During the FIFA World Cup, several mathematicians were called upon to speak about the teams' chances of qualification. In the group stage, each group is formed by four teams, and each team faces each of the other teams exactly once. In case of a win, a team earns 3 points; in case of a draw, 1 point; and in case of a loss, 0 points. It is known that the top two teams qualify for the next round. If two teams tie with the same number of points, the tiebreaker is the goal difference. What is the minimum number of points for a team to qualify without depending on the results of other matches?

Note: Remember that to show that the number $k$ found is indeed the minimum, in addition to showing that such a quantity is sufficient to guarantee victory, you must also ensure that there are examples of scores where teams can total no more than $k-1$ points and not advance to the next round.

Answer: 7


Example 4. Fold a square $ABCD$ with side length 3, with the crease being $EF$ (as shown in the figure), so that point $B$ lands on point $B'$ on $CD$, the extensions of $BA$ and $FE$ intersect at point $A'$, and $\angle B'BC=30^{\circ}$. Try to calculate the area of $\triangle A'EG$.

Answer: 7\sqrt{3} -12


Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$.
Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and
$A+B=\{0,1,2,\dots, 2019,2020\}$

Answer: 32


Onboard an airliner, there are $2n$ passengers, and the airline has loaded $n$ chicken meals and $n$ fish meals for them. It is known that each passenger prefers chicken with probability 0.5 and fish with probability 0.5. A passenger is considered dissatisfied if they are not left with their preferred meal.

a) Find the most probable number of dissatisfied passengers.

b) Find the expected number of dissatisfied passengers.

c) Find the variance of the number of dissatisfied passengers.

Answer: 1


A2. Given are two points $A$ and $B$ and two circles, one with center $A$ passing through $B$ and one with center $B$ passing through $A$. Point $C$ lies on the second circle and on line $A B$. Point $D$ also lies on the second circle. Point $E$ lies on the first circle and on line $B D$. Given that $\angle D=57^{\circ}$.

How large is $\angle A$ in degrees?

Note, the figure below is not drawn to scale.

![](https://cdn.mathpix.com/cropped/2024_04_17_998065f2d90ec501d25cg-3.jpg?height=395&width=571&top_left_y=919&top_left_x=751)

Answer: 48


An equilateral cone contains an equilateral cylinder. How does the volume of the cone compare to the volume of the cylinder?

Answer: \dfrac{26 + 15\sqrt{3}}{18}


A [circle](https://artofproblemsolving.com/wiki/index.php/Circle) is [inscribed](https://artofproblemsolving.com/wiki/index.php/Inscribe) in [quadrilateral](https://artofproblemsolving.com/wiki/index.php/Quadrilateral) $ABCD$, [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the [square](https://artofproblemsolving.com/wiki/index.php/Perfect_square) of the [radius](https://artofproblemsolving.com/wiki/index.php/Radius) of the circle.

Answer: 647


1. Variant 1.

Write the smallest number with a digit sum of 62, in the notation of which at least three different digits are used.

Answer: 17999999


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

Answer: 2


In $\triangle ABC$, the angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ on $AC$ and $AB$ respectively. The perpendicular from $I$ to $DE$ meets $DE$ at $P$. The extension of $PI$ meets $BC$ at $Q$. If $IQ = 2IP$, find $\angle A$.

---

The translation maintains the original format and line breaks as requested.

Answer: 60^\circ


Reflect the hyperbola given by the equation $x y=1$ over the line $y=2 x$. Write the equation of the reflection.

Answer: 12y^2 + 7xy - 12x^2 = 25


## Task 2 - 240512

Roland solved a division problem. He obtained the quotient 36.

To check his result, Roland multiplied the divisor by this quotient. However, he mistakenly read a 1 instead of a 7 in the divisor and obtained the result of this multiplication as 756 instead of the given dividend.

What was the division problem that Roland was supposed to solve?

Answer: 972 \div 27


# Problem 5. (3 points)

$A B C D$ is a cyclic quadrilateral. The extension of side $A B$ beyond point $B$ and the extension of side $C D$ beyond point $C$ intersect at point $P$. The extension of side $A D$ beyond point $D$ and the extension of side $B C$ beyond point $C$ intersect at point $Q$. It turns out that angles $B P C$ and $C Q D$ are equal. It is also known that $C Q=20, D Q=12, B P=3$. Find the square of the length of diagonal $A C$. If there are multiple possible answers, list them in ascending order separated by commas or semicolons.

Answer: 1040


Write the first 10 prime numbers in a row. How can you remove 6 digits to get the largest possible number?

Answer: 7317192329


In a $20 \times 19$ grid, use $3 \times 1$ dominoes to cover the grid without overlapping and without common points. Find the maximum possible number of dominoes needed.

Answer: 126


Show that in any set of 27 distinct odd positive numbers less than 100, we can always find two whose sum is 102. How many sets of 26 odd positive numbers less than 100 can be found with no two having a sum of 102?

Answer: 2^{24}


A graph has 30 points and each point has 6 edges. Find the total number of triples such that each pair of points is either joined or each pair of points is not joined.

Answer: 1990


For four non-coplanar points, an equalizer plane is a plane such that the respective distances from each of the points to that plane are all equal. Given a set of four non-coplanar points, how many equalizer planes are there?

Answer: 7


Tom enters a digital maze composed of 20 rooms. Each room contains a card marked with a number as shown in the figure. Tom starts in the first room and collects card 1. Subsequently, he can enter an adjacent room, but each room can only be entered once. If the number on the card in the current room is greater than any number on the cards he has already collected, he collects that card. What is the maximum number of cards Tom can collect by navigating through the maze?

Answer: 20


It is given that \( m, n>0 \) and \( m+n=1 \). If the minimum value of \( \left(1+\frac{1}{m}\right)\left(1+\frac{1}{n}\right) \) is \( a \), find the value of \( a \).

If the roots of the equation \( x^{2}-(10+a) x+25=0 \) are the square of the roots of the equation \( x^{2}+b x=5 \), find the positive value of \( b \).

If \( (x y-2)^{b-1}+(x-2 y)^{b-1}=0 \) and \( c=x^{2}+y^{2}-1 \), find the value of \( c \).

If \( f(x) \) is a polynomial of degree two, \( f(f(x))=x^{4}-2 x^{2} \) and \( d=f(c) \), find the value of \( d \).

Answer: 15


In the NBA Finals between the Los Angeles Lakers and the Boston Celtics, the series is played in a best-of-seven format, meaning the first team to win 4 games will be crowned the champion. The games are divided into home and away games. Since the Los Angeles Lakers had a better regular-season record, games 1, 2, 6, and 7 are played in Los Angeles, while games 3-5 are played in Boston. The Lakers eventually win the championship on their home court. How many possible win-loss sequences are there during the series?

Answer: 30


A thermally insulated vessel is divided into two parts by a heat-conducting partition. The first part contains helium at a temperature of \(127{ }^{\circ} \mathrm{C}\) with a quantity of \(v_{1}=0.2\) moles. The second part contains helium at a temperature of \(7{ }^{\circ} \mathrm{C}\) with a quantity of \(v_{2}=0.8\) moles.

1) What temperature (in degrees Celsius) will be established in the vessel after reaching thermal equilibrium?
2) Find the ratio of the final to the initial pressures in the first part of the vessel.

Answer: 0.76


As shown in the figure, in the rectangular plate $ABCD$ where $AB=3$ and $BC=2$, side $AD$ stands upright on the ground. There is another wooden pole $MN$ standing upright on the ground such that $NE$ is the perpendicular bisector of $AB$, where $E$ is the foot of the perpendicular and $NE=3$, $MN=5$. There is a point light source at $M$. Determine the area of the shadow of the rectangular plate on the ground.

Answer: 8


## Task B-1.4.

How many natural numbers less than 10000 have exactly three equal digits?

Determine the sum of all such numbers whose unit digit is 1.

Answer: 117695


The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?

Answer: 120^\circ


Let \(a\) and \(b\) be coprime positive integers. Let positive integers \(m\) and \(n\) satisfy \(n > m\). Prove that:
$$
\operatorname{lcm}[am + b, a(m+1) + b, \ldots, an + b] \geq (n + 1) \binom{n}{m}.
$$

Answer: \operatorname{lcm}[am + b, a(m+1) + b, \ldots, an + b] \geq (n + 1) \binom{n}{m}


## 6. Gold Coins

Ante, Branimir, Celestin, and Dubravko are sitting around a round table (in that order). Together, they have 1600 gold coins. First, Ante divides half of his gold coins into two equal parts and gives one part to his left neighbor and the other part to his right neighbor, while keeping the other half for himself. Then, Branimir does the same, followed by Celestin, and finally Dubravko. In the end, all four of them have an equal number of gold coins. How many gold coins did Branimir have at the beginning?

Result: $\quad 575$

Answer: 575


8. Given $O$ as the origin,

curve $C_{1}: x^{2}-y^{2}=1$ intersects with curve $C_{2}: y^{2}=2 p x$ at points $M$ and $N$. If the circumcircle of $\triangle O M N$ passes through point $P\left(\frac{7}{2}, 0\right)$, then the equation of curve $C_{2}$ is $\qquad$ .

Answer: y^2 = \dfrac{3}{2}x


]

[ The area of a figure is equal to the sum of the areas of the figures into which it is divided ]

The diagonals of quadrilateral $A B C D$ intersect at point $O, M$ and $N$ are the midpoints of sides $B C$ and $A D$ respectively. Segment $M N$ divides the area of the quadrilateral in half. Find the ratio $O M: O N$, if $A D=2 B C$.

#

Answer: \dfrac{1}{2}


32. [15] Pirate ships Somy and Lia are having a tough time. At the end of the year, they are both one pillage short of the minimum required for maintaining membership in the Pirate Guild, so they decide to pillage each other to bring their counts up. Somy by tradition only pillages $28 \cdot 3^{k}$ coins for integers $k$, and Lia by tradition only pillages $82 \cdot 3^{j}$ coins for integers $j$. Note that each pillage can have a different $k$ or $j$. Soma and Lia work out a system where Somy pillages Lia $n$ times, Lia pillages Somy $n$ times, and after both sets of pillages Somy and Lia are financially even.
What is the smallest $n$ can be?

