EDF = 35^\circ$. Determine the sum $\angle DAC + \angle ADE$.

Answer: 150^\circ


A map of a region is depicted by the Cartesian plane. John is located at $(3, -15)$ and Linda is located at $(-2, 20)$. They decide to meet at the closest point equidistant from both their starting points, and then walk upwards together to the location of their friend Maria at $\left(0.5, 5\right)$. How many units do John and Linda walk upwards together to get to Maria's location?

Answer: 2.5


Define a function \( f(x) \) for \( 0 \leq x \leq 1 \) with the following properties:

(i) \( f(0) = 0 \).
(ii) If \( 0 \leq x < y \leq 1 \), then \( f(x) \leq f(y) \).
(iii) \( f(1 - x) = \frac{3}{4} - \frac{f(x)}{2} \).
(iv) \( f \left( \frac{x}{3} \right) = \frac{f(x)}{3} \) for \( 0 \leq x \leq 1 \).

Determine the value of \( f \left( \frac{2}{9} \right) \).

Answer: \frac{5}{24}


The area of the floor in a rectangular room is 320 square feet. The room is 16 feet long. The homeowners plan to cover the floor with tiles that are each 1 foot by 1 foot. How many tiles will be in each row?

Answer: 20


A region \(S\) in the complex plane is defined by:
\[
S = \{x + iy: -1 \leq x \leq 1, -1 \leq y \leq 1\}.
\]
A complex number \(z = x + iy\) is chosen uniformly at random from \(S\). What is the probability that \((\frac{1}{2} + \frac{1}{2}i)z\) is also in \(S\)?

Answer: 1


In a right-angled triangle $PQR$ with angle $PQR = 90^{\circ}$, suppose $\cos Q = \frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$.

Answer: 12


The Chess Club at a high school has 25 members and needs to select 3 officers: president, treasurer, and secretary. Each member may hold at most one position. Two members, Alice and Bob, will only accept the positions if both are chosen as officers. Another member, Charlie, will only agree to be an officer if Alice is not chosen. In how many ways can the club select its officers?

Answer: 10758


In the diagram, each circle is divided into unequal areas by a horizontal line, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi.$ The line divides the larger circle such that the shaded lower semi-circle is $\frac{2}{3}$ of its total area. The smaller circle’s center is located 2 units above $O$ and just touches the boundary of the larger circle. What is the total area of the shaded regions? [asy]
size(120);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,2),1,0,180)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((0,2),1));

dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,2)--(1,2));
[/asy]

Answer: \frac{296\pi}{3}


12. (10 points) From five number cards $0, 2, 4, 6, 8$, select 3 different cards to form a three-digit number. How many different three-digit numbers can be formed (6 when flipped is 9)?

Answer: 78


1. How many five-digit numbers have the first digit equal to the last, the digit in the thousands place equal to the digit in the tens place, and the product of all digits equal to the square of a natural number?

Answer: 243


16. (15 points) Use two of the numbers $0, 1, 2, 3, 4, 5$ to form a five-digit even number, where one number appears 2 times and the other number appears 3 times. How many five-digit numbers satisfy this condition?

Answer: 118


1st Centromerican 1999 Problem A1 A, B, C, D, E each has a unique piece of news. They make a series of phone calls to each other. In each call, the caller tells the other party all the news he knows, but is not told anything by the other party. What is the minimum number of calls needed for all five people to know all five items of news? What is the minimum for n people?

Answer: 2n-2


1. An army of mice attacked a grain warehouse. To fight the mice, one cat was released into the warehouse on the first day, a second cat on the next day, and each subsequent day, one more cat was released. A mouse eats 100 grams of grain per day, and a cat eats 10 mice per day. After several days, there were no mice left. After calculating the losses, it was found that the mice had eaten 55 kg of grain over these few days. How many mice were there? (A mouse manages to eat 100 grams of grain even on the day it is caught by a cat.)

Answer: 150


3. The 2008 Olympic Games will be held in Beijing. Figure 3 is the Olympic five-ring logo, where $a, b, c, d, e, g, h, i$ are filled with different numbers from $1 \sim 9$. If the sum of the numbers in each circle is equal, then the maximum value of this sum is $\qquad$

Answer: 14


All integers from 1 to 100 are written in a string in an unknown order. With one question about any 50 numbers, you can find out the order of these 50 numbers relative to each other. What is the minimum number of questions needed to definitely find out the order of all 100 numbers?

#

Answer: 5


Example 5 As shown in Figure 6, square $A B C D$ and rectangle $B E F G$ share a common vertex $B$, the side length of square $A B C D$ is $6 \mathrm{~cm}$, and the length of rectangle $B E F G$ is $9 \mathrm{~cm}$. What is the width of the rectangle in $\mathrm{cm}$?

Answer: 4


# Task 14. (1 point)

Calculate the amount of personal income tax (NDFL) paid by Sergey for the past year, if he is a Russian resident and during this period had a stable income of 30000 rub./month and a one-time vacation bonus of 20000 rub. In the past year, Sergey sold a car he inherited two years ago for 250000 rub. and bought a land plot for building a residential house for 300000 rub. Sergey claimed all the tax deductions he was entitled to. (Provide the answer without spaces and units of measurement.)

Answer: 10400


C2. Fix an integer $n \geq 2$. A fairy chess piece leopard may move one cell up, or one cell to the right, or one cell diagonally down-left. A leopard is placed onto some cell of a $3 n \times 3 n$ chequer board. The leopard makes several moves, never visiting a cell twice, and comes back to the starting cell. Determine the largest possible number of moves the leopard could have made.

## Russia, Dmitry Khramtsov

Answer: 9n^2-3


On the board, a three-digit natural number was written. We added to it all other three-digit numbers that can be obtained by changing the order of its digits. Then, in addition to the original number, there were three new ones on the board. The sum of the two smallest of all four numbers is 1088. What digits does the original number contain?

(L. Hozová)

Hint. Determine whether the original number contains zero and whether any digits are repeated.

Answer: 0,5,8


49*. The hostess baked a pie for the guests. At the table, there can be either $p$ people or $q$, where $p$ and $q$ are coprime. Into what minimum number of pieces (not necessarily equal) should the pie be cut in advance so that in any case it can be distributed equally?

Answer: p+q-1


5. A school has 3 teachers who can teach English, 2 teachers who can teach Japanese, and 4 teachers who can teach both English and Japanese. Now, 3 English teachers and 3 Japanese teachers are to be selected to participate in off-campus tutoring during the holiday. How many ways are there to select them?