Answer: 2


A circle with its center on the diagonal $A C$ of trapezoid $A B C D (B C \| A D)$ passes through vertices $A$ and $B$, is tangent to side $C D$ at point $C$, and intersects the base $A D$ at point $E$. Find the area of trapezoid $A B C D$, if $C D=6 \sqrt{13}$, $A E=8$.

Answer: 204


Example 2. The Louka Problem: A shipping company has a ship leaving Harvard for New York and a ship leaving New York for Harvard every noon. The journey takes seven days and seven nights in both directions. How many ships leaving New York will the ship that departs from Harvard at noon today encounter on its journey?

Answer: 13


Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns taking 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, without violating the rules of the game. The computer plays according to the following algorithm: on each turn, it takes as many stones as needed to be in the most advantageous position. Petya always goes first. What is the probability that Petya will win?

Answer: \dfrac{1}{256}


13.077. The older brother on a motorcycle and the younger brother on a bicycle made a two-hour non-stop trip to the forest and back. During this time, the motorcyclist traveled each kilometer 4 minutes faster than the cyclist. How many kilometers did each of the brothers travel in 2 hours, if it is known that the distance covered by the older brother in this time is 40 km more?

Answer: 60


5. Xiaoming's honor stand has 3 gold medals, 2 silver medals, and 1 bronze medal, arranged from left to right in the order of gold, silver, and bronze. At the sports meet, he won 1 more gold medal and 1 more silver medal, and he needs to place these two medals on the honor stand while ensuring all medals are still arranged from left to right in the order of gold, silver, and bronze. He needs to move at least $\qquad$ of the original medals.

Answer: 2


10.4. A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n=a_{1} \pm a_{2} \pm a_{3} \pm \cdots \pm a_{k}$, where $a_{1}, \ldots, a_{k}$ are simple.

Answer: 9


Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space.
Let $V$ be the cone obtained by rotating the triangle around the $x$-axis.
Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

Answer: \dfrac{8}{3}\pi


16. For a plane region $D$, let $N(D)$ denote the number of all integer points (i.e., points on the $X O Y$ plane where both coordinates $x, y$ are integers) within $D$. If $A$ represents the region enclosed by the curve $y=x^{2} (x \geqslant 0)$ and the two lines $x=10, y=1$ (including the boundaries); $B$ represents the region enclosed by the curve $y=x^{2} (x \geqslant 0)$ and the two lines $x=1, y=100$ (including the boundaries). Then $N(A \cup B)+N(A \cap B)=$

Answer: 1010


$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find  the smallest possible value of $N $.

Answer: 2


1. Given the quadratic function
$$
\begin{aligned}
f(x)= & a(3 a+2 c) x^{2}-2 b(2 a+c) x+ \\
& b^{2}+(c+a)^{2}(a, b, c \in \mathbf{R}).
\end{aligned}
$$

Assume that for any $x \in \mathbf{R}$, we have $f(x) \leqslant 1$. Find the maximum value of $|a b|$. (Yang Xiaoming, Zhao Bin)

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


3.1. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at equal speeds and at equal distances from each other. Another part is flying the same way but along another straight line parallel to the first, at the same speeds but in the opposite direction, and at the same distance from each other. The ship is flying parallel to these lines. Astronaut Gavrila noted that the ship meets meteorites flying towards the ship every 7 s, and meteorites flying in the same direction as the ship every 13 s. He wondered how often meteorites would fly past him if the ship were to stand still. It seemed to him that he should take the arithmetic mean of the two given times. Is Gavrila right? If yes, write down this arithmetic mean in the answer. If not, specify the correct time in seconds, rounded to tenths.

Answer: 9.1


5. For the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, the right focus is $F$, and the right directrix is $l$. Points $P_{1}, P_{2}, \cdots, P_{24}$ are 24 points arranged in a counterclockwise order on the ellipse, where $P_{1}$ is the right vertex of the ellipse, and
$$
\angle P_{1} F P_{2}=\angle P_{2} F P_{3}=\cdots=\angle P_{23} F P_{24}=\angle P_{24} F P_{1} .
$$

Then the sum of the reciprocals of the distances from these 24 points to $l$ is

Answer: 6\sqrt{5}


For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$ , let $s(p)$ denote the sum of the three $3$ -digit numbers $a_1a_2a_3$ , $a_4a_5a_6$ , and $a_7a_8a_9$ . Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$ . Let $n$ denote the number of permutations $p$ with $s(p) = m$ . Find $|m - n|$ .
Please give the answer directly without any intermediate steps.

Answer: 162


18. In the Seven-Star Country, each citizen has an ID code, which is a seven-digit number composed of the digits $1 \sim 7$ without repetition. The ID code of Citizen 1 is 1234567, the ID code of Citizen 2 is 1234576, the ID code of Citizen 3 is $1234657, \cdots \cdots$, and the citizen numbers and ID codes increase sequentially. What is the ID code of Citizen 2520? $\qquad$

Answer: 4376521


17. (12 points) Draw a line through point $P(3+2 \sqrt{2}, 4)$ that intersects the $x$-axis and $y$-axis at points $M$ and $N$, respectively. Find the maximum value of $OM + ON - MN$ (where $O$ is the origin).

Answer: 6


1. Let $a, b$ and $c$ be non-zero real numbers and

$$
a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1
$$

Calculate the value of the expression $a b+b c+c a$.

Answer: 0


We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-rectangles, one form one rectangle of size $1993 \times 2000$ (figures don’t overlap each other). Let $s$ denote the sum of numbers of squares and $S$-rectangles used in such tiling. Find the maximal value of $s$.

Answer: 996500


6. Toss a coin independently $n$ times (where $n$ is a positive integer), each time the coin only shows heads or tails. Let $a(n)$ denote the total number of scenarios where the number of times heads appear is exactly a multiple of 3, and $b(n)$ denote the total number of scenarios where the number of times heads appear is exactly a multiple of 6.
(1) Find $a(2016), b(2016)$;
(2) When $n \leqslant 2016$, find the number of positive integers $n$ such that $2 b(n) - a(n) > 0$.
(Wu Genxiu, problem contributor)

Answer: \dfrac{2^{2016} + 2}{3}


10. Given five points $A, B, C, D, E$ on a plane, no three of which are collinear. By connecting these points with 4 lines, each point is an endpoint of at least one line segment. The number of different connection methods is $\qquad$ kinds.

Answer: 135


Misha's father always picked him up from kindergarten by car at the same time every day. However, today, after confirming that dinner would be pea porridge, Misha decisively left kindergarten and started walking towards his father. After walking 5 km, Misha met his father, got into the car, and they arrived home 10 minutes earlier than usual. What is Misha's father's driving speed?

Answer: 60


A merchant bought several sacks of salt in Tver and sold them in Moscow with a profit of 100 rubles. With all the money he gained, he bought salt again in Tver (at the Tver price) and sold it in Moscow (at the Moscow price). This time, the profit amounted to 120 rubles. How much money did he spend on the first purchase?

Answer: 500


Example 7.4 Let $(G, \circ)$ be the vertex permutation group derived from the normal rotation group of a regular hexagon. Find the cycle index of $(G, \circ)$.

Answer: \dfrac{1}{6} \left( x_1^6 + x_2^3 + 2x_3^2 + 2x_6 \right)


11. Six candidates are sitting in the same row of seats with a passage on each side to take an exam. The order in which the candidates finish their papers is uncertain, and each person leaves their seat and exits the classroom immediately after finishing. Then the probability that at least one person will disturb other students still taking the exam when they hand in their paper is $\qquad$ .

Answer: \dfrac{43}{45}


[ Trapezoids (miscellaneous). $[$ Application of trigonometric formulas (geometry).]

In trapezoid $A B C D$, angles $A$ and $D$ are right angles, $A B=1, C D=4, A D=5$. A point $M$ is taken on side $A D$ such that $\angle C M D=$ $2 \angle B M A$.

In what ratio does point $M$ divide side $A D$?

Answer: \dfrac{2}{3}


## Task B-3.6.

Two isosceles triangles have the same perimeter and area, but they are not congruent. The side lengths of one triangle are 29, 29, 40. The side lengths of the other triangle are integers. Determine what these numbers are.

Answer: 24


8. Make a full permutation of the elements in the set $\{1,2, \cdots, 8\}$, such that except for the number at the far left, for each number $n$ on the right, there is always a number to the left of $n$ whose absolute difference with $n$ is 1. The number of permutations that satisfy this condition is $\qquad$

Answer: 128


In the sequence of natural numbers \(\{a_n\}, n=1,2,\ldots\), each natural number occurs at least once, and for any distinct \(n\) and \(m\), the inequality \(\frac{1}{1998}<\frac{|a_n-a_m|}{|n-m|}<1998\) holds. Prove that \(|a_n-n|<2000000\) for all natural \(n\).

Answer: |a_n - n| < 2000000


Let \( n \in \mathbb{N}, k \geq 2 \), and let \( a_{1}, a_{2}, \ldots, a_{k} \) be distinct elements from the set \( \{1, 2, \ldots, n\} \) such that \( n \) divides \( a_{i}(a_{i+1} - 1) \) for \( i = 1, 2, \ldots, k-1 \). Show that \( n \) does not divide \( a_{k}(a_{1} - 1) \).

Answer: n \text{ does not divide } a_{k}(a_{1} - 1)


40 people entered a room with 40 black and white chairs and sat down. They all claimed to be sitting on black chairs. Then they moved around somehow, and exactly 16 claimed to be sitting on a white chair. Each person either lied both times or told the truth both times. How many people who lied were sitting on a black chair the second time?

Answer: 8


On the reverse sides of 2005 cards, different numbers are written (one number on each card). In one query, you are allowed to point to any three cards and learn the set of numbers written on them. What is the minimum number of queries needed to determine the numbers written on each card?

Answer: 1003


Let $ABCD$ be a parallelogram that is not rhombus. We draw the symmetrical half-line of $(DC$ with respect to line $BD$. Similarly we draw the symmetrical half- line of $(AB$ with respect to $AC$. These half- lines intersect each other in $P$. If $\frac{AP}{DP}= q$ find the value of $\frac{AC}{BD}$ in function of $q$.