Answer: 216


6. A bent thin homogeneous rod $ABC$, with small loads $m_{1}=2$ kg and $m_{2}$ located at its ends, is in equilibrium relative to a support placed at point $B$. The mass per unit length of the rod $\lambda=2$ kg. It is known that $AB=7$ m, $BC=5$ m, $BO=4$ m, $OC=3$ m. Find $m_{2}$.

(10 points)

![](https://cdn.mathpix.com/cropped/2024_05_06_f940deb12dfd641283d0g-3.jpg?height=300&width=899&top_left_y=2437&top_left_x=607)

Answer: 10.75


5. (6 points) As shown in the figure, from a rectangular piece of paper that is 50 cm long and 20 cm wide, two squares with side lengths of 12 cm and 4 cm are cut out, respectively. The perimeter of the remaining part of the figure is $\qquad$ cm.

Answer: 172


(7) Given that $C$ and $F$ are two points on line segment $AB$, $AB=12$, $AC=6$, $D$ is any point on the circle with center $A$ and radius $AC$, the perpendicular bisector of line segment $FD$ intersects line $AD$ at point $P$. If the locus of point $P$ is a hyperbola, then the range of the eccentricity of this hyperbola is $\qquad$.

Answer: (1,2]


Student A, B, C, and D participated in a science mini-test. The test only has 5 true or false questions, with 1 point awarded for each correct answer. The table below provides the answers of each student and the scores of A, B, and C. How many points did D get?

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

Answer: 2


21. Prime Minister Turtle ordered the shrimp soldiers and crab soldiers to stand in a solid square formation facing the Dragon Palace. Little shrimp Nini has 4 people directly in front of her, and 3 people directly to her left; Little crab Momo has 3 people directly behind her, and 4 people directly to her right; There are 7 people between Nini and Momo (in a straight line). The entire formation has $\qquad$ people.

Answer: 256


29. There are 4 cards, each with a different number written on it. The doctor first distributes these 4 cards to four children: Damao, Ermao, Sanmao, and Simaoyi, then collects them back and redistributes them. This process is repeated 3 times. If the sum of the numbers on the cards received by Damao over the three rounds is 0, by Ermao is 3, by Sanmao is 11, and by Simaoyi is 16, what is the largest number on the cards? $\qquad$

Answer: 7


3. Provide an example of three different integers, one of which is equal to the difference of the other two, and another is the quotient of the other two.

Answer: 2,-2,-4


1. Konstantin pronounced the names of all natural numbers from 180 to 220 inclusive, while Mikhail - from 191 to 231 inclusive. Who pronounced more words and by how many?

Answer: 1


8.5. On an $8 \times 8$ chessboard, $k$ rooks and $k$ knights are placed such that no figure attacks any other. What is the largest $k$ for which this is possible?

Answer: 5


Example 5 Let $S=\{1,2,3, \cdots, 98,99,100\}$, if a three-element subset $A=\{a, b, c\}$ of $S$ satisfies $a+b=3 c$, then $A$ is said to have property $P$. Find the number of all three-element subsets of $S$ that have property $P$.

Answer: 1600


The $ABCD$ square-shaped field, measuring 19.36 hectares, has a point $E$ on the $AD$ side, 110 meters from $D$, to which a straight path leads from $B$. Antal and Béla are having a race. Antal starts from point $A$, and Béla from point $B$, both starting at the same time and running at a constant speed towards the common goal, point $D$. Béla reaches $D$ via the segments $BE + ED$. When Béla reaches $E$, Antal has a 30 m lead. Who wins the race and by how many meters?

Answer: 8


Problem 6.4. On her birthday, Katya treated her classmates with candies. After giving out some candies, she noticed that she had 10 more candies left than Artem received. After that, she gave everyone one more candy, and it turned out that all the children in the class (including Katya) had the same number of candies. How many classmates does Katya have?

Answer: 9


## Task 3

The pioneers of a Berlin school presented cosmonauts Valentina Tereshkova and Yuri Gagarin with a folder of drawings as a memento. From seven classes, six drawings were each selected.

In addition, the art teacher of the school and a member of the parent council contributed a drawing for the cosmonauts.

How many pictures does the folder contain?

Answer: 44


Shapovalov A.V.

Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them receives nine handfuls, after which the other takes all the remaining coins (the division may also end if the coins are divided before anyone gets nine handfuls). Hapok can grab any number of coins in a handful. What is the maximum number of coins he can guarantee himself regardless of Glazok's actions?

Answer: 46


A bus and a truck start from locations A and B, respectively, at the same time, heading towards each other. The two vehicles meet after 6 hours, and at the meeting point, the bus is $240 \mathrm{~km}$ away from location B. The bus arrives at location B, rests for one hour, and then returns to location A. It is known that the truck takes 15 hours to travel from location B to location A, and after arriving at location A, it also rests for one hour before returning to location B. How many hours have passed from the first meeting to the second meeting?

Answer: 13


The number 81 has the following property: it is divisible by the sum of its digits $8+1=9$. How many two-digit numbers satisfy this property?

Answer: 23


4. There were plates on the shelf. First, from all the plates except two, $1 / 3$ part was taken, and then $1 / 2$ of the remaining plates. After this, 9 plates were left on the shelf. How many plates were on the shelf?

Answer: 26


7. (10 points) The sum of the numbers on opposite faces of each cube is 7. For the three dice placed as shown in the figure, you can only see the numbers on seven faces. What is the sum of the numbers on all the faces that you cannot see from this figure? $\qquad$ .

Answer: 41


A cooperative has 5 sites with the following known distances (the distances are understood to be between the entrances of the sites): From Almás, Bárkány is 2 km away, from there Cseresznye is $1650 \mathrm{~m}$ away. From Cseresznye to Dinnye, the distance is 8 and a half km, and from there to Epres, it is 3 and $3 / 4 \mathrm{~km}$. Finally, from Epres to Almás, it is 1100 m. How far is it from Epres to Bárkány?

Answer: 3100


4. Arrange several balls of two colors, red and black, in a row, requiring that both colors of balls must appear, and any two balls separated by 5 or 10 balls must be of the same color. Arrange according to this requirement, the maximum number of balls that can be placed is.

Answer: 15


4. Perpendiculars AP and AK are dropped from vertex A to the bisectors of the external angles ABC and ACB of triangle ABC. Find the length of segment PK, if the perimeter of triangle ABC is p.

#

Answer: \dfrac{p}{2}


[ Arithmetic of residues (miscellaneous). ]  [ Examples and counterexamples. Constructions ]

What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that they do not attack each other?

Answer: 64


8. Given 8 points $A_{1}, A_{2}, \cdots, A_{8}$ on a circle. Find the smallest positive integer $n$, such that among any $n$ triangles with these 8 points as vertices, there must be two triangles that share a common side.
(Tao Pingsheng, problem contributor)

Answer: 9


380 The four planes in space are mutually parallel, and the distance between each adjacent pair of planes is $h$. A regular tetrahedron has its vertices on the four planes. Find the edge length of the regular tetrahedron.

Answer: h\sqrt{10}


10. (10 points) On September 21, 2015, the Ministry of Education announced the list of national youth campus football characteristic schools. Xiao Xiao's school was on the list. To better prepare for the elementary school football league to be held in the city next year, the school's team will recently play a football friendly match with three other teams. The matches will be played in a round-robin format (i.e., each team will play one match against every other team). The rules are: 3 points for a win, 0 points for a loss, and 1 point for each team in a draw. The ranking will be determined by the total points, and if two teams have the same points, the goal difference, head-to-head results, and other factors will be considered to determine the ranking. To ensure that Xiao Xiao's school team finishes in the top two in this friendly match, they need to score at least $\qquad$ points.

Answer: 7


On a 1-square-meter (1 m²) ceiling in the shape of a square, there is a spider and a stationary fly. The spider can jump to the midpoint of any of the four line segments connecting it to the four vertices of the ceiling within 1 second. Prove that within 8 seconds, the spider can get within 1 centimeter (0.01 meters) of the fly.

Answer: 0.01


Let $n \geqslant 3$ be an integer. For each pair of prime numbers $p$ and $q$ such that $p<q \leqslant n$, Morgane writes the sum $p+q$ on the board. She then notes $\mathcal{P}(n)$ as the product of all these sums. For example, $\mathcal{P}(5)=(2+3) \times(2+5) \times(3+5)=280$.
Find all values of $n \geqslant 3$ for which $n$ ! divides $\mathcal{P}(n)$.
Note: If two sums $p+q$ formed from two different pairs are equal to each other, Morgane writes them both. For example, if $n=13$, she writes both sums $3+13$ and $5+11$.

Answer: 7


11. A. As shown in Figure 4, the parabola $C: y=a x^{2}+b x-3$, with vertex $E$, intersects the $x$-axis at points $A$ and $B$, and the $y$-axis at point $C$. It is given that $O B=O C=3 O A$. The line $y=-\frac{1}{3} x+1$ intersects the $y$-axis at point $D$. Find $\angle D B C-\angle C B E$.

Answer: 45^\circ


Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$.  Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right.  What is the largest possible area of quadrilateral $EFGH$?

[i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

Answer: 98


8. If a positive integer $n$ makes the equation $x^{3}+y^{3}=z^{n}$ have positive integer solutions $(x, y, z)$, then $n$ is called a "good number". Then, the number of good numbers not exceeding 2,019 is $\qquad$ .

Answer: 1346


Determine for how many natural numbers greater than 900 and less than 1001 the digital sum of the digital sum of their digital sum is equal to 1.

(E. Semerádová)

Hint. What is the largest digital sum of numbers from 900 to 1001?

Answer: 12


9. 44 Let $k$ be a natural number. Determine for which value of $k$, $A_{k}=\frac{19^{k}+66^{k}}{k!}$ attains its maximum value.

Answer: 65


Problem 11.7. All vertices of a regular tetrahedron $ABCD$ are on the same side of the plane $\alpha$. It turns out that the projections of the vertices of the tetrahedron onto the plane $\alpha$ form the vertices of a certain square. Find the value of $AB^2$ if it is known that the distances from points $A$ and $B$ to the plane $\alpha$ are 17 and 21, respectively.

![](https://cdn.mathpix.com/cropped/2024_05_06_20f31b6c2b9751326bf1g-6.jpg?height=484&width=438&top_left_y=91&top_left_x=500)

Answer: 32


31. The large garden at the Totoro's house is a parallelogram. As shown in the figure, line segments $A E$ and $B D$ divide the garden into four parts, among which the areas of $\triangle A B F$ and $\triangle A F D$ are 40 and 64, respectively. Then the area of quadrilateral $D F E C$ is $\qquad$

Answer: 79


Task 1. In a confectionery store, three types of candies are sold: caramels for 3 rubles, toffees for 5 rubles, and chocolates for 10 rubles. Varya wanted to buy exactly 8 candies of each type and took 200 rubles with her. In the morning, she saw announcements in the store: “When paying for three chocolates, get a free toffee at the cash register.” and “When paying for three toffees, get a free caramel at the cash register.” How much money will Varya have left?

Answer: 72


117*. The square of a prime number, different from $2,3,5$, when divided by 30, gives a remainder equal to 1 or 19.

Answer: 1


43rd Putnam 1982 Problem B3 Let p n be the probability that two numbers selected independently and randomly from {1, 2, 3, ... , n} have a sum which is a square. Find lim n→∞ p n √n. Solution

Answer: \dfrac{4(\sqrt{2} - 1)}{3}


Problem 3. Find all functions $f(x)$ with integer values and defined in the set of the integers, such that

$$
3 f(x)-2 f(f(x))=x
$$

for all integers $x$.

Answer: f(x) = x


Problem 11.3. In a football tournament, 15 teams participated, each playing against each other exactly once. For a win, 3 points were awarded, for a draw - 1 point, and for a loss - 0 points.

After the tournament ended, it turned out that some 6 teams scored at least $N$ points each. What is the greatest integer value that $N$ can take?

![](https://cdn.mathpix.com/cropped/2024_05_06_86512448fd32765ac040g-42.jpg?height=494&width=460&top_left_y=94&top_left_x=499)

Fig. 13: to the solution of problem 11.2

#

Answer: 34


Two circles touch each other internally at point \( M \). Let \( AB \) be a chord of the larger circle, touching the smaller circle at point \( T \). Prove that \( MT \) is the angle bisector of \(\angle AMB\).

Answer: MT \text{ is the angle bisector of } \angle AMB


[

[ Tangent Circles $]$

Two circles touch each other internally. A line passing through the center of the smaller circle intersects the larger circle at points $A$ and $D$, and the smaller circle at points $B$ and $C$. Find the ratio of the radii of the circles if $A B: B C: C D=2: 4: 3$.

#

Answer: 3


## Task $8 / 80$

We are looking for all natural numbers $n$ with the following properties:

1. It is $n=p_{1} \cdot p_{2}$; the product of two (proper) two-digit prime numbers $p_{1}$ and $p_{2}$.
2. For the cross sum $Q(n)$, it holds that $Q(n)=p_{1}$ with $p_{1}<p_{2}$.
3. The units digits of $p_{1}$ and $p_{2}$ are the same.
4. Also $p_{1} \pm 6$ is a (proper) two-digit prime number.

Answer: 629


(15) A $0, 1$ matrix with 20 rows and several columns satisfies: all columns are distinct and the number of rows where any two columns both have a 1 is at most 2. Find the minimum number of 1's in the matrix when the number of columns is maximized.