Answer: \sqrt{q}


In quadrilateral $A B C D$, the acute angle between the diagonals is $\alpha$. Through each vertex, a line is drawn perpendicular to the diagonal that does not contain this vertex. Find the ratio of the area of the quadrilateral bounded by these lines to the area of quadrilateral $A B C D$.

Answer: 2 \cot^2 \alpha


4.94. Solve the Cauchy problem: $y^{\prime \prime}-8 y^{3}=0 ; y(0)=-1 ; y^{\prime}(0)=2$.

Answer: -\dfrac{1}{2x + 1}


$\mathbb{R}^2$-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the $xy$ plane.  The red player moves first.  The first player to get $3$ of their points in a line without any of their opponent's points in between wins.  What is the least number of moves in which Red can guarantee a win?  (We count each time that Red places a point as a move, including when Red places its winning point.)

Answer: 4


Bi 4. For any real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the inequality $x_{1}+x_{2}+\cdots+x_{n} \leqslant \frac{1}{2}$. Find the minimum value of $\left(1-x_{1}\right)\left(1-x_{2}\right) \cdots\left(1-x_{n}\right)$.

Answer: \dfrac{1}{2}


Points \( A_{1}, \ldots, A_{n} \) lie on a circle with center \( O \) such that \( \overrightarrow{O A_{1}}+\ldots+\overrightarrow{O A_{n}}=\overrightarrow{0} \). Prove that for any point \( X \), the inequality \( X A_{1}+\ldots+X A_{n} \geqslant n R \) holds, where \( R \) is the radius of the circle.

Answer: X A_{1} + X A_{2} + \ldots + X A_{n} \geqslant n R


In a singing competition, the participants were Rooster, Crow, and Cuckoo. Each jury member voted for one of the three singers. The Woodpecker counted that there were 59 judges, with a total of 15 votes for Rooster and Crow combined, 18 votes for Crow and Cuckoo combined, and 20 votes for Cuckoo and Rooster combined. The Woodpecker is bad at counting, but each of the four numbers he named differs from the correct number by no more than 13. How many judges voted for Crow?

Answer: 13


In our class club afternoon, 31 of us participated, and after the program, we danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9; the girls can be listed further in such a way that each danced with 1 more boy than the one mentioned before; finally, Márta only did not dance with 3 boys. How many boys were there at the club afternoon?

Answer: 20


10. (10 points) Mom decides to take Xiao Hua on a car trip to 10 cities during the holiday. After checking the map, Xiao Hua is surprised to find: among any three of these 10 cities, either all have highways connecting them, or only two cities are not connected by a highway. Therefore, there are at least $\qquad$ highways connecting these 10 cities. (Note: There is at most one highway between any two cities)

Answer: 40


4. As shown in Figure 4, given that the two medians $B D$ and $C E$ of $\triangle A B C$ intersect at point $G$, and points $A, D, G, E$ are concyclic, $B C=6$. Then the length of $A G$ is . $\qquad$

Answer: 2\sqrt{3}


Consider a volleyball net with sides of 10 and 20, divided into $10 \times 20$ square cells, where each cell is divided into four triangular cells by diagonals: the vertices of the cell are connected with strings to the center of the cell. The nodes of the net are the vertices and centers of the cells. What is the maximum number of strings connecting neighboring nodes of the net that can be cut so that the net does not fall apart into separate pieces?

Answer: 800


َA natural number $n$ is given. Let $f(x,y)$ be a polynomial of degree less than $n$ such that for any positive integers $x,y\leq n, x+y \leq n+1$ the equality $f(x,y)=\frac{x}{y}$ holds. Find $f(0,0)$.

Answer: \dfrac{1}{n}


Kalinin D.A.

Find the maximum number of colors in which the edges of a cube can be painted (each edge in one color) so that for each pair of colors there are two adjacent edges painted in these colors. Adjacent are considered edges that have a common vertex.

#

Answer: 6


Exercise 5. In a football tournament, each team plays exactly twice against each of the others. There are no draws; a win earns two points, and a loss earns none. It turns out that only one team won the tournament with 26 points, and there are two teams tied for last with 20 points each. Determine the number of teams, and provide an example of a tournament where such results occur.

Answer: 12


[ Median to the hypotenuse ] [ Pythagorean Theorem (direct and inverse) ]

Outside the right triangle $A B C$ on its legs $A C$ and $B C$, squares $A C D E$ and $B C F G$ are constructed.

The extension of the altitude $C H$ of triangle $A B C$ intersects the line $D F$ at point $K$. Find $H K$, if the legs are equal to 2 and 3.

Answer: \dfrac{25\sqrt{13}}{26}


Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.

Answer: (2, 5)


. Let \( n \geqslant 3 \) be an integer. A frog is to jump along the real axis, starting at the point 0 and making \( n \) jumps: one of length 1, one of length 2, \(\ldots\), one of length \( n \). It may perform these \( n \) jumps in any order. If at some point the frog is sitting on a number \( a \leqslant 0 \), its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number \( a > 0 \), its next jump must be to the left (towards the negative numbers). Find the largest positive integer \( k \) for which the frog can perform its jumps in such an order that it never lands on any of the numbers \( 1, 2, \ldots, k \).

Answer: \left\lfloor \dfrac{n-1}{2} \right\rfloor


In a class of 33 students, 22 swim and 22 play soccer every day. Everyone participates in at least one of the exercises every day. Of those who play soccer today, 15 swam and 15 played soccer yesterday, and the same situation applies to those who swim today. How many students swam only every day?

Answer: 4


3. Let $P(x)=x^{2}+\frac{1}{2} x+b$ and $Q(x)=x^{2}+c x+d$ be two polynomials with real coefficients such that $P(x) Q(x)=Q(P(x))$ for all real $x$. Find all the real roots of $P(Q(x))=0$.

Answer: -1


Example 6. The center of square $ABCD$ is $O$, and its area is $1989 \mathrm{~cm}^{2} . P$ is a point inside the square, and $\angle O P B=45^{\circ}, P A : P B=5 : 14$. Then $P B=$ $\qquad$ . (1989, National Junior High School League)

Answer: 42


Let $p$ be a prime number such that $p^{2} + 2$ is prime. Show that the same holds for $p^{3} + 2$, $p^{3} + 10$, $p^{4} + 2$, and $p^{4} - 2$.

Answer: p = 3


Problem 7. In the given quadrilateral $A B C D$, no two sides are parallel. The angles between diagonal $A C$ and the sides of the quadrilateral are equal (in some order) to $16^{\circ}, 19^{\circ}, 55^{\circ}$ and $55^{\circ}$. What can the acute angle between the diagonals $A C$ and $B D$ be?

Answer: 87^\circ


4. (6) 2015 coins are placed in 5 chests. The number of coins in the chests are 5 consecutive numbers. Ivan the Fool can take 4 coins from any chest and distribute them among the remaining chests. He can repeat this operation as many times as he wants. At any point, Ivan can take all the coins from one chest. What is the maximum number of coins Ivan the Fool can take?

Answer: 2009


Problem 7.5. In each room of the hotel, no more than 3 people can be accommodated. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. In one room, only men or only women can be accommodated; also, fans of different teams cannot be accommodated together. How many rooms need to be reserved to definitely accommodate all the fans?

Answer: 37


Given circles $\odot O_{1}$ and $\odot O_{2}$ with radii $r$ and $R$ respectively, they are externally tangent at point $P$. A square $ABCD$ is inscribed in $\odot O_{1}$, and point $P$ lies on the minor arc $\overparen{AB}$. Tangents $AA_{1}$, $BB_{1}$, $CC_{1}$, and $DD_{1}$ are drawn from points $A$, $B$, $C$, and $D$ to $\odot O_{2}$, with the points of tangency being $A_{1}$, $B_{1}$, $C_{1}$, and $D_{1}$ respectively. Find the value of $\frac{AA_{1} + BB_{1}}{CC_{1} + DD_{1}}$.

Answer: \sqrt{2} - 1


10. The Forest Hotel has 180 rooms, numbered 1, 2, 3, ..., 180. During the Forest Conference, the first batch of arriving animals filled all the rooms with the number 6, the second batch of arriving animals filled all the remaining rooms with the number 8, and the third batch of arriving animals filled all the remaining rooms with the number 9. At this point, the Forest Hotel still has $\qquad$ rooms left.

Answer: 97


3. As shown in the figure, the side lengths of the two squares $\mathrm{ABCD}$ and $\mathrm{DEFG}$ are both integer centimeters, point $\mathrm{E}$ lies on line segment $\mathrm{CD}$, and $\mathrm{CE}<\mathrm{DE}$, line segment $\mathrm{CF}=5$ centimeters, then the area of pentagon $\mathrm{ABCFG}$ is $\qquad$ square centimeters.

Answer: 71


9. (14 points) (1) The quadratic function $f(x)=a x^{2}+b x + c(a, b, c \in \mathbf{R}, a \neq 0)$ satisfies:
(1) For $x \in \mathbf{R}$, $4 x \leqslant f(x) \leqslant \frac{1}{2}(x+2)^{2}$ always holds;
(2) $f(-4+2 \sqrt{3})=0$. Find $f(x)$.
(2) Let $f_{1}(x)=\frac{3}{2+x}, f_{n+1}(x)=$ $f_{1}\left(f_{x}(x)\right)$, find $f_{2009}(0)$.

Answer: \dfrac{1}{3}x^2 + \dfrac{8}{3}x + \dfrac{4}{3}


5. The given figure is an isosceles triangle $ABC (|AB|=|AC|)$, such that $|\angle BAC| > 50^{\circ}$. On the side $\overline{BC}$, a point $M$ is chosen such that $|\angle BAM| = 50^{\circ}$, and on the side $\overline{AC}$, a point $N$ is chosen such that $|AM| = |AN|$. What is the measure of the angle $\angle CMN$?

Answer: 25


Answer: \dfrac{2y}{y - 2}


4.008. Find four numbers that form a geometric progression, where the sum of the extreme terms is -49, and the sum of the middle terms is 14.

Answer: -56


There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1$ . These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$ , where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldots + \theta_{2n}$ .
Please give the answer directly without any intermediate steps.

Answer: 840


14. As shown in Figure 6, it is known that quadrilateral $ABCD$ is inscribed in a circle $\odot O$ with a diameter of 3, diagonal $AC$ is the diameter, the intersection point of diagonals $AC$ and $BD$ is $P$, $AB=BD$, and $PC=0.6$. Find the perimeter of quadrilateral $ABCD$.