Answer: 3820


11. Let real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy
$$
\left|x_{1}\right|=99,\left|x_{n}\right|=\left|x_{n-1}+1\right| \text {, }
$$

where, $n=2,3, \cdots, 2014$. Find the minimum value of $x_{1}+x_{2}+\cdots+x_{2014}$.

Answer: -5907


Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.

Find the greatest possible value of $S$.

Answer: \frac{11}{2}


5. Given that $A, B, C$ are three different points on the circle $O$ (with $O$ as the origin), and $\angle A O B=\frac{2 \pi}{3}$, if $\overrightarrow{O C}=$ $\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}(\lambda, \mu \in \mathbf{R})$,
then when $\omega=\sqrt{3} \lambda+\lambda+\mu$ takes the maximum value, $\frac{\lambda}{\mu}=$ $\qquad$ .

Answer: \dfrac{1 + \sqrt{3}}{2}


2. Inside the tetrahedron $ABCD$ there is a point $O$, satisfying $OA=OB=OC=4, OD=1$.
Find the maximum volume of the tetrahedron $ABCD$.

Answer: 9\sqrt{3}


Point $D$ is the midpoint of side $A C$ of triangle $A B C$. On side $B C$, a point $E$ is chosen such that $\angle B E A = \angle C E D$. Find the ratio $A E: D E$.

Answer: 2


Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$.
For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.

Answer: 65


11.5. A natural number $n$ is called good if each of its natural divisors, increased by 1, is a divisor of the number $n+1$. Find all good natural numbers.

(S. Berlov)

Answer: 1


13.1.1 $\star$ In the first quadrant of the Cartesian coordinate system, number the points (lattice points) with integer coordinates in the following manner: (1) $(0,0)$, (2) $(1,0)$, (3) $(1,1)$, (4) $(0,1)$, (5) $(0,2)$, (6) $(1,2)$, (7) $(2,2)$, (8) $(2,1)$, (9) $(2,0), \cdots \cdots$ (as shown in the order of the arrows in the figure). Find the coordinates of the 2000th point.

Answer: (44, 25)


2. Let A, B be sets of positive integers such that the sum of any two different numbers from $A$ belongs to $B$ and the quotient of any two different numbers from $B$ (the larger divided by the smaller) lies in $A$. Determine the maximum possible number of elements in the set $A \cup B$.

Answer: 5


Ggrmanova T.A.

Tanya sequentially wrote down numbers of the form $\{n \wedge 7-1\}$ for natural numbers $n=2,3, \dots$ and noticed that for $n=8$ the resulting number is divisible by 337. For what smallest $n>1$ will she get a number divisible by $2022?$

Answer: 79


Task 2. A poultry farm delivered 720 eggs, which were packed in boxes of 6 eggs each. During transportation, 140 eggs were broken. The remaining eggs were packed in boxes of 12 eggs each. How many boxes of eggs were packed. Were there any unbroken eggs left that could not be packed?

Answer: 48


265 It is known that for all positive integers $n$,
$$
\prod_{i=1}^{n}\left(1+\frac{1}{3 i-1}\right) \geqslant \frac{k}{2} \sqrt[3]{19 n+8}
$$

always holds. Try to find the maximum value of $k$.

Answer: 1


Let $x_1,\ldots,x_n$ be a sequence with each term equal to $0$ or $1$. Form a triangle as follows: its first row is $x_1,\ldots,x_n$ and if a row if $a_1, a_2, \ldots, a_m$, then the next row is $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$ where the addition is performed modulo $2$ (so $1+1=0$). For example, starting from $1$, $0$, $1$, $0$, the second row is $1$, $1$, $1$, the third one is $0$, $0$ and the fourth one is $0$.

A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is $1010$ and the corresponding sequence from last terms is $0010$ and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

Answer: 2^{\lceil n/2 \rceil}


2nd Australian 1981 Problem B3 In an archery contest, the two contestants A and B start at x = 0 and walk towards the target at x = 1 together. Each chooses the distance from which he shoots. A has probability x 2 of hitting the target from x. B has probability x of hitting the target from x. If both hit the target, then the first to shoot wins. What is A's best strategy?

Answer: \dfrac{\sqrt{5} - 1}{2}


Answer: 2012


12.235. The angle between a lateral edge of a regular quadrilateral pyramid and the plane of the base is equal to the magnitude of the plane angle at the vertex of the pyramid. Find the angle between a lateral face and the plane of the base.

Answer: \arctan \sqrt{1 + \sqrt{5}}


5. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{0}=\sqrt{6}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}} \text {, }
$$

where, $[a]$ denotes the greatest integer not exceeding the real number $a$, and $\{a\}=a-[a]$. Then $a_{2020}=$ $\qquad$ .

Answer: 6060 + \sqrt{6}


15. (15 points) In a certain year, there are 53 Fridays and 53 Saturdays. What day of the week was March 1st in that year?

Answer: Tuesday


Given the parabola \( y^{2} = 2px \) and fixed points \( A(a, b) \) and \( B(-a, 0) \) with the conditions \( ab \neq 0 \) and \( b^{2} \neq 2pa \). Let \( M \) be a point on the parabola. Suppose \( A M \) and \( B M \) intersect the parabola again at points \( M_{1} \) and \( M_{2} \), respectively.

Prove: When the point \( M \) moves along the parabola (as long as \( M_{1} \) and \( M_{2} \) exist and \( M_{1} \neq M_{2} \)), the line \( M_{1} M_{2} \) always passes through a fixed point. Also, determine the coordinates of this fixed point.

Answer: \left( a, \dfrac{2pa}{b} \right)


5. (7 points) Lёsha did not hesitate to calculate the sum

$$
9+99+999+\ldots+\underbrace{9 \ldots 9}_{2017}
$$

and wrote it on the board. How many times is the digit 1 written in the final result?

Answer: 2013


Problem 10.8. Rectangle $ABCD$ is such that $AD = 2AB$. Point $M$ is the midpoint of side $AD$. Inside the rectangle, there is a point $K$ such that $\angle AMK = 80^{\circ}$ and ray $KD$ is the bisector of angle $MKC$. How many degrees does angle $KDA$ measure?