Answer: 1 + \sqrt{3} + \sqrt{6} + 2\sqrt{2}


3. For a finite graph, the following operation can be performed: choose any cycle of length 4, select any edge in this cycle, and remove it from the graph. For a fixed integer $n(n \geqslant 4)$, if the complete graph with $n$ vertices is repeatedly subjected to the above operation, find the minimum number of edges in the resulting graph $q$.

Answer: n


## - Problem 3

Let $A B C D$ be a convex quadrilateral. Let $P$ be the intersection of $A C$ and $B D$. The angle $\angle A P D=60^{\circ}$. Let $E, F, G$ and $H$ be the midpoints of the sides $A B, B C, C D$ and $D A$ respectively. Find the largest positive real number $k$ such that

$$
E G+3 H F \geq k d+(1-k) s
$$

where $s$ is the semiperimeter of the quadrilateral $A B C D$ and $d$ is the sum of the lengths of its diagonals. When is equality achieved?

Answer: \sqrt{3} + 1


1. [3] Joe B. first places the black king in one corner of the board. In how many of the 35 remaining squares can he place a white bishop so that it does not check the black king?

Answer: 30


(12) As shown in the figure, given two points $A(-\sqrt{5}, 0)$ and $B(\sqrt{5}, 0)$, the incenter of $\triangle ABC$ moves on the line $x=2$.
(1) Find the equation of the locus of point $C$;
(2) Draw two rays from point $M(2,0)$, intersecting the locus found in (1) at points $P$ and $Q$, respectively, and $\overrightarrow{M P} \cdot \overrightarrow{M Q}=0$. Prove that the line $PQ$ must pass through a fixed point.

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


11.183. A regular triangular prism is described around a sphere, and a sphere is described around it. Find the ratio of the surfaces of these spheres.

Answer: 5


12. (5 points) $A, B, C, D$ four people are carrying a flashlight and need to cross a dark tunnel that only allows 2 people to walk at a time. Each time, 2 people take the flashlight to cross, then one person returns with the flashlight, and again 2 people take the flashlight to cross $\cdots$, if $A, B, C, D$ four people take $3, 4, 5, 6$ minutes respectively to cross the tunnel alone, then the minimum time required for all 4 to cross is $\qquad$ minutes.

Answer: 21


## Task B-1.4.

In trapezoid $A B C D$, $A B \| C D, |A B|=22, |B C|=10, |C D|=38, |D A|=14$. The angles of the trapezoid at vertices $A$ and $B$ are obtuse. The angle bisectors at vertices $A$ and $D$ intersect at point $E$, and the angle bisectors at vertices $B$ and $C$ intersect at point $F$. Calculate the area of the hexagon $A B F C D E$.

Answer: 120\sqrt{3}


Jirka had three identical rectangles. First, he placed them together as shown in the picture and got a rectangle with a perimeter of $20 \mathrm{~cm}$. Then he placed them together differently and got a rectangle with a different perimeter. What was the perimeter of this rectangle?

(When placing the rectangles together, no two rectangles overlapped and there were no gaps between them.)

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_c6cec8a95bc7df296d45g-1.jpg?height=305&width=211&top_left_y=1401&top_left_x=928)

Answer: 28


\left.\begin{array}{l}{[\text { Arithmetic. Mental calculation, etc. }]} \\ {[\text { Arithmetic progression }}\end{array}\right]

When little Clive approached his grandfather's cuckoo clock, it was 12:05.

Clive started turning the minute hand until the hour hand returned to its original position. How many "cuckoos" did his grandfather count during this time from the next room?

Answer: 78


17. Two thirds in addition (to the unknown), $\frac{1}{3}$ in subtraction, 10 remains (i.e., two thirds of the unknown are added to it, and one third is subtracted from the resulting sum. The remainder is 10).

Answer: 9


$\left.\begin{array}{l}\text { [ Homothety helps solve the problem. } \\ \text { [Two tangents drawn from one point] }\end{array}\right]$

On the sides $A C$ and $B C$ of triangle $A B C$, points $P$ and $Q$ are marked, respectively. It turned out that $A B=A P=B Q=1$, and the point of intersection of segments $A Q$ and $B P$ lies on the inscribed circle of triangle $A B C$. Find the perimeter of triangle $A B C$.

Answer: 4


19. Choose three different numbers from $1,2,3,4,5,6$ to replace $a, b, c$ in the linear equation $a x+b y+c=0$. The number of different lines that can be drawn in the Cartesian coordinate system is $\qquad$.

Answer: 114


## Task A-1.5.

In how many ways can we color the cells of a $2 \times 2016$ board in two colors so that there are no three cells of the same color that can be simultaneously covered by a tile of the shape shown in the figure? The tile can be rotated.

![](https://cdn.mathpix.com/cropped/2024_05_30_1a4376e76c5095e3db07g-06.jpg?height=149&width=140&top_left_y=1916&top_left_x=1552)

Answer: 2^{2016} + 2


Answer: \sqrt{3}


In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point.
What is the largest possible number of red points?

[i]Proposed by Art Waeterschoot, Belgium[/i]

Answer: 1012


Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010?

Answer: 503


A regular pentagon is drawn in the plane, along with all its diagonals.  All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded.  An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region.  If the ant enters an unbounded region, it explodes.  After first leaving the central region of the pentagon, let $x$ be the expected number of times the ant re-enters the central region before it explodes.  Find the closest integer to $100x$.

Answer: 200


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

Answer: 4


Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$.  How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$?

Answer: 120


24th CanMO 1992 Problem 5 There are 2n+1 cards. There are two cards with each integer from 1 to n and a joker. The cards are arranged in a line with the joker in the center position (with n cards each side of it). For which n < 11 can we arrange the cards so that the two cards with the number k have just k-1 cards between them (for k = 1, 2, ... , n)?

Answer: 3


The product of three consecutive numbers is in the form of $87 X X X X X 8$. Determine the 5 missing digits. (The $X$s do not necessarily represent the same digits.)

This was the original problem statement, the following is the corrected version that appeared in January 1981:

The product of three consecutive even numbers is in the form of $87 X X X X X 8$. Determine the 5 missing digits. (The $X$s do not necessarily represent the same digits.)

Answer: 52660


2. Given three positive numbers: $a, b, c$. Petya wrote on the board the numbers $\frac{1}{a}+b c, \frac{1}{b}+a c, \frac{1}{c}+a b, a$ and Vasya wrote the numbers $2 a^{2}, 2 b^{2}, 2 c^{2}$. It turned out that both wrote the same three numbers (possibly in a different order). What is the product $a b c?$ (N. Agakhanov)

Answer: 1


4. At a point $R$ on a line, there is a robot that moves along this line to the left or right as it wishes. It is programmed to take 2 steps on the first move, 4 steps on the second move, 6 steps on the third move, and in general, $2n$ steps on the $n$-th move.

a) Describe a variant of the robot's movement such that it starts from point $R$ and ends its movement at point $R$ after exactly 4 moves.

b) What is the minimum number of moves the robot must make so that it starts from point $R$ and ends up back at $R$ at the end of its movement?

c) Prove that there is a variant of the robot's movement such that it starts from $R$ and ends up at $R$ after 179 moves.

## NOTE

- All questions are mandatory;
- Each question is worth 7 points;
- No points are awarded by default;
- The actual working time is 2 hours from the moment the question is received.

## Mathematics Olympiad  Regional Phase - February 16, 2013

## Grade 6 - grading rubric

Answer: 3


Let \( P_{1}(x)=x^{2}-2 \) and \( P_{j}(x)=P_{1}\left(P_{j-1}(x)\right) \) for \( j=2, 3, \ldots \). Show that for arbitrary \( n \), the roots of the equation \( P_{n}(x)=x \) are real and different from one another.

Answer: \text{All roots of } P_n(x) = x \text{ are real and distinct for any } n


5. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=a>2, a_{2017}=$ 2017, and for any positive integer $n, a_{n+1}=a_{n}^{2}-2$. Then $\left[\frac{\sqrt{a-2}}{10^{6}} a_{1} a_{2} \cdots a_{2017}\right]=$ $\qquad$, where $[x]$ denotes the greatest integer not exceeding the real number $x$.

Answer: 2


## Task 1

An LPG is picking up seed potatoes. The first wagon already has 35 full sacks. Nine full sacks are still prepared on the ramp. Two wagons are to be loaded with 40 sacks of seed potatoes each.

How many sacks still need to be filled?

Answer: 36


6. Replace the letters with digits to get a true equality: $\overline{13 a b c d e}: 5=\overline{a b c d e 6}$.

Answer: 1326530


From the vertex of the obtuse angle \( A \) of triangle \( ABC \), the height \( AD \) is dropped. A circle with center \( D \) and radius \( DA \) is drawn, which intersects the sides \( AB \) and \( AC \) again at points \( M \) and \( N \) respectively. Find \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).

Answer: \dfrac{cm}{n}


Divide a regular $n$-gon $(n \geq 5)$ into the smallest number of:

a) acute-angled triangles,

b) obtuse-angled triangles.

Each side of any triangle in the partition must simultaneously be a side of another triangle in the partition or a side of the polygon.

It is important that no vertex of one triangle divides the side of another triangle into parts. Such a partition is usually called a triangulation.

Answer: n


7. The positive integer solutions of the equation $x+y^{2}+(x, y)^{3}=x y(x, y)$ are $\qquad$ groups $((x, y)$ represents the greatest common divisor of integers $x, y)$.

Answer: 4


It is clear that if you take a random natural number from a sufficiently long segment of the natural number series, the probability that this number is divisible by 10 will approach 0.1 as the segment length increases.

Let's pose another question: how much can this probability differ from 0.1? It can be equal to 0 if, for example, we specified the segment $1,2,3$ where no number is divisible by 10. What is:

a) the greatest possible probability;

b) the smallest possible non-zero probability?

Answer: 1


Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$ . Find the remainder when $x$ is divided by $1000$ .
Please give the answer directly without any intermediate steps.

Answer: 896


8.4. The shorter leg $AC$ of the right triangle $ABC$ has a length of $b$. A point $D$ is chosen on the hypotenuse $AB$ such that $BD = BC$. A point $E$ is taken on the leg $BC$ such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.

---

Translation:
8.4. The shorter leg $AC$ of the right triangle $ABC$ has a length of $b$. A point $D$ is chosen on the hypotenuse $AB$ such that $BD = BC$. A point $E$ is taken on the leg $BC$ such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.

Answer: 2b


17. Let the complex number $z$ satisfy $|z|=1$. Try to find the maximum and minimum values of $\left|z^{3}-3 z-2\right|$.

Answer: 3\sqrt{3}


During a training shooting exercise, each soldier fired 10 shots. One of them succeeded by scoring 90 points. How many times did he score 9 points if he scored a full ten 4 times, with the remaining hits being sevens, eights, and nines, and no miss shots?

Answer: 3


Example 8 A positive integer, if it can be expressed as the difference of squares of two positive integers, is called a "wise number". Arrange all the wise numbers in ascending order. Find the wise number at the 2009th position.

Answer: 2681


## Task B-2.1.

Determine all real numbers $k$ for which one solution of the equation

$$
(x-2 k)^{2}=k(x+2)
$$

is 1 greater than the other.

Answer: -1


Problem 7. Let's introduce the notation as shown in the figure. By extending the lateral sides, denote their point of intersection as $E$. The point of intersection of the diagonals is called $F$. Right triangles $B F C$ and $D F A$ are similar, and if the legs of the first are denoted as $x$ and $y$, then the corresponding legs of the second are $4 x$ and $4 y$.

The height of the trapezoid $h$ is composed of the heights of triangles $B F C$ and $A F D$:

$$
h=\frac{B F \cdot F C}{1}+\frac{A F \cdot F D}{4}=x y+\frac{4 x \cdot 4 y}{4}=5 x y
$$

The area of the trapezoid is $\frac{15}{16}$ of the area of triangle $A E D$. We get:

$$
\frac{1}{2} A C \cdot B D=\frac{15}{16} \cdot \frac{1}{2} \cdot A E \cdot E D \cdot \sin 60^{\circ}=\frac{15 \sqrt{3}}{64} \cdot \frac{4}{3} A B \cdot \frac{4}{3} C D=\frac{15 \sqrt{3}}{36} A B \cdot C D
$$

from which[^0]

$$
\frac{25}{2} x y=\frac{15 \sqrt{3}}{36} \sqrt{x^{2}+16 y^{2}} \cdot \sqrt{y^{2}+16 x^{2}}
$$

Considering that $x^{2}+y^{2}=1$, we find: $\frac{30}{\sqrt{3}} x y=\sqrt{1+15 y^{2}} \cdot \sqrt{1+15 x^{2}}$.

Performing obvious transformations:

$$
300 x^{2} y^{2}=1+15\left(x^{2}+y^{2}\right)+225 x^{2} y^{2} ; \quad 75 x^{2} y^{2}=16 ; \quad 5 x y=\frac{4}{\sqrt{3}}
$$

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


Subject (4). For each non-empty subset $A=\left\{a_{1}, a_{2}, \ldots, a_{k}\right\}$ of the set $\{1,2, \ldots, 10\}, k=1,2, \ldots, 10$, consider the sum

$$
S(A)=a_{1}-a_{1} a_{2}+a_{1} a_{2} a_{3}-\cdots-(-1)^{k} a_{1} a_{2} \cdots a_{k}
$$

where $a_{1}<a_{2}<\cdots<a_{k}$. Determine the sum of all these sums.

Answer: 512


10. A kilogram of one type of candy is 80 k more expensive than another. Andrey and Yura bought 150 g of candies, which included both types of candies, with Andrey having twice as much of the first type as the second, and Yura having an equal amount of each. Who paid more for their purchase and by how much?

14

Answer: 2


Example 1 The scoring rules for a football match are: 3 points for a win, 1 point for a draw, and 0 points for a loss. Then, how many different ways can a team score 19 points in 14 matches?

Answer: 4


51st Putnam 1990 Problem B6 C is a non-empty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let B α be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support lines. What is the smallest α such that ∩ B α contains a point of C, where the intersection is taken over all possible directions for the support line L? Solution

Answer: \dfrac{1}{3}


12. Let $P$ be an interior point of $\triangle A B C$, and extend $A P, B P, C P$ to intersect the opposite sides. In the figure, $a, b, c$, $d$ are the lengths of the respective line segments. Given that $a+b+c=43$, $d=3$, find $a b c=$ ?

Answer: 441


20. As shown in the figure, with the three sides of the isosceles right triangle $A B C$ as sides, squares are constructed outward. Given that the area of triangle $A B C$ is 4, then the area of the hexagon $E F G H I J$ formed by the vertices of the squares is $\qquad$

Answer: 48


## Zadatak B-2.3.

Zadan je trokut $A B C$. Ako je $|A B|=3 \sqrt{3} \mathrm{~cm},|A C|-2|B C|=3 \mathrm{~cm}$, a mjera kuta nasuprot stranici $\overline{A C}$ iznosi $150^{\circ}$, odredite sinus kuta nasuprot stranici $\overline{A B}$.

Answer: \dfrac{3\sqrt{3}}{14}


Task 66. Find the distance from the point of tangency of two circles with radii $R$ and $r$ to their common external tangent.

## Java 9. Lemma about the perpendicular from the point of tangency

##

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

---

Task 66. Find the distance from the point of tangency of two circles with radii $R$ and $r$ to their common external tangent.

## Java 9. Lemma about the perpendicular from the point of tangency

Answer: \dfrac{2Rr}{R + r}


9.2. On the board, there are $n$ different integers, any two of which differ by at least 10. The sum of the squares of the three largest of them is less than three million. The sum of the squares of the three smallest of them is also less than three million. For what largest $n$ is this possible?

Answer: 202


The periods of two sequences, \( m \) and \( n \), are coprime numbers. What is the maximum length of the initial segment that they can have in common?

Answer: m + n - 2


Answer: 0


5. (7 points) Three schoolgirls entered a store. Anya bought 2 pens, 7 pencils, and 1 notebook, Varya - 5 pens, 6 pencils, and 5 notebooks, Sasha - 8 pens, 4 pencils, and 9 notebooks. They all paid equally, but one of them used a discount when paying. Who? (Explain your answer).

Answer: Varya


One, (50 points) In $\triangle A B C$, $\angle A=45^{\circ}, \angle B=60^{\circ}, O$ is the circumcenter of the triangle. A line through point $A$ parallel to $O B$ intersects the extension of $C B$ at point $D$. Find the value of $\frac{B C}{D B} \sin D$.

Answer: \sqrt{2}


Given real numbers \(a_{1}, a_{2}, \cdots, a_{n}\), define \(a_{n+1}=0\) and \(M=\max _{1 \leqslant k \leqslant n}\left|a_{k}-a_{k+1}\right|\). Prove that \(\left|\sum_{k=1}^{n} k a_{k}\right| \leqslant \frac{1}{6} M \cdot n(n+1)(n+2)\).

Answer: \left|\sum_{k=1}^{n} k a_{k}\right| \leqslant \frac{1}{6} M \cdot n(n+1)(n+2)


1. Can the number 8 be represented as the sum of eight integers such that the product of these numbers is also equal to 8?

Answer: Yes


The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous?

Answer: 107


16. Elsa makes ice sculptures with ice blocks, 1 ice block can make 1 small ice sculpture, 3 ice blocks can make 1 large ice sculpture. The leftover shavings from making 2 small ice sculptures or 1 large ice sculpture are just enough to make 1 ice block. Given 30 ice blocks, to make the number of small ice sculptures more than the number of large ice sculptures, the maximum number of large ice sculptures that can be made is $\qquad$.

Answer: 11


An event critic is attending a series of upscale events to review unique locations and services. On the first day, they visit a luxurious art gallery where each ticket costs $45. The gallery hosts a special event, and the critic receives two complimentary tickets. On the second day, the critic attends a gourmet dining experience where the cost per plate is $70. Since the critic is reviewing the service, they receive a 50% discount on the meal. On the third day, the critic explores an exclusive rooftop party at an iconic building. The entry fee for this event is $120, but as a special guest, the critic gets a 25% discount. Calculate the total amount the critic would have spent if they had to pay for everything without any discounts or complimentary tickets.

Answer: 280


The owner of a modern auto parts chain wants to expand into the nostalgia market by selling classic car parts. They decide to launch their new product line with 5 different types of vintage hubcaps, each priced at $25. On the first day of the launch, the store sells 3 sets of each type of hubcap.

Additionally, the owner offers a 10% discount on the total purchase for customers who buy all 5 types of hubcaps together. On that day, 4 customers take advantage of this discount.

How much total revenue did the store earn from the hubcap sales on the first day, including the discounts?

Answer: 825


Alex is a risk management specialist who helps poker players optimize their winnings and minimize their losses. One of his clients, Jamie, played three rounds of poker. In the first round, Jamie won $120. In the second round, Jamie lost $45. In the third round, Jamie won double the amount he lost in the second round. Alex advised Jamie to save 25% of his total winnings after these three rounds. How much money should Jamie save based on Alex's advice?

Answer: 52.5


Alex is a tech enthusiast who values originality and has a keen eye for spotting counterfeits. He recently attended a tech convention where he found a booth selling authentic new gadgets and another booth selling counterfeit copies. At the authentic booth, a new gadget costs $120 each, while at the counterfeit booth, they sell imitations for $40 each. Alex decides to buy 3 original gadgets and then notices that for every original gadget he buys, he can spot 2 fake ones.

How much more money would Alex spend on the original gadgets compared to the amount he would have spent if he had bought the same number of fake ones instead?

Answer: 240


Dur Muhammad Baloch and his old friend, who was also his former boxing trainee partner, decided to reminisce about their training days by doing a workout together. During their session, they planned to do a series of exercises: 3 rounds of boxing sparring, 5 sets of push-ups, and a 2-mile jog.

For boxing sparring, they spent 4 minutes per round with a 1-minute rest between rounds. For push-ups, they did 15 push-ups per set, taking a 2-minute break between each set. Finally, they jogged at a pace of 8 minutes per mile.

How many minutes in total did Dur Muhammad Baloch and his friend spend training together during this session?