![](https://cdn.mathpix.com/cropped/2024_05_06_c49ecf27446af41a8bcdg-43.jpg?height=291&width=495&top_left_y=595&top_left_x=479)

Answer: 35


10. As shown in Figure 4, $AB$ is the diameter of $\odot O_{1}$, $AO_{1}$ is the diameter of $\odot O_{2}$, chord $MN \parallel AB$, and $MN$ is tangent to $\odot O_{2}$ at point $C$. If the radius of $\odot O_{1}$ is 2, then the area of the figure enclosed by $O_{1}B$, $\overparen{BN}$, $NC$, and $\overparen{CO_{1}}$ is equal to

Answer: 1 + \dfrac{\sqrt{3}}{2} + \dfrac{\pi}{12}


One. (20 points) Given that $a$, $b$, and $c$ are three distinct real numbers, and the three quadratic equations
\[
\begin{array}{l}
x^{2} + a x + b = 0, \\
x^{2} + b x + c = 0, \\
x^{2} + c x + a = 0
\end{array}
\]

each pair of which has exactly one common root. Find the value of $a^{2} + b^{2} + c^{2}$.

Answer: 6


A tournament will take place with 100 competitors, all with different skill levels. The most skilled competitor always wins against the least skilled competitor. Each participant plays exactly twice, with two randomly drawn opponents (once against each). A competitor who wins two matches receives a medal. Determine the minimum number of medals that can be distributed in the tournament.

Answer: 1


João is going to paint figures composed of squares and triangles. Each square can be painted blue, red, or green, and each triangle can be painted blue, red, or yellow, in such a way that polygons with a common side do not share the same color. For each of the following items, determine how many ways João can paint the corresponding figure.

#### a)

#### b)

#### c)

Answer: 7


7. Solve the following problem using as many methods as possible. Methods are considered different if they use different mathematical ideas, as well as various technical approaches to implementing the same idea. Indicate the place of each method used in the school mathematics curriculum.

Compare the numbers $2+\log _{2} 6$ and $2 \sqrt{5}$.

## Sixth Olympiad (2012). I (correspondence) round

Answer: 2 + \log_{2} 6 > 2\sqrt{5}


2. In $\square A B C D$, $A B<A C<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. If segment $A D$ intersects $C E$, and $\angle A B F=\angle D C E$, find $\angle A B C$.

Answer: 60^\circ


Task 1. The arc above the base of an isosceles triangle is $\frac{2}{5}$ of the circumscribed circle of that triangle. Find the angles of that triangle.

Answer: 72^\circ


4. There are 2015 coins on the table. Two players play the following game: they take turns; on a turn, the first player can take any odd number of coins from 1 to 99, and the second player can take any even number of coins from 2 to 100. The player who cannot make a move loses. How many coins should the first player take on the first move to then guarantee a win with an unchanging strategy? (20 points)

#

Answer: 95


4. Let $l(n)$ denote the the greatest odd divisor of any natural number $n$. Find the sum

$$
l(1)+l(2)+l(3)+\cdots+l\left(2^{2013}\right) .
$$

(Michal Rolínek)

Answer: \dfrac{4^{2013} + 2}{3}


24. The blackboard has 11 39 written on it. Each time, Teacher Li erases 2 numbers and writes down their difference (the larger number minus the smaller number) on the blackboard. When only one number remains on the blackboard, the smallest possible number is $\qquad$.

Answer: 1


In the right triangle $ABC$, the hypotenuse $BC = a$ is divided into $n$ equal segments, where $n$ is any odd natural number. Let $h$ denote the height corresponding to the hypotenuse, and $\alpha$ the angle at which the segment containing the midpoint of the hypotenuse appears from vertex $A$. Prove that

$$
\operatorname{tg} \alpha = \frac{4nh}{(n^2 - 1)a}
$$

Answer: \operatorname{tg} \alpha = \dfrac{4nh}{(n^2 - 1)a}


Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the large circle. Calculate the area of quadrilateral $ABCD$.

Answer: 24


A rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of the existing surfaces is cut by a straight line, resulting in two new areas. What is the minimum number of cuts needed such that among the obtained polygons, there are $251$ polygons with $11$ sides?

Answer: 2007


Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even.

Initially, there are $2021$ integers on the board, each one sampled uniformly at random from the set $\{0, 1, 2, 3, \ldots, 2021\}$. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\gcd(m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.

Answer: 3


Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.

Answer: 70


On triangle $ABC$, let $D$ be the point on $AB$ such that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ such that $AE$ bisects angle $BAC$. Let $G$ be the intersection of $AE$ and $CD$, and let point $F$ be the intersection of side $AC$ and the ray $BG$. If $AB$ has length $28$, $AC$ has length $14$, and $CD$ has length $10$, then the length of $CF$ can be written as $\frac{k-m\sqrt{p}}{n}$ where $k$, $m$, $n$, and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $k - m + n + p$.

Answer: 623


Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
- She rolled a $6$ exactly three times.
- The product of her first three rolls is the same as the product of her last three rolls.

How many distinct sequences of six rolls could Kara have rolled?

Answer: 72


The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers.

Answer: 20


Each of the 2009 distinct points in the plane is colored either blue or red. On every unit circle centered at a blue point, there are exactly two red points. Find the greatest possible number of blue points.

Answer: 1964


We say that a positive integer $m$ covers the number $1998$ if the digits $1, 9, 9, 8$ appear in this order as digits of $m$. For instance, $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.

Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n \ge 5$), with all digits different from $0$. What is the remainder of $k(n)$ when divided by $8$?

Answer: 1


There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

Answer: 2017


Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.

Answer: 167


$\text{HOW}$, $\text{BOW}$, and $\text{DAH}$ are equilateral triangles in a plane such that $WO = 7$ and $AH = 2$. Given that $D$, $A$, $B$ are collinear in that order, find the length of $BA$.

Answer: 11


Positive numbers are written in the squares of a $10 \times 10$ table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to $10^2$. Then the frogs jumped again, and the sum changed to $10^3$ and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?

Answer: 1000000


12. (15 points) Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ and an inscribed parallelogram with one pair of opposite sides passing through the foci $F_{1}$ and $F_{2}$ of the ellipse. Find the maximum area of the parallelogram.

Answer: 6


We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?

Answer: 859


Answer: 2


It is well known that the value of the polynomial $(x+1)(x +2) \dots (x + n)$ is exactly divisible by n for every integer x. Given n, let $k=k(n)$ be the minimal degree of any monic integral polynomial.

$$f(x)=x^k+a_1x^{k-1}+\dots +a_k$$

(with integer coefficients and leading coefficient 1) such that the value of f(x) is exactly divisible by n for every integer x.

Find the relationship between n and $k=k(n)$. In particular, find the value of k corresponding to n=1 000 000.

Answer: 25


Two distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is called a C-gap if for every numbering of the squares of C with all the integers $1,2,\ldots,64$ there exist two adjacent squares whose numbers differ by at least g. Determine the largest C-gap g.

Answer: 9


Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A{ }^{{ }^{1}} B^{n_{2}} A^{n_{3}} \cdots B^{n_{k}}$, where $n_{1}, \ldots, n_{k}$ are any integers, and $A^{\prime \prime}=B^{\prime \prime}=1$ as usual. Also, suppose that $A^{4}=B^{7}=A B A^{-1} B=1, A^{2} \neq 1$, and $B \neq 1$.

(a) How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?

(b) Write each such square as a word in $A$ and $B$.

Answer: 8


Two distinct squares of the 8 by 8 chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1,2, \ldots, 64$ there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.

Answer: 9


A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are 12 possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$ ?

Answer: 578


Let $a_0=1, a_1=2$, and $a_n=4 a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.

Answer: 181


Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
$$
f\left(\frac{1}{n}\right)=\frac{n^{2}}{n^{2}+1}, \quad n=1,2,3, \ldots
$$
compute the values of the derivatives $f^{(k)}(0), k=1,2,3, \ldots$.

Answer: 0


Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^{3}-x$. Let

$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)}
$$

for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.

Answer: 3984


We say a $2023$-tuple of nonnegative integers $\left(a_{1}, a_{2}, \ldots a_{2023}\right)$ is sweet if the following conditions hold:
- $a_{1}+a_{2}+\ldots+a_{2023}=2023$,
- $\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leqslant 1$.

Determine the greatest positive integer $L$ such that
$$
a_{1}+2 a_{2}+\ldots+2023 a_{2023} \geqslant L
$$
holds for every sweet 2023 -tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$.

Answer: 22228


Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?

Answer: 6


Color the surfaces of a cube of dimension $5\times 5\times 5$ red, and then cut the cube into smaller cubes of dimension $1\times 1\times 1$. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?

Answer: 96


There are 62 boxes in total, with small boxes being 5 units per ton and large boxes being 3 units per ton. A truck needs to transport these boxes. If the large boxes are loaded first, after loading all the large boxes, exactly 15 small boxes can still be loaded. If the small boxes are loaded first, after loading all the small boxes, exactly 15 large boxes can still be loaded. How many large boxes are there among these boxes in total?

Answer: 27


4. In an isosceles triangle with an angle of $120^{\circ}$, the inscribed circle has a radius of $5 \mathrm{~cm}$. Determine the sides of the triangle.

Answer: \dfrac{10}{3}(3 + 2\sqrt{3})


15. For the function $f(x)$, if $f(x)=x$, then $x$ is called a "fixed point" of $f(x)$. If $f(f(x))=x$, then $x$ is called a "stable point" of $f(x)$; the sets of "fixed points" and "stable points" of the function $f(x)$ are denoted as $A$ and $B$, respectively, i.e., $A=\{x \mid f(x)=x\}, B=\{x \mid f(f(x))=x\}$.
(1) Prove that $A \subseteq B$;
(2) If $f(x)=a x^{2}-1(a \in \mathbf{R}, x \in \mathbf{R})$, and $A=B \neq \varnothing$, find the range of real number $a$.

Answer: [-\frac{1}{4}, \frac{3}{4}]


10. (6 points) Using 24 matchsticks of the same length, first use some of them to form 6 triangles on a plane, with each side of the equilateral triangle being one matchstick. Then use the remaining matchsticks to form squares on the plane, with each side of the square being one matchstick. How many such squares can be formed at most? $\qquad$

Answer: 4


a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be?
b)  Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?

Answer: -24.15


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, and 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


Five, color the numbers in $S=\{0,1,2, \cdots, n\}$ with two colors arbitrarily. Find the smallest positive integer $n$, such that there must exist $x, y, z \in S$ of the same color, satisfying $x+y=2 z$.

Answer: 8


7.4. A number is called good if any two adjacent digits in its notation differ by at least 5. Vera wrote a good number, and then replaced identical digits with identical letters, and different digits with different letters. Could she have ended up with the word NOVOSIBIRSK?

Answer: \text{No}


3. (10 points) $A, B, C, D$ are assigned a permutation of the numbers $10, 20, 30, 40$, such that the value of the expression $\frac{1}{\mathrm{~A}-\frac{1}{B+\frac{1}{C-\frac{1}{D}}}}$ is maximized. Then the value of $A+2B+3C+4D$ is $\qquad$

Assign $A, B, C, D$ using a permutation of the numbers $10, 20, 30, 40$ to maximize the value of the expression $\frac{1}{\mathrm{~A}-\frac{1}{B+\frac{1}{C-\frac{1}{D}}}}$. Then, the value of $A+2B+3C+4D$ is $\qquad$

Answer: 290


$[$ The Pigeonhole Principle (continued) ]
$[\quad$ Estimation + example $\quad]$

The hostess baked a pie for her guests. She may have either 10 or 11 guests. Into what minimum number of pieces should she cut the pie in advance so that it can be evenly divided among either 10 or 11 guests?

Answer: 20


14. Teacher $D$ placed cards with the numbers $1, 2, \cdots, 13$ in a clockwise order on a circle. Initially, all cards are face up. Each time, one card can be flipped to face down (once a card is face down, it cannot be flipped again). Teacher $D$'s rule for flipping cards is: if a face-up card has the number $A$, and the card that is 2 cards away is also face up, then Teacher $D$ can flip the card with the number $A$. For example: as long as the cards with the numbers 9 or 2 are face up, Teacher $D$ can flip the card with the number 12 (of course, provided that the card with the number 12 is still face up). In the end, as long as Teacher $D$ flips 12 cards to face down, it is considered a success. How many different sequences of flipping can Teacher $D$ use to achieve success?

Answer: 26624


3. (7 points) From the airplane window, I can see part of an island, part of a cloud, and a bit of the sea. The cloud occupies half of the landscape visible from the window and thus hides a quarter of the island, which therefore takes up only a quarter of the observed landscape. What part of the landscape does the part of the sea hidden by the cloud constitute?

#

Answer: \dfrac{5}{12}


On a clearing, ladybugs gathered. If a ladybug has six spots on its back, it always tells the truth, and if it has four spots, it always lies, and there were no other ladybugs on the clearing. The first ladybug said: "Each of us has the same number of spots on our backs." The second said: "All of us together have 30 spots on our backs." - "No, all of us together have 26 spots on our backs," - the third one objected. "Exactly one of these three told the truth," - each of the other ladybugs declared. How many ladybugs gathered on the clearing?

Answer: 5


A $10 \times 10$ square is divided into unit squares. How many triangles are formed after drawing one diagonal?

Answer: 110


## Task 5 - 280835

Let $ABC$ be a triangle, $\alpha$ be the measure of the angle $\angle BAC$, and $\beta$ be the measure of the angle $\angle ABC$. The incircle of the triangle touches the side $AB$ at $D$, the side $BC$ at $E$, and the side $AC$ at $F$.

Determine the measure of the angle $\angle FDE$ in terms of $\alpha$ and $\beta$!

Hint: The incircle of a triangle is the circle that touches all three sides of the triangle from the inside.

Answer: \dfrac{\alpha + \beta}{2}


3A. The numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ are consecutive terms of an arithmetic progression, and the numbers $\cos \alpha_{1}, \cos \alpha_{2}, \ldots, \cos \alpha_{n}$ are also consecutive terms of an arithmetic progression. Determine $n$, if $\cos \alpha_{1}=\frac{1}{2}$, and $\cos \alpha_{n}=-\frac{1}{2}$.

Answer: 3


$\underline{\text { Khachaturyan A.V. }}$

13 children sat at a round table and agreed that boys would lie to girls, but tell the truth to each other, and girls, on the contrary, would lie to boys, but tell the truth to each other. One of the children said to their right neighbor: "The majority of us are boys." That child said to their right neighbor: "The majority of us are girls," and that one to their right neighbor: "The majority of us are boys," and so on, until the last child said to the first: "The majority of us are boys." How many boys were at the table?

Answer: 7


Which of the fractions have a value of $\frac{45}{56}$ and whose numerators are square numbers, and denominators are cube numbers? Among these fractions, which one has the simplest form?

Answer: \frac{525^2}{70^3}


4. Determine the minimum value of the quantity $|x+y|$ given that the numbers $x$ and $y$ satisfy the relation $5 \cos (x+4 y)-3 \cos (x-4 y)-4 \sin (x-4 y)=10$.

Answer: \dfrac{\pi - 3\arctan\left(\dfrac{4}{3}\right)}{8}


The set $\{1,2, \ldots, 100\}$ has a subset $H$ with the property that the tenfold of any element in $H$ is not in $H$. What is the maximum number of elements $H$ can have?

Answer: 91


As shown in Figure 1, a solid metal sphere with a radius of 20 has a circular hole with a radius of 12 drilled through it along one of its diameters. The remaining part is then melted down and cast into a new solid sphere. What is the radius of the new sphere?

Answer: 16


27. Determine the type of section of a cube by a plane passing through its center and perpendicular to its diagonal.

Answer: regular hexagon


The carillon on the courtyard plays a short piece every hour, starting at 8 and ending at 22. There are a total of eighteen pieces, and only one is played at each hour, and after all eighteen have been played, the cycle starts again in the same order. Olga and Libor were in the courtyard on Monday at 3 PM. Later that week, they came to listen to the carillon again at noon, to their disappointment, however, the same melody played that they heard on Monday.

On which day did Olga and Libor visit the courtyard for the second time?

(L. Simünek)

Hint. How many of the eighteen pieces will sound in one day and how many will not get a chance to play?

Answer: Saturday


Problem 1. Consider the set $A=\left\{1,2,3,4, \ldots, 2^{n}\right\}, n \geq 2$. Find the number of the subsets $B$ of $A$, such that if the sum of two elements of $A$ is a power of 2 then exactly one of them belongs to $B$.

Aleksandar Ivanov

Answer: 2^{n+1}


8. A small bug crawls on the surface of a regular octahedron,

each second it moves from one vertex to one of the four adjacent vertices with equal probability. Then the probability that the bug returns to the initial position at the 8th second is $\qquad$ .

Answer: \dfrac{43}{256}


Problem 2. On a shelf, there are three books. The first has 90, the second 110, and the third 150 pages. The covers of the books are of equal thickness, and each of them is 2 mm thick. How many millimeters thick are the books together if it is known that 10 pages have a thickness of $1 \mathrm{~mm}$?

Answer: 47


3. Ivan Semenovich leaves for work at the same time every day, drives at the same speed, and arrives exactly at 9:00. One day he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich drove at a speed 60% faster than usual and arrived at 8:35. By what percentage should he have increased his usual speed to arrive exactly at 9:00?

Answer: 30\%


8,9
[ Equilateral (regular) triangle ]

A line is drawn through vertex $C$ of the square $A B C D$, intersecting the diagonal $B D$ at point $K$, and the perpendicular bisector of side $A B$ at point $M$ ( $M$ is between $C$ and $K$ ). Find $\angle D C K$, if $\angle A K B=\angle A M B$.

Answer: 15^\circ


1. Karlson filled a conical glass with lemonade and drank half of it by height (measuring from the surface of the liquid to the apex of the cone), and Little Man drank the second half. How many times more lemonade did Karlson drink compared to Little Man?

Answer: 7


1. Several boys and girls gathered around a round table. It is known that exactly for 7 girls, the person sitting to their left is a girl, and for 12 - a boy. It is also known that for $75\%$ of the boys, the person sitting to their right is a girl. How many people are sitting at the table?

Answer: 35


145. Three friends had a monkey. They bought a certain number of mango fruits and hid them. At night, one of the friends wanted to indulge, went to the storeroom, and wanted to take a third of the fruits, but it turned out that one fruit was left over, and he gave it to the monkey, and ate his share. After some time, the second one came and, not suspecting that the first one had already taken his share, divided the remainder into three equal parts, and again one fruit was left over, and he gave it to the monkey and ate his share. Finally, the third one came and also divided the remainder into three equal parts, and again one fruit was left over, which he gave to the monkey, and ate his share.

The next morning, all three came to the storeroom and divided the remaining fruits equally among themselves, and again one fruit was left over, which they gave to the monkey.

Determine the smallest number of fruits that satisfies the above conditions of distribution.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-029.jpg?height=268&width=394&top_left_y=1491&top_left_x=837)

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-030.jpg?height=297&width=1307&top_left_y=248&top_left_x=383)

## Problem from the tractate "Bava Batra".

Talmud.

Answer: 79


## Problem A1

The numbers $1,2, \ldots, 2002$ are written in order on a blackboard. Then the 1 st, 4 th, 7 th, $\ldots$, $3 \mathrm{k}+1$ th, $\ldots$ numbers in the list are erased. Then the 1 st, $4 \mathrm{th}, 7 \mathrm{th}, \ldots 3 \mathrm{k}+1$ th numbers in the remaining list are erased (leaving $3,5,8,9,12, \ldots$ ). This process is carried out repeatedly until there are no numbers left. What is the last number to be erased?

Answer: 1598


On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.

(i) One may give cookies only to people adjacent to himself.
(ii) In order to give a cookie to one's neighbor, one must eat a cookie.

Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.

Answer: 2^n


One face of a tetrahedron has sides of length $3$, $4$, and $5$. The tetrahedron’s volume is $24$ and surface area is $n$. When $n$ is minimized, it can be expressed in the form $n = a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers and b is not divisible by the square of any prime. Evaluate $a + b + c$.

Answer: 157


Mitya is going to fold a square sheet of paper $A B C D$. Mitya calls a fold beautiful if the side $A B$ intersects the side $C D$ and the four resulting right triangles are equal.

Before that, Vanya selects a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold that passes through point $F$.

Answer: \frac{1}{2}


A firework was thrown vertically upward with a speed of $20 \, \mathrm{m/s}$. One second after the start of the flight, it exploded into two unequal parts, with their masses in the ratio of $1:2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \, \mathrm{m/s}$. Find the magnitude of the speed of the second fragment (in $\mathrm{m/s}$) immediately after the explosion. Assume the acceleration due to gravity is $10 \, \mathrm{m/s}^2$.

Answer: 17\,


Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye Moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, with 100 grams of each type?

Answer: 45


A supermarket purchases 500 kg of sugar oranges at a cost price of 4.80 yuan per kilogram, with an expected weight loss of 10%. If the supermarket intends to achieve a profit of 20% on the sale of these sugar oranges, what should be the retail price per kilogram?

Answer: 6.4\,


In a table tennis singles tournament, it was initially planned that each pair of players would play one match. However, 3 players each played 2 matches and then withdrew from the tournament. As a result, 50 matches were played in total. How many matches were played among these 3 players?

Answer: 1


On side \( AC \) of triangle \( ABC \), a point \( M \) is taken. It turns out that \( AM = BM + MC \) and \(\angle BMA = \angle MBC + \angle BAC \). Find \(\angle BMA\).

Answer: 60


Cube \(ABCD EFGH\) has an edge length of 100. Point \(P\) is on \(AB\), point \(Q\) is on \(AD\), and point \(R\) is on \(AF\) such that \(AP = x\), \(AQ = x + 1\), and \(AR = \frac{x + 1}{2x}\) for some integer \(x\). For how many integers \(x\) is the volume of the triangular-based pyramid \(APQR\) between 0.04% and 0.08% of the volume of cube \(ABCDEFGH\)? (The volume of a pyramid is equal to one-third of the area of its base times its height.)

Answer: 28


A $3 \times 3$ grid of 9 dots labeled $A, B, C, D, E, F, K, L$, and $M$ is shown in the figure. There is one path connecting every pair of adjacent dots, either orthogonal (horizontal or vertical) or diagonal. A turtle walks on this grid, alternating between orthogonal and diagonal moves. Any sequence of paths can be described in terms of the letters $A$ to $M$. For example, $A-B-F$ describes a sequence of two paths: $A$ to $B$ and $B$ to $F$.

What is the maximum number of paths the turtle could traverse, given that it does not traverse any path more than once?

Answer: 17


Suppose we have an infinite cone \(\mathcal{C}\) with apex \(A\) and a plane \(\pi\). The intersection of \(\pi\) and \(\mathcal{C}\) is an ellipse \(\mathcal{E}\) with major axis \(BC\), such that \(B\) is closer to \(A\) than \(C\), and \(BC = 4\), \(AC = 5\), \(AB = 3\). Suppose we inscribe a sphere in each part of \(\mathcal{C}\) cut by \(\mathcal{E}\) with both spheres tangent to \(\mathcal{E}\). What is the ratio of the radii of the spheres (smaller to larger)?

Answer: \frac{1}{3}


a) Prove that a bounded figure cannot have more than one center of symmetry.

b) Prove that no figure can have exactly two centers of symmetry.

c) Let $M$ be a finite set of points on a plane. A point $O$ is called an "almost center of symmetry" of the set $M$ if one point can be removed from $M$ such that $O$ becomes the center of symmetry of the remaining set. How many "almost centers of symmetry" can $M$ have?

Answer: 3


Long ago, nine identical books cost 11 rubles and some kopecks, and thirteen such books cost 15 rubles and some kopecks. How much did one book cost?

Answer: 1


There are 2011 numbers written on a board. It turns out that the sum of any three of these numbers is also a number written on the board. What is the minimum number of zeros that can be among these numbers?

Answer: 2009


A group of 101 Dalmatians participate in an election, where they each vote independently for either candidate \( A \) or \( B \) with equal probability. If \( X \) Dalmatians voted for the winning candidate, the expected value of \( X^{2} \) can be expressed as \( \frac{a}{b} \) for positive integers \( a, b \) with \( \operatorname{gcd}(a, b)=1 \). Find the unique positive integer \( k \leq 103 \) such that \( 103 \mid a - b k \).

Answer: 51


On a $7 \times 7$ grid, 14 cells are marked. We will call a pair of cells with a common side interesting if at least one cell from the pair is marked. What is the maximum number of interesting pairs possible?

Answer: 55


7.5. At each vertex of a cube lives a number, not necessarily positive. All eight numbers are distinct. If a number is equal to the sum of the three numbers living in the adjacent vertices, then it is happy. What is the maximum number of happy numbers that can live at the vertices of the cube?

Answer: 8


5. Write $(\sqrt{7}-\sqrt{6})^{6}$ in the form $\sqrt{N+1}-\sqrt{N}$, where $N$ is a natural number. Then $N=$ $\qquad$

Answer: 76545000


There are $n$ cards such that for each $i=1,2, \cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations:

[list]
[*] $A$ : One can take the top card of the pile and move it to the bottom;

[*] $B$ : One can remove the top card from the pile.
[/list]
The operation $ABBABBABBABB \cdots $ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$.

Answer: 1


3. An Indian engineer thought of a natural number, listed all its proper natural divisors, and then increased each of the listed numbers by 1. It turned out that the new numbers are all the proper divisors of some other natural number. What number could the engineer have thought of? Provide all options and prove that there are no others. Recall that a divisor is called proper if it is not equal to 1 and does not coincide with the number being divided.

Answer: 4


7. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$ there exist 5 edge-disjoint monochromatic triangles.

Answer: 11


Let $k$ be the perimeter of the convex quadrilateral $ABCD$, and $d$ be the sum of the lengths of its two diagonals. Let $A_{1}, B_{1}, C_{1}, D_{1}$ be the points that divide the sides $AB$, $BC$, $CD$, and $DA$ respectively in the ratio $(1-\lambda): \lambda$, where $1/2 \leq \lambda \leq 1$. Denote by $k_{\lambda}$ the perimeter of the convex quadrilateral $A_{1}B_{1}C_{1}D_{1}$. Prove that

\[ |2 \lambda d - (2 \lambda - 1) k| \leq k_{\lambda} \leq (2 \lambda - 1) k + 2(1 - \lambda) d. \]

Answer: |2 \lambda d - (2 \lambda - 1) k| \leq k_{\lambda} \leq (2 \lambda - 1) k + 2(1 - \lambda) d