Answer: 38


Aiman is a Malaysian university student majoring in economics, and he is concerned about the employment situation in Malaysia. He read a report stating that the current unemployment rate in Malaysia is 4.5%. In his economics class, Aiman learned that if the government successfully creates 100,000 new jobs, the unemployment rate is projected to decrease by 0.1%.

Currently, the workforce in Malaysia consists of 15 million people. Aiman wants to calculate how many new jobs would need to be created to potentially lower the unemployment rate to 4.0%. How many new jobs need to be created to achieve this goal?

Answer: 500000


Sarah is a data analyst who is responsible for analyzing user behavior on a website. She collects data every day to understand how users interact with the site. On Monday, she observes that 120 users visited the homepage, and each user, on average, viewed 5 additional pages. On Tuesday, the number of users who visited the homepage increased by 25%, and each of these users, on average, viewed 3 more pages than they did on Monday. How many total page views did Sarah record from Monday and Tuesday combined?

Answer: 2070


Alex is a dedicated fan of Professor Smith and loves to share the professor's work on social media. Every day, Alex retweets 5 of the professor's tweets. In addition, Alex shares 3 of these retweets with 4 of their own followers. After a week (7 days), how many total retweets and shares has Alex made in support of Professor Smith's work?

Answer: 119


Alex, an avid gamer and tech enthusiast, recently bought the latest virtual reality headset for $300. He decided to sell his old gaming console to help cover the cost. He sold the console for $150. As a tech enthusiast, Alex is always looking for the best deals, so he used a coupon to get a 10% discount on the headset. Meanwhile, his friend offered him a psychic reading for $30, which Alex decided to skip, saving that amount. How much did Alex actually spend in total after selling his console and applying the discount, while also considering the psychic reading he didn't pay for?

Answer: 120


Engineer Alex is working on a new hydraulic system for a machine. The system has three main components: a pump, a valve, and a cylinder. The pump can move 50 liters of hydraulic fluid per minute. The valve controls the flow and reduces it by 20%. The cylinder requires 30 liters of hydraulic fluid per minute to operate efficiently.

If Alex runs the system for 10 minutes, how much hydraulic fluid is delivered to the cylinder by the end of that time?

Answer: 400


3.4. Five students Ana, Bojan, Vesna, Goran, and Dejan took a test consisting of five questions. For the first two questions, the possible answers were $\alpha, \beta$, and $\gamma$, while for the last three questions, the possible answers were YES and NO. The students gave the following answers to these five questions:

|  | 1 | 2 | 3 | 4 | 5 |
| :--- | :---: | :---: | :---: | :---: | :---: |
| ANA | $\alpha$ | $\alpha$ | YES | YES | YES |
| BOJAN | $\beta$ | $\beta$ | YES | NO | YES |
| VESNA | $\alpha$ | $\beta$ | YES | YES | NO |
| GORAN | $\beta$ | $\gamma$ | YES | YES | NO |
| DEJAN | $\gamma$ | $\alpha$ | NO | YES | YES |

If no two students had the same number of correct answers, determine which student performed the best on the test.

Answer: Goran


9. (12 points) Three people, A, B, and C, depart from location $A$ to location $B$. A departs at 8:00, B at 8:20, and C at 8:30. They all travel at the same speed. 10 minutes after C departs, the distance from A to $B$ is exactly half the distance from B to $B$. At this moment, C is 2015 meters away from $B$. What is the distance between $A$ and $B$ in meters? $\qquad$

Answer: 2418


6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from twenty-seven consecutive natural numbers to some number $a$ is 1926, and the sum of the distances from these same twenty-seven numbers to the number $a^{2}$ is 1932. Find all possible values of $a$.

Answer: \dfrac{2}{3}


IMO 1988 Problem A3 A function f is defined on the positive integers by: f(1) = 1; f(3) = 3; f(2n) = f(n), f(4n + 1) = 2f(2n + 1) - f(n), and f(4n + 3) = 3f(2n + 1) - 2f(n) for all positive integers n. Determine the number of positive integers n less than or equal to 1988 for which f(n) = n. Solution

Answer: 92


## Task 5 - 250835

The following problem is from Leo Nikolayevich Tolstoy (1828 - 1910), a prominent Russian writer:

Reapers are to mow two meadows. In the morning, all of them started mowing the larger meadow. However, from noon of that day, they divided the work differently: half of the reapers continued mowing the first meadow, which they finished by evening. The other reapers moved to mow the second meadow, whose area was equal to half of the first, and worked until evening. Now, a remainder was left on the second meadow, which one reaper alone would need an entire day to complete.

How many reapers were at work on the first day?

Note: It is assumed that each reaper mows the same area in the morning as in the afternoon, and that the work performance of all reapers is the same.

Answer: 8


Let $\xi$ and $\zeta$ be independent random variables with $\mathrm{E} \xi = 0$. Show that

$$
\mathrm{E}|\zeta| \leqslant \mathrm{E}|\xi + \zeta|
$$

Generalize this property by establishing that

$$
\mathrm{E} \max _{1 \leqslant i \leqslant n}\left|\zeta_{i}\right| \leqslant \mathrm{E} \max _{1 \leqslant i \leqslant n}\left|\xi_{i}+\zeta_{i}\right|,
$$

if $\mathrm{E} \xi_{i} = 0, 1 \leqslant i \leqslant n$, and the families $\left\{\xi_{i}\right\}_{i=1}^{n}$ and $\left\{\zeta_{i}\right\}_{i=1}^{n}$ are independent of each other.

Answer: \mathrm{E}|\zeta| \leqslant \mathrm{E}|\xi + \zeta|


174. \(n^{2}\) integers from 1 to \(n^{2}\) are written in a square table of size \(n \times n\): number 1 is in any position in the table; number 2 belongs to the row whose number is equal to the column number containing 1; number 3 belongs to the row whose number matches the column number containing 2, and so on. By how much does the sum of the numbers in the row containing number 1 differ from the sum of the numbers in the column containing the number \(n^{2}\)?

Answer: n^{2} - n


5. (6 points) 12 question setters make guesses about the answer to this question, with their guesses being “not less than 1”, “not greater than 2”, “not less than 3”, “not greater than 4”, “not less than 11”, “not greater than 12” (“not less than” followed by an odd number, “not greater than” followed by an even number). The number of question setters who guessed correctly is $\qquad$ people.

Answer: 7


4. How many natural numbers less than 10000 have exactly three equal digits?

Answer: 333


A group of ten people went to the cinema. They received tickets for 5 adjacent seats in two different rows. Ábel and Bendegúz want to sit next to each other, while Zsuzsi and Anikó want to sit separately. In how many ways can they be seated?

Answer: 518400


1. In a tennis tournament, 254 schoolchildren are participating. A win earns 1 point, a loss earns 0 points. If the winner initially had fewer points than the opponent, the winner additionally receives one point from the loser. In each round, participants with the same number of points compete, but in one of the pairs, a difference of 1 point between opponents is allowed. The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 5 points?

Answer: 56


In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are  $a$  particles on the left side of the tube, some number  $n$  of these particles move to the right side, where  $n \in \{0,1,\dots,a\}$  is chosen uniformly at random. In a similar manner, some number of the particles from the right side of the tube move to the left, at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is  $\tfrac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ .  Compute  $100a+b$ .

*Proposed by Alvin Zou*

Answer: 102


678. The number 43 is written in the septenary (base-7) numeral system. In which system is it written with the same digits but in reverse order?

Answer: 9


A chord of a circle is at a distance \( h \) from the center. In each of the segments subtended by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc, and the other two lie on the chord or its extension. What is the difference in the side lengths of these squares?

Answer: \dfrac{8h}{5}


2.3. Given an empty graph on $n>2$ vertices and a natural number $k<4$. Vasya and Petya play the following game: Vasya chooses 3 vertices and draws edges between them that do not yet exist. Then Petya erases any $k$ edges of the graph. Vasya makes the first move. Vasya wins if after Petya's turn the graph on $n$ vertices is connected. For which $k$ does Vasya win?

## Third Round

Answer: 1


Example 6 Three equal cylindrical surfaces are pairwise tangent, and their axes are perpendicular to each other. If the radius of each cylindrical surface is $r$, find the radius of the smallest sphere that is tangent to these three cylindrical surfaces.

 untranslated text remains the same as the source text in terms of line breaks and format.

Answer: (\sqrt{2} - 1)r


1. From Central Square to the station, there is a straight street divided by 11 intersections into 12 equal blocks. At each intersection, there is a traffic light. All traffic lights simultaneously turn green for 3 minutes, then red for 1 minute. It takes the bus two minutes to travel one block (from intersection to intersection), while a car takes one minute to travel one block. The bus and the car start from the square at the same time, at which moment all traffic lights turn green. Which vehicle will arrive at the station first, and how many minutes earlier?

Answer: 9


Example 4. For what value of $a$ does the curve $y=\sqrt{x+1}$ and the line $y=x+a$ have two intersection points?

Answer: [1, \frac{5}{4})


$P$ and $Q$ are points on the semicircular arc with diameter $AB$. $R$ is a point on the radius $OB$, such that $O P R \angle = O Q R \angle = 10^{\circ}$. If $P O A \angle = 40^{\circ}$, what is the measure of $Q O B \angle$?

Answer: 20^\circ


## Task A-2.2.

Determine all real numbers $x$ for which

$$
\left\lfloor\frac{x^{2}+1}{x+2}\right\rfloor+\left\lfloor\frac{x-1}{2}\right\rfloor=\frac{x(3 x+1)}{2(x+2)}
$$

For a real number $t, \lfloor t\rfloor$ is the greatest integer not greater than $t$.

For example, if $t=3.14$, then $\lfloor t\rfloor=3$.

Answer: -7


1. A four-digit number (in the decimal system) is given. By changing the order of its digits, exactly eight other four-digit numbers can be formed. The sum of the smallest three of all these nine numbers is 12528. Determine the digits of the given number.

Answer: 0


p1. The sides of a triangle $ABC$ have lengths $AB = 26$ cm, $BC = 17$ cm, and $CA = 19$ cm. The bisectors of the angles $B$ and $C$ intersect at the point $I$. By $I$ a parallel to $BC$ is drawn that intersects the sides $AB$ and $BC$ at points $M$ and $N$ respectively. Calculate the perimeter of the triangle $AMN$.

p2. Let $n$ be a positive integer. Determine the exact value of the following sum
$$\frac{1}{1+\sqrt3}+\frac{1}{\sqrt3+\sqrt5}++\frac{1}{\sqrt5+\sqrt7}+...+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}$$

p3. Find the product of all positive integers that are less than $150$ and that have exactly $4$ positive divisors. Express your answer as a product of prime factors.

p4. There is a board of $ n \times n$, where n is a positive integer and the numbers from $ 1$ to $n$ are written in each of its rows in some order, so that the arrangement obtained is symmetric with respect to the diagonal. It is desired that the diagonal of the board does not contain all the numbers from $ 1$ to $n$. Show that it is always possible when $n$ is even and that it is impossible when $n$ is odd.

p5. There are six circles of radius $\frac12$, tangent to each other and to the sides of the rectangle, as shown in the figure. Also the centers $A$, $B$, $C$ and $D$ are collinear. Determine the lengths of the sides of the rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/e/1/c8c23f796a73aaab25b78d8f0539750dc3f159.png[/img]

Answer: 45


1. Milan moved 10 steps forward, then returned 2 steps back, then 10 forward and 1 back, then again 10 forward and 2 back, etc. How many steps does Milan need to take to be 1000 steps away from the starting point?

Answer: 1356


In a table $4\times 4$ we put $k$ blocks such that
i) Each block covers exactly 2 cells
ii) Each cell is covered by, at least, one block
iii) If we delete a block; there is, at least, one cell that is not covered.
Find the maximum value of $k$.
Note: The blocks can overlap.

Answer: 12


2. Three towns $X, Y$ and $Z$ lie on a plane with coordinates $(0,0),(200,0)$ and $(0,300)$ respectively. There are 100,200 and 300 students in towns $X, Y$ and $Z$ respectively. A school is to be built on a grid point $(x, y)$, where $x$ and $y$ are both integers, such that the overall distance travelled by all the students is minimized. Find the value of $x+y$.

Answer: 300


Say a positive integer $n$ is [i]radioactive[/i] if one of its prime factors is strictly greater than $\sqrt{n}$. For example, $2012 = 2^2 \cdot 503$, $2013 = 3 \cdot 11 \cdot 61$ and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime factors less than $30$?

[i]Proposed by Evan Chen[/i]

Answer: 119


Four, among the 10-digit numbers where each digit is different, how many are multiples of 11111? Prove your conclusion.

In the 10-digit numbers where each digit is different, how many are multiples of 11111? Prove your conclusion.

Answer: 3456


A function $f$ is given by $f(x)+f\left(1-\frac{1}{x}\right)=1+x$ for $x \in \mathbb{R} \backslash\{0,1\}$.
Determine a formula for $f$.

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


## Task 6 - 311236

Consider all pyramids $A B C S$ that satisfy the following conditions (1), (2), (3):

(1) The base $A B C$ of the pyramid has an area of 1.

(2) It holds that $A B=A C=S B=S C$.

(3) It holds that $B C=S A$.

Investigate whether among all pyramids that satisfy these conditions, there is one with the largest volume. If this is the case, determine the size of the angle $\angle B A C$ for such a pyramid.

Answer: 60^\circ


4. From $1,2, \cdots, 10$ choose 3 different numbers $a, b, c$ as the coefficients of the quadratic equation $a x^{2}+b x=c$. Then the number of equations with different solutions is $\qquad$

Answer: 654


A number was written on the board. At each step, we add to it the largest of its digits (for example, if we have written 142, the next number will be 146). What is the maximum number of odd numbers that can be written consecutively by proceeding in this way?

## High school statements

Answer: 5


Let $ABC$ be a triangle such that $AB = 55$, $AC = 35$, and $BC = 72$. Consider a line $\ell$ that intersects side $BC$ at $D$ and side $AC$ at $E$, dividing the triangle into two figures with equal perimeters and equal areas. Determine the length of segment $CD$.

Answer: 60


Five. (Full marks: 15 points) From the 91 natural numbers $1,2,3, \cdots, 90,91$, select $k$ numbers such that there must be two natural numbers $p, q$ satisfying $\frac{2}{3} \leqslant \frac{q}{p} \leqslant \frac{3}{2}$. Determine the minimum value of the natural number $k$, and explain your reasoning.

Answer: 10


2. 8 girls and 25 boys stand in a circle, with at least two boys standing between any two girls. How many different arrangements are there (arrangements that can be made to coincide by rotating the circle are considered the same)?

Answer: 7! \times 25! \times \dbinom{16}{7}


In triangle \( ABC \) where \( AB < BC \), point \( I \) is the incenter, \( M \) is the midpoint of side \( AC \), and \( N \) is the midpoint of the arc \( ABC \) of the circumcircle.

Prove that \( \angle IMA = \angle INB \).

Answer: \angle IMA = \angle INB


7.13. In the report on the ski race, it is stated that $96 \%$ of its participants met the standard. It is known that this figure is given with an accuracy of $0.5 \%$. What is the smallest number of participants in this race?

Answer: 23


Let \( a \) and \( b \) be arbitrary real numbers. The equation \( x^{2} + a|x| + b = 0 \) has a set of solutions \( \text{A} \) in the complex number set \( \mathbb{C} \). If \( |\mathbf{A}| = n \), determine the set of all possible values of \( n \), forming the set \( B \).

Answer: \{1, 2, 3, 4, 6\}


Example 1 (kp. $1^{\circ}$ ). Find $\int \sin 5 x \cos 7 x d x$.

Answer: -\dfrac{1}{24}\cos 12x + \dfrac{1}{4}\cos 2x + C


Let \(O\) be the center of a circle \(k\). Points \(A, B, C, D, E, F\) on \(k\) are chosen such that the triangles \(OAB, OCD, OEF\) are equilateral. Let \(L, M, N\) be the midpoints of \(BC, DE, FA\) respectively. Prove that triangle \(LMN\) also is equilateral.

Answer: \triangle LMN \text{ is equilateral}


3. Divide 5 boys and 3 girls into three groups, with each group having at least 1 boy and 1 girl. Then there are a total of different grouping methods.

Answer: 150


【Example 3】To distribute 11 papers to 4 experts for review, one expert should review 4 papers, one should review 3 papers, and the other two should each review 2 papers. How many different ways are there to distribute the papers?

Answer: 831600


Problem 4. In a singing competition, a Rooster, a Crow, and a Cuckoo participated. Each member of the jury voted for one of the three performers. The Woodpecker calculated that there were 59 judges in total, and that the Rooster and the Crow received a total of 15 votes, the Crow and the Cuckoo received 18 votes, and the Cuckoo and the Rooster received 20 votes. The Woodpecker is a poor counter, but each of the four numbers he mentioned differs from the correct one by no more than 13. How many judges voted for the Crow? [6 points] (I.V. Raskina)

Answer: 13


6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. For the sake of secrecy, he will not visit the same village twice, and he will not visit villages in a row that are directly connected by a road. How many villages can he guarantee to fly over? He can start from any village.

Answer: 2019


Example 6. Calculate approximately $1.02^{3.01}$.

Answer: 1.06


8,9

Point $K$ is located on edge $AD$ of tetrahedron $ABCD$, point $N$ is on the extension of edge $AB$ beyond point $B$, and point $M$ is on the extension of edge $AC$ beyond point $C$, such that $AK:KD=3:1$, $BN=AB$, and $CM:AC=1:3$. Construct the section of the tetrahedron by the plane passing through points $K, M, N$. In what ratio does this plane divide the volume of the tetrahedron?

Answer: 2:33


*Five, (20 points) 100 matchboxes, numbered 1 to 100. We can ask whether the total number of matches in any 15 boxes is odd or even. What is the minimum number of questions needed to determine the parity (odd or even) of the number of matches in box 1?

Answer: 3


1. Vitya Pereperepkin always calculates percentages incorrectly in surveys: he divides the number of people who answered in a certain way by the number of all the others. For example, in the survey "What is your name?", conducted among 7 Anyas, 9 Ols, 8 Yuls, Vitya counted $50 \%$ Yuls.

Vitya conducted a survey in his school: what kind of triangle is one with sides $3,4,5$? According to his calculations, $5 \%$ answered "acute-angled", $5 \%$ - "obtuse-angled", $5 \%$ - "such a triangle does not exist", $50 \%$ - "right-angled", and the remaining $a \%$ - "it depends on the geometry". What is $a$?

Answer: 110


## 12. Figure

On a plane, there are five points such that no three of them lie on the same line. The points are connected by straight lines. What is the number of intersection points of the lines (excluding the original five points), if none of the drawn lines are parallel to each other?

Answer: 15


2. The numbers from 1 to 2019 were written in a row. Which digit was written more: ones or twos, and by how many?

Answer: 990


# Task 7. (14 points)

In a company, there are 168 employees. Among any four people, at least one can be chosen who is acquainted with the other three. What is the minimum possible number of people who are acquainted with everyone? #

Answer: 165


1. Zhenya drew several different lines on the plane. How many lines did Zhenya draw if it is known that they divided the plane into 211 parts? In your answer, indicate the smallest and largest possible values in any order, separated by a semicolon.

Answer: 20;210


Ya Chenenko I.V.

In the store, they sell DVD discs - individually and in two types of packages (packages of different types differ in quantity and cost). Vasya calculated how much money is needed to buy $N$ discs (if it is more profitable to buy more discs than needed - Vasya does so):

| $N$ | 1 | 2 | 3 | 4 | 5 | $6-10$ | 11 | 12 | 13 | 14 | 15 | $16-20$ | 21 | 22 | $23-25$ | 26 | 27 | 28 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| c. | 20 | 40 | 60 | 80 | 100 | 111 | 131 | 151 | 171 | 191 | 211 | 222 | 242 | 262 | 265 | 285 | 305 | 325 |

How many discs were in the packages and at what price were the packages sold?

How much money does Vasya need to buy at least 29 discs?

Answer: 333


3. In square $A B C D$, diagonals $A C$ and $B D$ intersect at point $O$. Point $K$ is the midpoint of side $A B$. Points $M$ and $N$ are chosen on sides $A D$ and $B C$ respectively such that rays $O K, O M$, and $O N$ divide the square into three parts of equal area. In what ratio does point $M$ divide side $A D$?

Answer: 5:1


Let \(a\) and \(b\) be positive reals such that:
\[ a = 1 + \frac{a}{b} \]
\[ b = 3 + \frac{4 + a}{b - 2} \]
The original answer is in the form \(\sqrt[3]{k}\). Please find the value of \(k\).

Answer: 2


Suppose \(S\) cans of soda can be purchased from a vending machine for \(Q\) quarters. Find the expression that describes the number of cans of soda that can be purchased for \(D\) dollars, where \(1\) dollar is worth \(4\) quarters. The original answer is in \(\frac{kDS}{mQ}\) format, please give the value of k + m + n.

Answer: 5


A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest distance that the grasshopper can be away from the point $2010$.

Answer: 5


Let \(ABCDA'B'C'D'\) be a rectangular prism with \(|AB|=2|BC|\). \(E\) is a point on the edge \([BB']\) satisfying \(|EB'|=6|EB|\). Let \(F\) and \(F'\) be the feet of the perpendiculars from \(E\) at \(\triangle AEC\) and \(\triangle A'EC'\), respectively. If \(m(\widehat{FEF'})=60^{\circ}\), find the value of \(|BC|/|BE|\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). Please give the value of k + m + n.

Answer: 20


In a circle, chords \( AC \) and \( BD \) intersect at point \( E \). The tangent to the circle passing through point \( A \) is parallel to \( BD \). It is known that \( CD:ED = 3:2 \) and the area of triangle \( \Delta ABE \) is 8. Find the area of triangle \( \Delta ABC \).

Answer: 18


A number from 1 to 144 is hidden. You are allowed to define a subset of the set of numbers from 1 to 144 and ask whether the hidden number belongs to this subset. For an affirmative answer, you must pay 2 rubles; for a negative answer, you must pay 1 ruble. What is the...

Answer: 11


Consider all the real triangles whose vertices' coordinates in a plane Cartesian coordinate system are chosen from the numbers $-1, 0, 1$. What is the possible distance of the centroid of these triangles from the origin?

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


Given a 29-digit number \(X=\overline{a_{1} a_{2} \ldots a_{29}}\) (where \(0 \leq a_{k} \leq 9\) and \(a_{1} \neq 0\)), it is known that for every \(k\), the digit \(a_{k}\) occurs in the number exactly \(a_{30-k}\) times (for example, if \(a_{10}=7\), then the digit \(a_{20}\) occurs seven times). Find the sum of the digits of the number \(X\).

Answer: 201


A set of 1985 points is distributed around the circumference of a circle and each of the points is marked with 1 or -1. A point is called "good" if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with -1 is less than 662, there must be at least one good point.

Answer: 1


Paul is painting a wall that measures 1920 square meters (correct to the nearest ten). Each tin of paint covers 18 square meters (correct to the nearest integer). He needs to paint the entire wall and still have at least half a tin of paint left over. What is the smallest number of tins he must buy to ensure he has enough paint?

Answer: 111


A dart is thrown at a square dartboard of side length 2 so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance of 1 of a corner?

Answer: \frac{\pi-2}{4}


Connecting the right-angled vertex of a right triangle and the two trisection points on the hypotenuse, the lengths of the two resulting line segments are $\sin \alpha$ and $\cos \alpha$ (where $0 < \alpha < \frac{\pi}{2}$). What is the length of the hypotenuse?

A. $\frac{4}{3}$

B. $\sqrt{5}$

C. $\frac{2}{\sqrt{5}}$

D. $\frac{3}{\sqrt{5}}$

Answer: \frac{3}{\sqrt{5}}


Plane figure $W$ represents the set of all points whose coordinates $(x, y)$ satisfy the inequality $|2 - |x| - ||y| - 2|| \leqslant 1$. Draw the figure $W$ and find its area.

Answer: 30


Let \( f(x) \) be an increasing continuous function defined on the interval \([0, 3]\), and let \( g(x) \) be its inverse function, such that \( g(x) > f(x) \) for all positive \( x \) where both functions are defined. Additionally, \( f(0) = 0 \) and \( f(3) = 2 \).

The area under the graph of \( f(x) \) on the interval \([0, 3]\) is 2. Find the area of the figure bounded by the graphs of \( f(x) \) and \( g(x) \), as well as the segment connecting the points \((3, 2)\) and \((2, 3)\).

Answer: 4.5


A square board is divided by a grid of horizontal and vertical lines into \(n^2\) cells, each with a side length of 1. What is the largest \(n\) such that you can mark \(n\) cells in a way that every rectangle with an area of at least \(n\) and sides along the grid lines contains at least one marked cell?

Answer: 7


As shown in the figure, $A$ and $B$ are the two endpoints of a diameter of a circular track. Three miniature robots, labeled as A, B, and C, start at the same time from point $A$ and perform uniform circular motion on the track. Robots A and B start from $A$, while robot C starts from $B$. Robot B moves in a clockwise direction, while robots A and C move in a counterclockwise direction. After 12 seconds, robot A reaches $B$. After an additional 9 seconds, robot A catches up with robot C for the first time and simultaneously meets robot B for the first time. Calculate how many seconds after robot C first reaches $A$ it will take for robot B to reach $B$ for the first time.

Answer: 56


Positive integers \(a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}\) satisfy \(2 \leq a_{i} \leq 166\) and \(a_{i}^{b_{i}} \equiv a_{i+1}^{2} \pmod{167}\) for each \(1 \leq i \leq 7\) (where \(a_{8}=a_{1}\)). Compute the minimum possible value of \(b_{1} b_{2} \cdots b_{7} \left(b_{1}+b_{2}+\cdots+b_{7}\right)\).

Answer: 675


In the table on the left, an operation consists of adding 1 or subtracting 1 to the numbers in any two adjacent cells sharing a common edge. After performing a finite number of operations, the table on the left becomes the table on the right. What is the number in cell $A$ in the table on the right?

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute Nationality for the person whose attribute Height is average? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - The people keep unique animals: `cat`, `horse`, `bird`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`

## Clues:
1. The person who keeps horses and the photography enthusiast are next to each other.
2. The bird keeper is the person who likes Cherry smoothies.
3. The person who loves cooking is the Desert smoothie lover.
4. The person who enjoys gardening is the person who loves a carnations arrangement.
5. The person who loves cooking is directly left of Peter.
6. The person who loves a bouquet of daffodils is the Desert smoothie lover.
7. The Watermelon smoothie lover is the person who keeps horses.
8. The photography enthusiast is Eric.
 What is the value of attribute Name for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

## Clues:
1. The person with a doctorate and Bob are next to each other.
2. There is one house between Carol and the person who has black hair.
3. Peter is the person with a doctorate.
4. The person who loves science fiction books is Eric.
5. The one who only drinks water is the person who has an average height.
6. Alice is the person who loves mystery books.
7. The person who loves romance books is Bob.
8. The person who has brown hair is in the second house.
9. The person who has an average height is in the fourth house.
10. The boba tea drinker is the person who is very tall.
11. The person with a high school diploma is Bob.
12. The person who is short is somewhere to the left of the person who is super tall.
13. The person who loves historical fiction books and the person with a bachelor's degree are next to each other.
14. The person who attended trade school is the person who has red hair.
15. Carol is the person who has auburn hair.
16. The person who is very short is somewhere to the right of the person who has gray hair.
17. There are two houses between Eric and the person who loves fantasy books.
18. There are two houses between the tea drinker and the person who is very short.
19. Eric is somewhere to the left of the person with a bachelor's degree.
20. The root beer lover is Alice.
21. The person with an associate's degree is Arnold.
22. The person who is tall is not in the second house.
23. The person who loves science fiction books is the boba tea drinker.
24. The person who has black hair is not in the fourth house.
25. The person who is short is the coffee drinker.
26. The coffee drinker and the person with a high school diploma are next to each other.
27. The coffee drinker is not in the second house.
 What is the value of attribute House for the person whose attribute Drink is boba tea? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute Nationality for the person whose attribute Food is stir fry? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`, `Bob`
 - People use unique phone models: `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`, `google pixel 6`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `watermelon`, `cherry`, `lime`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Penny`, `Aniya`, `Kailyn`
 - Each person has an occupation: `doctor`, `lawyer`, `engineer`, `teacher`, `artist`

## Clues:
1. The person whose mother's name is Kailyn is Peter.
2. The Dragonfruit smoothie lover is not in the fourth house.
3. The person who uses an iPhone 13 is the person who is an artist.
4. The person whose mother's name is Aniya is Arnold.
5. The person whose mother's name is Aniya is the Desert smoothie lover.
6. The person whose mother's name is Aniya is in the second house.
7. The person who drinks Lime smoothies is in the first house.
8. The person who uses a Huawei P50 is the person who is a doctor.
9. Eric is the person who drinks Lime smoothies.
10. There is one house between the person who is a teacher and Bob.
11. The person whose mother's name is Aniya is the person who is a lawyer.
12. The person who uses an iPhone 13 is the person who likes Cherry smoothies.
13. Arnold is the person who uses a OnePlus 9.
14. The person who is a doctor is The person whose mother's name is Penny.
15. Peter is the person who uses a Samsung Galaxy S21.
16. Bob is not in the third house.
17. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute Mother for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - The people keep unique animals: `fish`, `horse`, `cat`, `bird`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `classical`

## Clues:
1. The person with an associate's degree is The person whose mother's name is Aniya.
2. Peter is the person whose birthday is in September.
3. The person with a bachelor's degree is directly left of the person who loves pop music.
4. Arnold is The person whose mother's name is Janelle.
5. The person with a master's degree is Arnold.
6. The person whose mother's name is Janelle is the person who loves classical music.
7. The person whose birthday is in April is somewhere to the right of the person who loves rock music.
8. The cat lover is in the second house.
9. The fish enthusiast is in the third house.
10. Eric is the person whose birthday is in April.
11. The person who loves pop music is the person whose birthday is in January.
12. The person whose mother's name is Holly is the person with a bachelor's degree.
13. The person who keeps horses is the person with a master's degree.
14. The person whose mother's name is Aniya is the fish enthusiast.
15. Arnold is somewhere to the left of the fish enthusiast.
 What is the value of attribute Mother for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn