14. In $\triangle A B C$, $B D=D C$. There is a piece of cheese on side $A C$, located at the point closest to point $C$ among the four equal segments. On $A D$, there are three mirrors $W_{1}, W_{2}, W_{3}$, which divide $A D$ into four equal segments. A very suspicious mouse is crawling on $A B$ (from $A$ to $B$), with $A B=400$ meters. The mouse can observe the cheese when the mouse, a certain mirror, and the cheese are aligned. Due to its suspicion, the mouse hopes to see the cheese multiple times to ensure that the cheese has not been eaten by other mice before it reaches it. Therefore, in the first minute, it crawls forward 80 meters, in the second minute, it retreats 20 meters, in the third minute, it crawls forward 80 meters, in the fourth minute, it retreats 20 meters, and so on. When the mouse reaches point $B$, it directly rushes along $B C$ to eat the cheese. Question: How many times can the mouse see the cheese on segment $A B$?

Answer: 5


4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

Answer: 11


Task B-3.4. A group of children found a wooden board in the shape of a quadrilateral and decided to use it for the game "pick-a-point". Since the target was of an unusual shape, they had to adapt the game rules. They asked for advice from Mark's older brother, a good mathematician. He measured, calculated, and wrote something... and came to an important conclusion. He established that there is a point $O$ equally distant from all sides of the quadrilateral. Then he divided the board into areas by drawing lines from point $O$ to all vertices $A, B, C, D$. He assigned points to the areas inversely proportional to their area. Which area yields the most points and why? Mark's brother added a condition that points are only awarded if the distance from the hit point $T$ to point $O$ is not greater than the distance from point $O$ to any side of the board. Determine the ratio of the area of the part of the board that does not yield points to the area of the part that does yield points.

Mark's brother measured these values $|A B|=5 \mathrm{dm},|B C|=3 \mathrm{dm},|A D|=4 \mathrm{dm}$, $|\angle B A D|=60^{\circ}$.

Answer: \dfrac{49 - \pi(5\sqrt{3} + \sqrt{5})}{\pi(5\sqrt{3} + \sqrt{5})}


A grid of points with 10 rows and 10 columns is given. Each point is colored either red or blue. Whenever two points of the same color are adjacent in the same row or column, they are connected by a segment of the same color as the points. If two adjacent points are of different colors, they are connected by a green segment. In total, there are 52 red points. Of these red points, 2 are in the corners and 16 are on the grid's edge. The rest of the red points are inside the grid.

There are 98 green segments. Determine the number of blue segments.

Answer: 37


Problem 3. When Maria woke up and looked at the clock, she found that one quarter of the day and night had passed. Her first class will start in 75 minutes, and she will stay at school for 4 hours and 50 minutes. What time will Maria leave school and head home?

Answer: 12:05 \text{ PM}


Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$ . From any vertex of the heptagon except $E$ , the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$ , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$ .
Please give the answer directly without any intermediate steps.

Answer: 351


4. 100 balls of the same mass move along a trough towards a metal wall with the same speed. After colliding with the wall, a ball bounces off it with the same speed. Upon collision of two balls, they scatter with the same speed. (The balls move only along the trough). Find the total number of collisions between the balls.

Answer: 4950


5.1. The sequence is defined by the relations $a_{1}=1$,

$$
a_{2 n}=\left\{\begin{array}{ll}
a_{n}, & \text { if } n \text { is even, } \\
2 a_{n}, & \text { if } n \text { is odd; }
\end{array} \quad a_{2 n+1}= \begin{cases}2 a_{n}+1, & \text { if } n \text { is even, } \\
a_{n}, & \text { if } n \text { is odd. }\end{cases}\right.
$$

Find the smallest natural $n$ for which $a_{n}=a_{2017}$.

Answer: 5


[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]

[u]Set 4[/u]

[b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$.

[b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers.

[b]G.18 / C.23[/b]  How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.

[b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint?

[b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization.

[u] Set 5[/u]

[b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$?

[b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up?

[b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$.

[b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible?

[b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle.

PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]

Answer: 258


1. Given points $A, B, C, D$ lie on the same circle, and $BC = DC = 4, AC$ intersects $BD$ at point $E, AE = 6$. If the lengths of segments $BE$ and $DE$ are both integers, find the length of $BD$.

Answer: 7


# Task 8.2

For a natural number $N$, all its divisors were listed, and then the sum of digits for each of these divisors was calculated. It turned out that among these sums, all numbers from 1 to 9 were found. Find the smallest value of $\mathrm{N}$.

## Number of points 7

Answer: 288


Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

Answer: 4


Task B-1.3. Two small and two large circles are inscribed in a rectangle as shown in the figure. The radius of the large circles is $10 \mathrm{~cm}$. Calculate the area of the part of the rectangle that is not covered by the circles.

![](https://cdn.mathpix.com/cropped/2024_05_30_6e5f8ba1616e2c4b87f4g-02.jpg?height=244&width=460&top_left_y=423&top_left_x=800)

Answer: 800 - \frac{425}{2}\pi


8.6. First solution. Divide all the coins into two parts of 20 coins each and weigh them. Since the number of counterfeit coins is odd, one of the piles will weigh more. This means that there is no more than one counterfeit coin in it. Divide it into two piles of 10 coins and weigh them. If the scales are in balance, then all 20 weighed coins are genuine. If one of the pans outweighs, then there are 10 genuine coins on that pan, and among the other 10 coins, there is exactly one counterfeit. Divide these 10 coins into three piles of 4, 4, and 2 coins. In the third weighing, compare the two piles of 4 coins. If they balance, then all 8 coins are genuine and we have found 18 genuine coins. If one of the piles outweighs, then there are 4 genuine coins in it, the other pile contains the counterfeit coin, and the 2 set aside coins are genuine. In total, 16 genuine coins have been found.

Answer: 16


6. (7 points) Petya showed Vasya 37 identical-looking cards laid out in a row. He said that on the hidden sides of the cards, all numbers from 1 to 37 are written (each exactly once) such that the number on any card starting from the second is a divisor of the sum of the numbers written on all preceding cards. Then Petya showed Vasya that the number on the first card is 37, and on the second card is 1. Vasya said that he then knows what number is written on the third card. What is it?

Answer: 2


Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $ABC$. The circle centered at $A$ of radius $AI$ intersects the circumcircle of $ABC$ at $H$ and $J$. Let $L$ be a point that lies on both the incircle of $ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.

Answer: 17


4. In $\triangle A B C$, $\angle A=30^{\circ}, \angle B=105^{\circ}$, a line $D E$ is drawn through a point $D$ on side $A C$, intersecting side $A B$ or side $B C$ at point $E$, such that $\angle C D E=60^{\circ}$, and $D E$ bisects the area of $\triangle A B C$. Then $\left(\frac{C D}{A C}\right)^{2}=$ $\qquad$ -

Answer: \dfrac{\sqrt{3}}{6}


In the acute triangle \( \triangle ABC \) with \( AC > AB \), let \( \odot O \) be its circumcircle. The tangent to \(\odot O\) at point \( A \) intersects the extension of side \( CB \) at point \( P \). Let point \( X \) be on line segment \( OP \) such that \( \angle AXP = 90^\circ \). Points \( E \) and \( F \) are on sides \( AB \) and \( AC \) respectively (on the same side of \( OP \)), such that:
\[ \angle EXP = \angle ACX \quad \text{and} \quad \angle FXO = \angle ABX. \]

If line \( EF \) intersects \(\odot O\) at points \( K \) and \( L \), prove that \( OP \) is tangent to the circumcircle of \( \triangle KLX \).

Answer: OP \text{ is tangent to the circumcircle of } \triangle KLX


Khachaturyan A.V.

Petr was born in the 19th century, and his brother Pavel - in the 20th century. Once, the brothers met to celebrate their shared birthday. Petr said: "My age is equal to the sum of the digits of the year of my birth." - "Mine too," replied Pavel. How much younger is Pavel than Petr?

Answer: 9


Five. (20 points) For what value of $a$ does the graph of the function $f(x)=(x+a)(|x+1-a|+|x-3|)-2x+4a$ have a center of symmetry?

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

Answer: -\dfrac{2}{3}


14. In $\triangle A B C$, it is known that $A B=\sqrt{30}, A C=$ $\sqrt{6}, B C=\sqrt{15}$, point $D$ makes $A D$ bisect $B C$, and $\angle A D B=90^{\circ}$, the ratio $\frac{S_{\triangle A D B}}{S_{\triangle A B C}}$ can be written in the form $\frac{m}{n}$, where $m, n$ are coprime positive integers. Then $m+n=$

Answer: 65


3. Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. Points $D, E$ on its hypotenuse $B C$ satisfy $C D=C A, B E=B A$. Let $F$ be such a point inside $\triangle A B C$ that $D E F$ is an isosceles right triangle with hypotenuse DE. Find $\angle B F C$. (Patrik Bak)

Answer: 135^\circ


41. The road from home to school takes Seryozha 30 minutes. Once on the way, he remembered that he had forgotten a pen at home. Seryozha knew that if he continued on to school at the same speed, he would arrive there 9 minutes before the bell, but if he returned home for the pen, he would, walking at the same speed, be 11 minutes late for the start of the lesson. What part of the way had he walked?

Answer: \dfrac{1}{3}


Example 3 As shown in Figure 1.4.4, in quadrilateral $A B C D$, $\angle B=\angle C=60^{\circ}, B C=1$, a circle is drawn with $C D$ as its diameter intersecting $A B$ at point $M$ and tangent to $A B$ at point $M$, and intersects $B C$ at point $E$. Find the length of segment $B E$.

Answer: 4 - 2\sqrt{3}


6. The axial cross-section of a large round wine glass is the graph of the function $y=x^{4}$. A cherry, a small sphere with radius $r$, is placed into the wine glass. What is the maximum radius $r$ for the cherry to touch the lowest point of the bottom of the glass (in other words, what is the maximum $r$ such that a circle with radius $r$ located in the region $y \geqslant x^{4}$ can pass through the origin)?

Answer: \dfrac{3}{4} \sqrt[3]{2}


Equilateral $\triangle ABC$ has side length $\sqrt{111}$ . There are four distinct triangles $AD_1E_1$ , $AD_1E_2$ , $AD_2E_3$ , and $AD_2E_4$ , each congruent to $\triangle ABC$ ,
with $BD_1 = BD_2 = \sqrt{11}$ . Find $\sum_{k=1}^4(CE_k)^2$ .
Please give the answer directly without any intermediate steps.

Answer: 677


12. In $\triangle A B C$, $\angle B A C=30^{\circ}, \angle A B C=70^{\circ}, M$ is a point inside the triangle, $\angle M A B=\angle M C A=20^{\circ}$, find the degree measure of $\angle M B A$.

Answer: 30


Points \( C_{1} \) and \( C_{2} \) are chosen on side \( AB \) of triangle \( ABC \). Similarly, points \( A_{1} \) and \( A_{2} \) are chosen on side \( BC \), and points \( B_{1} \) and \( B_{2} \) are chosen on side \( AC \). The segments \( A_{1} B_{2} \), \( B_{1} C_{2} \), and \( C_{1} A_{2} \) are found to have equal lengths, intersect at a single point, and any two of them form a \( 60^\circ \) angle. Prove that \(\frac{A_{1} A_{2}}{BC} = \frac{B_{1} B_{2}}{CA} = \frac{C_{1} C_{2}}{AB}\).

Answer: \frac{A_{1}A_{2}}{BC} = \frac{B_{1}B_{2}}{CA} = \frac{C_{1}C_{2}}{AB}


Exercise 6. A set of $\mathrm{n}$ cells in an $\mathrm{n} \times \mathrm{n}$ grid is said to be distributed if it never includes two cells in the same row or column. In how many ways can one color some (possibly none) cells of an $\mathrm{n} \times \mathrm{n}$ grid so that all distributed sets contain the same number of colored cells?

Answer: 2^{n+1} - 2


Folkior

There is a set of tickets numbered from 1 to 30 (numbers can repeat). Each student drew one ticket. The teacher can perform the following operation: read out a list of several (possibly one) numbers and ask their owners to raise their hands. How many times does he need to perform this operation to find out the number of each student? (There are not necessarily 30 students.)

#

Answer: 5


In the rhombus $A_{1}BCD$ with a side length of 3 and $\angle A_{1}=60^{\circ}$, point $E$ is a moving point on the line segment $A_{1}B$ (excluding the points $A_{1}$ and $B$). By folding along $DE$, the triangle $A_{1}DE$ is folded so that the dihedral angle $A-DE-C$ becomes $60^{\circ}$.

(1) Prove that the projection $O$ of point $A$ on the plane $DEBC$ cannot be the orthocenter of $\triangle DBC$.

(2) When $A_{1}E=2EB$, find the angle between line $BC$ and plane $ADE$.

Answer: \arcsin \dfrac{3\sqrt{7}}{14}


4. In how many different ways can the faces of a cube be painted using six given colors (each face must be painted entirely with one color), if only those colorings are considered different that cannot be made to coincide by rotating the cube?

Answer: 30


7.2. Cut a $5 \times 6$ grid rectangle along the grid lines into five figures, the areas of which will be five consecutive natural numbers, and the perimeter of each figure will be twice its area.

Answer: 4, 5, 6, 7, 8


7. In a $100 \times 100$ square, 10000 cuts were made along the grid lines (each cut one cell long), and it split into 2500 four-cell figures. How many of them are $2 \times 2$ squares?

Answer: 2300


Danka had a paper flower with ten petals. On each petal, there was exactly one digit, and no digit was repeated on any other petal. Danka tore off two petals so that the sum of the numbers on the remaining petals was a multiple of nine. Then she tore off another two petals so that the sum of the numbers on the remaining petals was a multiple of eight. Finally, she tore off two more petals so that the sum of the numbers on the remaining petals was a multiple of ten.

Find the three sums that could have remained after each tearing. Determine all such triplets of sums.

(E. Novotná)

Answer: (36, 24, 10)


[

Isosceles, inscribed, and circumscribed trapezoids [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part

An isosceles trapezoid with bases $A D$ and $B C (A D > B C)$ is circumscribed around a circle, which touches the side $C D$ at point $M$. The segment $A M$ intersects the circle at point $N$. Find the ratio of $A D$ to $B C$, if $A N: N M = k$.

#

Answer: 8k - 1


As shown in Figure 1, two circles with centers $O_{1}$ and $O_{2}$ intersect at points $A$ and $B$. Point $X$ is on circle $\omega_{2}$. Let point $Y$ be on circle $\omega_{1}$ such that $\angle X B Y = 90^{\circ}$. Let $O_{1} X$ intersect $\omega_{2}$ at another point $X'$, and let $X' Y$ intersect $\omega_{2}$ at another point $K$. Prove that $X$ is the midpoint of the arc $\overparen{A K}$.

Answer: X \text{ is the midpoint of the arc } \overparen{AK}


Juca is a scout exploring the surroundings of his campsite. After collecting fruits and wood, he must get water from the river and return to his tent. Let's represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance between the feet of the perpendiculars $C$ and $E$, on $r$ from points $J$ and $B$, is $180 m$.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-14.jpg?height=304&width=873&top_left_y=652&top_left_x=457)

What is the shortest distance Juca can travel to return to his tent, passing by the river?

#

Answer: 180\sqrt{2}


We define  $\mathbb F_{101}[x]$  as the set of all polynomials in  $x$  with coefficients in  $\mathbb F_{101}$  (the integers modulo  $101$  with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of  $x^k$  are equal in  $\mathbb F_{101}$  for each nonnegative integer  $k$ . For example,  $(x+3)(100x+5)=100x^2+2x+15$  in  $\mathbb F_{101}[x]$  because the corresponding coefficients are equal modulo  $101$ .

We say that  $f(x)\in\mathbb F_{101}[x]$  is \emph{lucky} if it has degree at most  $1000$  and there exist  $g(x),h(x)\in\mathbb F_{101}[x]$  such that \[f(x)=g(x)(x^{1001}-1)+h(x)^{101}-h(x)\] in  $\mathbb F_{101}[x]$ . Find the number of lucky polynomials.

*Proposed by Michael Ren.*

Answer: 101^{954}


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

Answer: \frac{8}{13}


Let  $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$ .  When  $a$ ,  $b$ ,  $c$  run over all the real numbers, find the smallest possible value of  $F$ .

Answer: \frac{1}{4}


There are  $24$  participants attended a meeting. Each two of them shook hands once or not. A total of  $216$  handshakes occured in the meeting. For any two participants who have shaken hands, at most  $10$  among the rest  $22$  participants have shaken hands with exactly one of these two persons. Define a *friend circle* to be a group of  $3$  participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

Answer: 864


Let  $\mathcal{S}$  be the set of real numbers that can be represented as repeating decimals of the form  $0.\overline{abc}$  where  $a, b, c$  are distinct digits.  Find the sum of the elements of  $\mathcal{S}$ .

Answer: 360


The equation  $$ (x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016) $$  is written on the board, with  $2016$  linear factors on each side. What is the least possible value of  $k$  for which it is possible to erase exactly  $k$  of these  $4032$  linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Answer: 2016


Problem 3 (MAR CP 1992) :
From the digits  $1,2,...,9$ , we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows :  $123456789$ ,  $123456798$ , ...,  $987654321$ . What is the  $100000th$  number ?

Answer: 358926471


Consider a function  $f:\mathbb{R} \to \mathbb{R}$ . Given that  $f(0)=0$ .  $\lim_{x\to0} \frac{f(h)}{h}=7$ , and  $f(x+y)=f(x)+f(y)+3xy$  for all  $x,y\in{\mathbb{R}}$ . Compute  $f(7)$

Answer: 122.5


A scanning code consists of a  $7 \times 7$  grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of  $49$  squares. A scanning code is called *symmetric* if its look does not change when the entire square is rotated by a multiple of  $90 ^{\circ}$  counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes? $\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

Answer: 1022


Let  $A=(a_{ij})$  be the  $n\times n$  matrix, where  $a_{ij}$  is the remainder of the division of  $i^j+j^i$  by  $3$  for  $i,j=1,2,\ldots,n$ . Find the greatest  $n$  for which  $\det A\ne0$ .

Answer: 5


The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country.

Answer: 7


Let  $S$  be the number of bijective functions  $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$  such that  $f((m+n)\pmod{17})$  is divisible by  $17$  if and only if  $f(m)+f(n)$  is divisible by  $17$ . Compute the largest positive integer  $n$  such that  $2^n$  divides  $S$ .

Answer: 270


Let  $P(X,Y)=X^2+2aXY+Y^2$  be a real polynomial where  $|a|\geq 1$ . For a given positive integer  $n$ ,  $n\geq 2$  consider the system of equations:  \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions  $(x_1,x_2,\ldots,x_n)$  and  $(y_1,y_2,\ldots,y_n)$  of the system to be equivalent if there exists a real number  $\lambda \neq 0$ ,  $x_1=\lambda y_1$ ,  $\ldots$ ,  $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have?

*Mircea Becheanu*

Answer: 1


There is a lamp on each cell of a  $2017 \times 2017$  board. Each lamp is either on or off. A lamp is called *bad* if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.)

Answer: 1


**p1.** The least prime factor of  $a$  is  $3$ , the least prime factor of  $b$  is  $7$ . Find the least prime factor of  $a + b$ .**p2.** In a Cartesian coordinate system, the two tangent lines from  $P = (39, 52)$  meet the circle defined by  $x^2 + y^2 = 625$  at points  $Q$  and  $R$ . Find the length  $QR$ .**p3.** For a positive integer  $n$ , there is a sequence  $(a_0, a_1, a_2,..., a_n)$  of real values such that  $a_0 = 11$  and  $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$  for every  $k$  with  $0 \le k \le n-1$ . Find the maximum possible value of  $n$ . (Be careful that your answer isn’t off by one!)**p4.** Persons  $A$  and  $B$  stand at point  $P$  on line  $\ell$ . Point  $Q$  lies at a distance of  $10$  from point  $P$  in the direction perpendicular to  $\ell$ . Both persons intially face towards  $Q$ . Person  $A$  walks forward and to the left at an angle of  $25^o$  with  $\ell$ , when he is again at a distance of  $10$  from point  $Q$ , he stops, turns  $90^o$  to the right, and continues walking. Person  $B$  walks forward and to the right at an angle of  $55^o$   with line  $\ell$ , when he is again at a distance of  $10$  from point  $Q$ , he stops, turns  $90^o$  to the left, and continues walking. Their paths cross at point  $R$ . Find the distance  $PR$ .**p5.** Compute  $$ \frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}. $$  **p6.** There is a unique real value  $A$  such that for all  $x$  with  $1 < x < 3$  and  $x \ne 2$ ,   $$ \left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999. $$  Compute  $A$ .**p7.** Nine poles of height  $1, 2,..., 9$  are placed in a line in random order. A pole is called *dominant* if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly  $2$  dominant poles.**p8.**  $\tan (11x) = \tan (34^o)$  and  $\tan (19x) = \tan (21^o)$ . Compute  $\tan (5x)$ .

PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 2


For a sequence  $s = (s_1, s_2, . . . , s_n)$ , define  $$ F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2. $$  Consider the sequence  $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$ . Let  $R$  be the sum of all  $F(m)$  for all non-empty subsequences  $m$  of  $S$ . Find the remainder when  $R$  is divided by  $1000$ .

Note: A subsequence is a sequence that can be obtained from another sequence by deleting some non-negative number of values without changing the order.

Answer: 500


13.089. A train traveled a certain distance at a speed of 120 km/h. After that, it traveled a distance 75 km longer at a speed of 150 km/h, and the remaining distance, 135 km shorter than the distance traveled, at a speed of 96 km/h. How long is the entire journey if the average speed of the train turned out to be 120 km/h?

Answer: 415


## 292. Math Puzzle $9 / 89$

Good resolution! This school year, we will declare war on the "fours" and the "ones of the little man" - as the unloved three is called. The final grades in mathematics from the previous year had led the students of $8 \mathrm{a}$ and $8 \mathrm{~b}$ to this plan.

None of them received a five, but every ninth student could proudly present a one, every third a two, and every sixth had to receive a four for lack of diligence. More than twenty, but fewer than forty students learn together in these parallel classes.

How many of them want to improve their three with effort in the new school year?

Answer: 14


3. A number from 0 to 1 inclusive was written on the board. Each of the twenty-five students in the class approached the board, erased the number $x$ written there at the moment, and wrote $\frac{1}{2} \cos \pi x$ instead.

For which initial value of the number will the final result be the maximum?

If there are multiple answers, list them in any order separated by a semicolon.

Answer: 0


2. In $\triangle A B C$, it is known that $\angle C=100^{\circ}$, points $P$ and $Q$ are both on side $A B$, such that $A P=B C, B Q=A C$, the midpoints of segments $A B$, $C P$, and $C Q$ are $M$, $N$, and $K$ respectively. Find $\angle N M K$.

Answer: 40^\circ


5. In a certain month, there are more Mondays than Tuesdays, and more Sundays than Saturdays. What day of the week is the 5th of this month? Could this month be December?

Answer: Thursday


5. As shown in the figure, quadrilaterals $A B C D$ and $C E F G$ are two squares. Given that $\frac{S_{\triangle A B I}}{S_{\triangle E F I}}=27$, find $\frac{A B}{E F}$.

Answer: 3


1. 10 runners start simultaneously: five in blue T-shirts from one end of the running track, and five in red T-shirts from the other. Their speeds are constant and different, with each runner's speed being greater than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately turns around and runs back, and upon returning to their starting point, they finish running. The coach marks a check in the notebook each time two runners in different colored T-shirts meet (face to face or one overtakes the other), and no more than two runners meet at the same point during the run. How many checks will the coach have marked by the time the fastest runner finishes running? (I. Rubanov)

Answer: 50


The base and height of a right circular cylinder and a right circular cone are equal. The lateral surface areas of these solids are in the ratio $8: 5$. The area of the axial section of the cone is $588 \mathrm{~cm}^{2}$. What are the heights and the radii of the bases of these solids?

Answer: 28


## Task A-1.7.

Ivica and Marta have two identical sets of fifty cards with different symbols. Each of them shuffles their set of cards. Then Ivica puts his set of cards on the table, and then Marta puts her cards on top of Ivica's. Ivica then counts how many cards are between each pair of identical cards, for each of the 50 pairs of identical cards, and adds up the resulting numbers. What are all the possible results Ivica can get?

Answer: 2450


4. An equilateral triangle $T$, with side 2022, is divided by lines parallel to its sides into equilateral triangles with side 1. The triangle is covered with the figures shown in the diagrams below, each composed of 4 equilateral triangles with side 1, and during the covering, they can be rotated by an angle of $k \cdot 60^{\circ}, k=1,2,3,4,5$.

![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=130&width=103&top_left_y=252&top_left_x=540)
1
![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=144&width=528&top_left_y=250&top_left_x=722)

The covering satisfies the following conditions:

- There can be no figures of a certain type and there can be more figures of another type.
- The triangles of the figures match the triangles into which triangle $T$ is divided.
- No two figures overlap and no figure extends outside triangle $T$.
- The entire triangle $T$ is covered.

What is the smallest possible number of figures of type 1 used in such a covering?

Answer: 1011


10.121. Calculate the area of a rectangular trapezoid if its acute angle is $60^{\circ}$, the smaller base is $a$, and the larger lateral side is $b$.

Answer: \dfrac{(4a + b)b\sqrt{3}}{8}


KöMaL Survey on a total of 14 questions, with 4 possible answers for each question. The best result was 10 correct answers. What is the probability that we achieve this result by random filling out?

Answer: 3.02 \times 10^{-4}


1. How many five-digit numbers of the form $\overline{37 a b c}$ are there such that each of the numbers $\overline{37 a b c}, \overline{37 b c a}$, and $\overline{37 c a b}$ is divisible by 37?

Answer: 28


6. In the planar quadrilateral $ABCD$, $AC$ intersects $BD$ at point $O$. If
$$
\begin{array}{l}
\overrightarrow{AC} + 2 \overrightarrow{DO} = 4 \overrightarrow{AB}, \\
3 AB = 2 CD, \\
\overrightarrow{BA} \cdot \overrightarrow{BD} = 9,
\end{array}
$$

then the maximum value of the area of $\triangle ACD$ is $\qquad$.

Answer: 4\sqrt{3}


11. Given the set $A=\{1,2,3,4, \cdots, 101\}, B \subseteq A$, and the smallest element in set $B$ is an even number.
(1) If the smallest element in set $B$ is 2, and the largest element is 13, find the number of sets $B$ that satisfy the condition;
(2) Find the sum of the largest elements of all sets $B$ that satisfy the condition.

Answer: \dfrac{50(2^{102} -1)}{3}


## Task B-4.2.

The radius of the base, the slant height, and the height of a right circular cone are in some order three consecutive terms of an increasing arithmetic sequence. If the area of the axial section of the cone is $300 \mathrm{~cm}^{2}$, what is its volume?

Answer: 1500\pi


11. As shown in the figure, the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has its left focus at $F$. A line passing through point $F$ intersects the ellipse at points $A$ and $B$. When the line $AB$ passes through one of the vertices of the ellipse, its inclination angle is exactly $60^{\circ}$.
（1）Find the eccentricity of the ellipse;
（2）The midpoint of segment $AB$ is $G$, and the perpendicular bisector of $AB$ intersects the $x$-axis and $y$-axis at points $D$ and $E$, respectively. Let the area of $\triangle GDF$ be $S_{1}$, and the area of $\triangle OED$ (where $O$ is the origin) be $S_{2}$. Find the range of $\frac{S_{1}}{S_{2}}$.

Answer: (9, +\infty)


Task 2. Lazarot Petre has 151 liters of wine in a storm. Help Petre, as a friend, to give away 81 liters of wine if he has only 2 barrels that hold 51 and 91 liters.

Answer: 81


4. In the equality, digits were replaced with letters: identical digits with identical letters, and different ones with different letters. The result was

$$
Y!\cdot P!\cdot A!=K P O C C
$$

What could the original equality have looked like? Find all the options and prove that there are no others.

$N!$ denotes the product of numbers from 1 to $N$, and by definition, $0!=1$.

Answer: 5! \cdot 6! \cdot 1! = 86400


33 Find the largest positive integer that cannot be expressed as the sum of a positive multiple of 42 and a composite number.

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

Answer: 215


8,9

Through the lateral edge $P C$ of a regular triangular pyramid $A B C P$, a plane parallel to the side $A B$ of the base is drawn. The lateral edge $P A$ forms an angle $\arcsin \frac{\sqrt{2}}{3}$ with this plane. Find the angle between the lateral edge and the base plane of the pyramid.

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


Find \( x \) and \( y \) for which
\[
\begin{cases}
  x - 2y + [x] + 3\{x\} - \{y\} + 3[y] = 2.5 \\
  2x + y - [x] + 2\{x\} + 3\{y\} - 4[y] = 12
\end{cases}
\]
where \( [x] \) and \( [y] \) are the integer parts and \( \{x\} \) and \( \{y\} \) are the fractional parts of \( x \) and \( y \) respectively. The integer part of a number \( a \) is the greatest integer less than or equal to \( a \), and \( \{a\} = a - [a] \).

Answer: \left( \frac{53}{28}, -\frac{23}{14} \right)


4. If the edge length of the cube $A_{1} A_{2} A_{3} A_{4}-B_{1} B_{2} B_{3} B_{4}$ is 1, then the number of elements in the set $\left\{x \mid x=\overrightarrow{A_{1} B_{1}} \cdot \overrightarrow{A_{i} B_{j}}, i \in\right.$ $\{1,2,3,4\}, j \in\{1,2,3,4\}\}$ is $\qquad$.

Answer: 1


6. In a convex quadrilateral $A B C D$, the bisector of angle $B$ passes through the midpoint of side $A D$, and $\angle C=\angle A+\angle D$. Find the angle $A C D$. (S. Berlov)

Answer: 90


For the positive real numbers \( x \) and \( y \) the condition \( x y = 4 \) holds.

Prove that the inequality

\[
\frac{1}{x + 3} + \frac{1}{y + 3} \leq \frac{2}{5}
\]

holds. For which \( x \) and \( y \) does equality occur?

(Walther Janous)

Answer: x = 2 \text{ and } y = 2


55. A person worked for 24 consecutive days and earned 1900 yuan. He works full days from Monday to Friday, with a daily wage of 100 yuan; works half days on Saturday, with a wage of 50 yuan; and does not work on Sunday, with no wage. It is known that he started working from a certain day in late March. Given that March 1st is a Sunday, the day he finished working is April $\qquad$th.

Answer: 18


7. Let an "operation" be the act of randomly changing a known positive integer $n$ to a smaller non-negative integer (each number has the same probability). Then, the probability that the numbers $10$, $100$, and $1000$ all appear during the process of performing several operations to reduce 2019 to 0 is $\qquad$

Answer: \dfrac{1}{1112111}


Example 1.12 A and B are playing table tennis, and the final score is $20: 20$. How many scenarios are there in which A has been leading B throughout the game?

Answer: \dfrac{1}{20} \dbinom{38}{19}


## Task B-1.7.

Two Italians and three Englishmen are looking for accommodation in a hotel as loyal guests. The hotel has 8 rooms reserved for accommodating its loyal guests, of which 3 rooms have walls painted blue, 3 rooms are painted green, and the remaining 2 rooms are painted yellow. The Italians want to be in either a blue or a yellow room, while the Englishmen want to be in either a green or a yellow room. In how many ways can the two Italians and three Englishmen be assigned rooms so that each of them is alone in one room?

Answer: 660


The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells?

Answer: 1


The class teacher calculated the class average grades for each subject, and Kati helped her by recalculating the grades based on how many students received each consecutive grade. When comparing the results of the first subject, it turned out that Kati had used the data for consecutive fives in reverse order, taking the number of excellent grades as the number of poor grades. Feeling disheartened, Kati wanted to start the entire calculation over, but the class teacher said that this calculation would still be good for verification. Explain this statement.

Answer: 6


Evokimov M.A.
By the river, there lives a tribe of Mumbo-Jumbo. One day, with urgent news, a young warrior Mumbo and a wise shaman Yumbo set out simultaneously for the neighboring tribe. Mumbo ran at a speed of 11 km/h to the nearest raft storage and then sailed on a raft to the neighboring tribe. Meanwhile, Yumbo, without hurrying, walked at a speed of 6 km/h to another raft storage and then sailed to the neighboring tribe from there. In the end, Yumbo arrived before Mumbo. The river is straight, and the rafts float at the speed of the current. This speed is the same everywhere and is expressed as an integer km/h, not less than 6. What is the greatest possible value of this speed?

Answer: 26


Task 5. (5 points) On a new website, 2000 users have registered. Each of them invited 1000 (of the registered users) to become friends. Two people become friends only if each of them has invited the other. What is the minimum number of friend pairs on the site?

Answer: 1000


We say that an integer $m$ covers the number 1998 if $1,9,9,8$ appear in this order as digits of $m$. (For instance, 1998 is covered by 215993698 but not by 213326798 .) Let $k(n)$ be the number of positive integers that cover 1998 and have exactly $n$ digits $(n \geqslant 5)$, all different from 0 . What is the remainder of $k(n)$ in division by 8 ?

## Solutions

## Translation

We say that an integer $m$ covers the number 1998 if $1,9,9,8$ appear in this order as digits of $m$. (For instance, 1998 is covered by 215993698 but not by 213326798.) Let $k(n)$ be the number of positive integers that cover 1998 and have exactly $n$ digits $(n \geqslant 5)$, all different from 0. What is the remainder of $k(n)$ in division by 8?

Answer: 1


Problem 3. Determine the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which there exists a function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f(x)+f(y)=[g(x+y)]
$$

for any real numbers $x, y$.

We have denoted by $[a]$ the integer part of the real number $a$.

Answer: f(x) = \frac{k}{2}


4. Given two odd positive integers $k$ and $n$. For each two positive integers $i, j$ satisfying $1 \leq i \leq k$ and $1 \leq j \leq n$ Martin wrote the fraction $i / j$ on the board. Determine the median of all these fractions, that is a real number $q$ such that if we order all the fractions on the board by their values from smallest to largest (fractions with the same value in any order), in the middle of this list will be a fraction with value $q$.

(Martin Melicher)

Answer: \dfrac{k+1}{n+1}


Bogdanov I.I.

In a store, 21 white and 21 purple shirts are hanging in a row. Find the smallest $k$ such that for any initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts so that the remaining white shirts hang together and the remaining purple shirts also hang together.

Answer: 10


Alice and Bob want to play a game. In the beginning of the game, they are teleported to two random position on a train, whose length is  $ 1 $  km. This train is closed and dark. So they dont know where they are.

Fortunately, both of them have iPhone 133, it displays some information:
1. Your facing direction
2. Your total walking distance
3. whether you are at the front of the train
4. whether you are at the end of the train
Moreover, one may see the information of the other one. Once Alice and Bob meet, the game ends.

Alice and Bob can only discuss their strategy before the game starts. Find the least value  $ x $  so that they are guarantee to end the game with total walking distance  $ \le x $  km.

Answer: 1.5


Some of  $100$  towns of a kingdom are connected by roads.It is known that for each two towns  $A$  and  $B$  connected by a road there is a town  $C$  which is not connected by a road with at least one of the towns  $A$  and  $B$ . Determine the maximum possible number of roads in the kingdom.

Answer: 4900


Mary chose an even  $4$ -digit number  $n$ . She wrote down all the divisors of  $n$  in increasing order from left to right:  $1,2,...,\tfrac{n}{2},n$ . At some moment Mary wrote  $323$  as a divisor of  $n$ . What is the smallest possible value of the next divisor written to the right of  $323$ ? $\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$

Answer: 340


Three numbers,  $a_1$ ,  $a_2$ ,  $a_3$ , are drawn randomly and without replacement from the set  $\{1, 2, 3, \dots, 1000\}$ .  Three other numbers,  $b_1$ ,  $b_2$ ,  $b_3$ , are then drawn randomly and without replacement from the remaining set of 997 numbers.  Let  $p$  be the probability that, after a suitable rotation, a brick of dimensions  $a_1 \times a_2 \times a_3$  can be enclosed in a box of dimensions  $b_1 \times b_2 \times b_3$ , with the sides of the brick parallel to the sides of the box.  If  $p$  is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Answer: 21


For positive integer  $n$ , let  $s(n)$  denote the sum of the digits of  $n$ . Find the smallest positive integer  $n$  satisfying  $s(n)=s(n+864)=20$ .

Answer: 695


Line segment  $\overline{AB}$  is a diameter of a circle with  $AB=24$ .  Point  $C$ , not equal to  $A$  or  $B$ , lies on the circle.  As point  $C$  moves around the circle, the centroid (center of mass) of  $\triangle{ABC}$  traces out a closed curve missing two points.  To the nearest positive integer, what is the area of the region bounded by this curve? $\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$

Answer: 50


Each successive term in the sequence $4096, 1024, 256, 64, x, y, 4, 1, \frac{1}{4},...$ is obtained by multiplying the previous term by a constant until $y$, and then the pattern changes after $y$. What is the value of $x + y$ if the multiplication constant changes after $y$?

Answer: 20


Suppose I have 5 shirts, 4 pairs of pants, and 5 ties. An outfit requires a shirt and pants, and can either have a tie or not have a tie. Additionally, there are 2 different pairs of shoes to choose from. How many different outfits can I create?

Answer: 240


Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 500\}\,$. Another three numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining numbers where the sum of $b_1 + b_2 + b_3\,$ must be greater than the sum of $a_1 + a_2 + a_3$. Determine the probability $p\,$ that a brick of dimensions $a_1 \times a_2 \times a_3\,$ will fit into a box of dimensions $b_1 \times b_2 \times b_3\,$ after including the constraint on sums. If $p$ is in reduced form, what is the sum of the numerator and denominator?

Answer: 3


Let $p$ and $q$ be real numbers so that the roots of
\[ z^2 - (14 + pi)z + (48 + qi) = 0 \]
are complex conjugates. Determine the ordered pair $(p, q)$.

Answer: (0, 0)


In triangle $ABC$, where $a$ is the side length opposite $\angle A$, $b$ opposite $\angle B$, and $c$ opposite $\angle C$, the side lengths satisfy the relations $a + b + 2c = a^2$ and $a + b - 2c = -1$. Determine the largest angle of the triangle.

Answer: 120^\circ


Sandy is updating her backyard playhouse. She wants to replace siding on one exterior wall measuring 10 feet wide by 8 feet high and on two inclined roof sections, each inclined at a 30-degree angle with a base of 10 feet and an average height of 7 feet. The siding is only available in 10-foot by 10-foot sections at a cost of \$30 each. How much will the siding for these renovations cost?

Answer: \$90


Given an equilateral triangle $ABC$ and a square $BCFG$ such that they share a common side $BC$, find the measure of $\angle FAG$ where point $F$ is on the extension of side $AB$ and point $G$ is on the extension of side $AC$. Assume $F$ and $G$ are such that $BF = AB$ and $CG = AC$.

Answer: 60^\circ


An oreo shop now sells $6$ different flavors of oreos and $4$ different flavors of milk. Alpha and Beta decide to purchase some oreos and milk. Alpha will not order more than 1 of the same flavor and is willing to buy milk. Beta will only order oreos, with repeats of flavors allowed. How many ways could they have left the store with $4$ products collectively?

Answer: 2561


Tracy had a bag of candies. She first ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. Afterward, her mom and Tracy each ate 10 candies. Her cousin then took one to three candies, leaving Tracy with exactly 5 candies. How many candies did Tracy have at the beginning?

Answer: 56


Planet Zircon encircles a star in an elliptical path, where the star is at one of the focus points. The closest point (perigee) of Zircon is 3 astronomical units (AU) from the star, and the farthest point (apogee) is 10 AU away. Determine the distance from Zircon to the star when it is midway along its orbit.

Answer: 6.5 \text{ AU}


The volume of a sphere is $288\pi$ cubic inches. Find the total surface area of a spherical cap of height $h=2$ inches that sits atop this sphere, in terms of $\pi$.

Answer: 24\pi


A cutting tool is designed in the shape of a notched circle. The radius of the circle is $\sqrt{80}$ cm, with points $A$, $B$, and $C$ on its boundary such that $AB = 8$ cm and $BC = 3$ cm, forming a right angle at point $B$. Compute the square of the distance from $B$ to the center of the circle.

Answer: 80


Suppose that $x$ and $y$ are positive integers such that $(x-yi)^2 = 15 - 20i$. What is $x-yi$?

Answer: 5 - 2i


Modern widescreen television screens typically have a ratio of $16:9$. What is the horizontal length (in inches) of a "50-inch" widescreen television?

Answer: 43.6


Sarah rolls a six-sided dice 10 times. The dice is biased and has a $\frac{1}{3}$ probability of landing on six and a $\frac{2}{3}$ probability of landing on any number that is not six. What is the probability that she rolls exactly 4 sixes?

Answer: \frac{13440}{59049}


The measures of the interior angles of a triangle are in a 3:4:9 ratio. If the largest angle is decreased by $x$ degrees, and it still maintains a triangle with the sum of angles being 180 degrees, find $x$ such that the smallest angle doubles its initial value.

Answer: 33.75


In a school's Science Club, there are 18 members eligible for election to 6 different officer positions. Among these candidates, 8 have served as officers before. Determine how many different groups of officers can be elected such that at least 2 past officers are included.

Answer: 16338


Problem 9.3. Find the least number of colors with the following property: the integers $1,2 \ldots, 2004$ can be colored such that there are no integers $a<b<c$ of the same color for which $a$ divides $b$ and $b$ divides $c$.

Alexander Ivanov

Answer: 6


6. Two Skaters

Allie and Bllie are located
at points $A$ and $B$ on
a smooth ice surface, with $A$ and $B$
being 100 meters apart. If Allie starts from $A$
and skates at 8 meters/second
along a line that forms a $60^{\circ}$
angle with $A B$, while Bllie starts from $B$ at 7 meters/second
along the shortest line to meet Allie, how far will Allie have skated when they meet?

Answer: 160


7. [4] My friend and I are playing a game with the following rules: If one of us says an integer $n$, the opponent then says an integer of their choice between $2 n$ and $3 n$, inclusive. Whoever first says 2007 or greater loses the game, and their opponent wins. I must begin the game by saying a positive integer less than 10 . With how many of them can I guarantee a win?

Answer: 6


Schul D. ...

In the Family of Nameless, it is a tradition to calculate the age of the family, which is the sum of the ages (the number of full years) of the father, mother, and all the children. On December 31, the Nameless family celebrates the birthday of their family. In the year when the youngest child Vasya was born, the family was 101 years old. A few years later, the Nameless family celebrated their 150th anniversary. How many children are there in the Nameless family?

Answer: 5


Example 5.25. Calculate $e^{2}$ with an accuracy of 0.01.

Answer: 7.38


In a regular quadrilateral pyramid, there are two identical spheres of radius \( r \) touching the base of the pyramid at points belonging to the segment connecting the midpoints of opposite sides of the base. Each of the spheres touches a lateral face of the pyramid and the other sphere. Find the height of the pyramid that minimizes the volume of the pyramid.

Answer: \dfrac{5 + \sqrt{17}}{2} r


7. The "Hua Luogeng" Golden Cup Junior Mathematics Invitational Competition, the first session was held in 1986, the second session in 1988, the third session in 1991, and thereafter every 2 years. The sum of the digits of the year in which the first "Hua Cup Competition" was held is: $A 1=1+9+8+6=24$.
The sum of the digits of the years in which the first two sessions were held is: $A 2=1+9+8+6+1+9+8+8=50$.
What is the sum of the digits of the years in which the first 50 sessions of the "Hua Cup Competition" were held, $A 50=$?

Answer: 629


92. 50 chess pieces are arranged in a circle, and numbers $1, 2, 3, \cdots \cdots$ 50 are sequentially marked on the chess pieces in a clockwise direction. Then, in a clockwise direction, every other chess piece is removed until only one chess piece remains. If the number of the remaining chess piece is 42, then the first chess piece to be taken away is $\qquad$ number chess piece.

Answer: 7


A circle is inscribed in quadrilateral $ABCD$ , 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 of the radius of the circle.
Please give the answer directly without any intermediate steps.

Answer: 647


When Paulo turned 15, he invited 43 friends to a party. The cake was in the shape of a regular 15-sided polygon and had 15 candles on it. The candles were placed in such a way that no three candles were in a straight line. Paulo divided the cake into triangular pieces where each cut connected two candles or connected a candle to a vertex. Moreover, no cut crossed another already made. Explain why, by doing this, Paulo was able to give a piece of cake to each of his guests, but he himself was left without any.

#

Answer: 43


Tokaeva I.

Let $F_{1}, F_{2}, F_{3}, \ldots$ be a sequence of convex quadrilaterals, where $F_{k+1}$ (for $k=1,2,3, \ldots$) is obtained by cutting $F_{k}$ along a diagonal, flipping one of the parts, and gluing it back along the cut line to the other part. What is the maximum number of different quadrilaterals that this sequence can contain? (Different polygons are those that cannot be made to coincide by a movement.)

Answer: 6


In triangle \(ABC\), the side \(c\) is the longest and the side \(a\) is the shortest. Prove that \(l_{c} \leqslant h_{a}\).

Answer: l_c \leqslant h_a


4. Given that the center of an ellipse is at the origin, and its foci are on the coordinate axes, the line $y=x+1$ intersects the ellipse at points $P, Q$, and $O P \perp O Q,|P Q|=\frac{\sqrt{10}}{2}$. Find the equation of the ellipse.
(1991 National College Entrance Examination)

Answer: \dfrac{x^2}{2} + \dfrac{y^2}{\dfrac{2}{3}} = 1


10. Given that the product of a root of the quadratic trinomial $a x^{2}+b x+b$ and a root of the quadratic trinomial $a x^{2}+a x+b$ is equal to 1. Then the sum of the squares of these two roots is $\qquad$

Answer: 3


11. Several identical boxes are arranged in a row. Xiao Ming distributed more than fifty identical chess pieces into the boxes, with only one box left empty. Then he went out. Xiao Guang took one chess piece from each of the boxes that had chess pieces and placed them in the empty box, and then rearranged the boxes. When Xiao Ming returned, he carefully examined everything but did not notice that anyone had tampered with the boxes and chess pieces. How many boxes are there?

Answer: 11


8.2. In a convex quadrilateral $\mathrm{ABCD}$, the bisector of angle $\mathrm{B}$ passes through the midpoint of side $\mathrm{AD}$, and $\angle \mathrm{C}=\angle \mathrm{A}+\angle \mathrm{D}$. Find the angle $\mathrm{ACD}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b8f46b2c2d4464c00b2dg-1.jpg?height=274&width=423&top_left_y=2213&top_left_x=291)

Answer: 90^\circ


Archaeologists found a gold necklace made of plates in the shape of regular pentagons. Each of these plates is connected to two other plates, as illustrated in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_64d7a736a938f072abd8g-1.jpg?height=332&width=600&top_left_y=1002&top_left_x=565)

Figure 51.1

How many plates form the necklace?

Answer: 10


Let $x$ and $y$ be real numbers such that
\[\frac{\sin x}{\cos y} + \frac{\sin y}{\cos x} = 1 \quad \text{and} \quad \frac{\cos x}{\sin y} + \frac{\cos y}{\sin x} = 6.\]Compute
\[\frac{\tan x}{\tan y} + \frac{\tan y}{\tan x}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 137


Let $x_1, x_2, \dots , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \ge \frac{1}{540}$. Find the maximum value of
\[x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5 x_6 x_1 +x_6 x_1 x_2.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 559


Each of $\triangle PQR$ and $\triangle STU$ has an area of $1.$ In $\triangle PQR,$ $U,$ $W,$ and $V$ are the midpoints of the sides. In $\triangle STU,$ $R,$ $V,$ and $W$ are the midpoints of the sides. What is the area of parallelogram $UVRW?$ [asy]
draw((0,0)--(2,0)--(1,2)--cycle,linewidth(1));
draw((0,2)--(2,2)--(1,0)--cycle,linewidth(1));
label("$T$",(0,0),SW);
label("$R$",(1,0),S);
label("$S$",(2,0),SE);
label("$W$",(1.5,1),E);
label("$Q$",(2,2),NE);
label("$U$",(1,2),N);
label("$P$",(0,2),NW);
label("$V$",(0.5,1),W);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 3


Three cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 664


In the figure given, there is a point \( P \) within \(\angle M A N\), and it is known that \( \tan \angle M A N = 3 \). The distance from point \( P \) to the line \( A N \) is \( P D = 12 \) and \( A D = 30 \). A line passing through \( P \) intersects \( A N \) and \( A M \) at points \( B \) and \( C \) respectively. Find the minimum area of \(\triangle A B C\).

Answer: 624


As shown in the figure, consider the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a > b > 0\). The left and right foci of the ellipse are \(F_{1}\) and \(F_{2}\), respectively. The line passing through \(F_{2}\) intersects the ellipse at the points \(A(x_{1}, y_{1})\) and \(B(x_{2}, y_{2})\). If the area of the inscribed circle in triangle \(A F_{1} B\) is \(\pi\) and \(\left| y_{1} - y_{2} \right| = 4\), what is the eccentricity of the ellipse?

Answer: \frac{1}{2}


Inside the triangle \(ABC\), a point \(D\) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). It is known that \(AB = 15\) and \(CD = 8\). Find the length of the segment \(AD\). If necessary, round the answer to 0.01 or write the answer as a common fraction.

Answer: 3.5


Each of the 16 circles in the figure contains a student. A total of 3360 coins are distributed among the 16 students, in such a way that students equidistant from the center receive the same number of coins. The circles are arranged in groups of five, from the outside in, on the vertices of a regular pentagon, with the central circle equidistant to the vertices of one of these pentagons.

At a certain moment, each student gives all of their coins, dividing them equally among each of their neighbors. After the coin exchange, each student has the same number of coins as they had at the beginning. Determine the number of coins that the student in the central circle originally had.

Answer: 280


The 3rd term of an arithmetic sequence is 14, and the 18th term is 23. Determine which term(s) among the first 2010 terms of the sequence are integers.

Answer: 402


Let \(a, b \in \mathbf{R}\). Given that the equation \(\left(z^{2}+a z+b\right)\left(z^{2}+a z+2 b\right)=0\) has 4 distinct complex roots \(z_{1}, z_{2}, z_{3}, z_{4}\), and these roots correspond to the vertices of a square with side length 1 in the complex plane, find the value of \(\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|\).

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


Suppose \( a_{1}, \cdots, a_{10} \) and \( b_{1}, b_{2}, \cdots, b_{10} \) are sequences of distinct complex numbers. It is known that for \( i = 1,2, \cdots, 10 \), the equation \(\left(a_{1}+b_{i}\right)\left(a_{2}+b_{i}\right) \cdots \left(a_{10}+b_{i}\right) = 100\) holds. Prove that for any \( j = 1,2, \cdots, 10 \), the product \(\left(b_{1}+a_{j}\right)\left(b_{2}+a_{j}\right) \cdots \left(b_{10}+a_{j}\right)\) is equal to the same constant, and find this constant.

Answer: -100


Let \( X_{1}, X_{2}, \ldots, X_{100} \) be a sequence of mutually distinct nonempty subsets of a set \( S \). Any two sets \( X_{i} \) and \( X_{i+1} \) are disjoint and their union is not the whole set \( S \), that is, \( X_{i} \cap X_{i+1} = \emptyset \) and \( X_{i} \cup X_{i+1} \neq S \), for all \( i \in \{1, \ldots, 99\} \). Find the smallest possible number of elements in \( S \).

Answer: 8


A hemisphere with a radius of $10 \mathrm{~cm}$ protrudes from a horizontal plane (to the annoyance of drivers). A cylinder rolls on the plane and collides with the hemisphere with its side. Let $\alpha$ be the angle between the common tangent plane at the point of contact of the two bodies and the horizontal plane. What is the minimum radius of the cylinder if $\alpha \leq 30^{\circ}$?

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


Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $100m+n$.

[i]Proposed by Michael Ren[/i]

Answer: 81803


7. In triangle $A B C$, altitudes $A D, B E, C F$ are drawn. The length of side $A C$ is $1+\sqrt{3}$. The distances from the center of the inscribed circle in triangle $D E F$ to points $A$ and $C$ are $\sqrt{2}$ and 2, respectively. Find the length of side $A B$.

Answer: 2


Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$.  Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

Answer: 2\sqrt{2} + 3


Putnam 1993 Problem B3 x and y are chosen at random (with uniform density) from the interval (0, 1). What is the probability that the closest integer to x/y is even? Solution

Answer: \dfrac{5 - \pi}{4}


Let \( a_i (i = 1, 2, \cdots, p) \) be a decreasing non-negative sequence of real numbers such that \(\sum_{i=1}^{p} a_i = n \) (\(n>0\)), and \( a_i \leq M \) where \( M \) is a constant. Also, \( n = qM + r \) where \( q \in \mathbb{N}^* \) and \( 0 \leq r < M \). Prove that:
\[ \sum_{i=1}^{p} a_i^2 \leq pM^2 + r^2. \]

Answer: \sum_{i=1}^{p} a_i^2 \leq pM^2 + r^2


$1 \cdot 200$ Write the number $1234567891011 \cdots \cdots \cdot 19941995$ on the blackboard, forming the integer $N_{1}$, erase the digits in the even positions of $N_{1}$, the remaining digits form the integer $N_{2}$, erase the digits in the odd positions of $N_{2}$, the remaining digits form the integer $N_{3}$, erase the digits in the even positions of $N_{3}$, the remaining digits form the integer $N_{4}$. This process continues until only one digit remains on the blackboard. Determine this digit (Note: Count the positions from left to right, for example, in 12345, 1 is in the 1st position, 2 is in the 2nd position, and so on).

Answer: 9


Let $n,k$ be given natural numbers. Find the smallest possible cardinality of a set $A$ with the following property: There exist subsets $A_1,A_2,\ldots,A_n$ of $A$ such that the union of any $k$ of them is $A$, but the union of any $k-1$ of them is never $A$.

Answer: \dbinom{n}{k-1}


10.042. One of the angles of the trapezoid is $30^{\circ}$, and the lines containing the lateral sides of the trapezoid intersect at a right angle (Fig. 10.42). Find the length of the shorter lateral side of the trapezoid if its midline is 10 cm, and one of the bases is $8 \mathrm{~cm}$.

Answer: 2


2. Jovan and Goce were fishing with their sons. Jovan caught as many fish as his son, and Goce caught three times as many as his son. Together, they caught 25 fish. Jovan's son is named Risto. What is the name of Goce's son?

Answer: Jovan


Triangle $ABC$ has $BC = 7, CA = 8, AB = 9$. Let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively, and let $G$ be the intersection of $AD$ and $BE$. $G'$ is the reflection of $G$ across $D$. Let $G'E$ meet $CG$ at $P$, and let $G'F$ meet $BG$ at $Q$. Determine the area of $APG'Q$.

Answer: \dfrac{16\sqrt{5}}{3}


79. Oncoming Trains. At Wurzltown station, an old lady, looking out the window, shouted:

- Conductor! How long is it from here to Madville?
- All trains take 5 hours in either direction, ma'am, - he replied.
- And how many trains will I meet on the way?

This absurd question puzzled the conductor, but he readily answered:

- Trains from Wurzltown to Madville and from Madville to Wurzltown depart at five minutes past one, five minutes past two, and so on, with an interval of exactly one hour.

The old lady made one of her fellow passengers in the compartment find the answer to her question.

So how many trains will she meet on the way?

Answer: 9


5. Given that $[x]$ represents the greatest integer not exceeding $x$. Then the number of solutions to the equation
$$
3^{2 x}-\left[10 \times 3^{x+1}\right]+\sqrt{3^{2 x}-10 \times 3^{x+1}+82}=-80
$$
is

Answer: 2


9. In trapezoid $A B C D$, $A B \| C D$, and $A B$ $=92, B C=50, C D=19, A D=70$. A circle with center $P$ on $A B$ is tangent to sides $B C$ and $A D$. If $A P=\frac{m}{n}$, where $m, n$ are coprime positive integers. Find $m+n_{0} \quad$

Answer: 164


10.5. Find all pairs $(x ; y)$ of real numbers that satisfy the conditions: $x^{3}+y^{3}=1$ and $x^{4}+y^{4}=1$.

Answer: (1, 0)


For any integer \( d > 0 \), let \( f(d) \) be the smallest positive integer that has exactly \( d \) positive divisors (so for example we have \( f(1)=1 \), \( f(5)=16 \), and \( f(6)=12 \) ). Prove that for every integer \( k \geq 0 \) the number \( f(2^{k}) \) divides \( f(2^{k+1}) \).

Answer: f(2^{k}) \text{ divides } f(2^{k+1}) \text{ for all } k \geq 0


A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree?

[asy]
filldraw(circle((0,0),1),gray(0.7));
draw((0,0)--(0.81915, 0.57358));
dot((0.81915, 0.57358));
draw((0.81915, 0.57358)--(0.475006, 1.06507));
arrow((0.417649,1.14698), dir(305), 12);
draw((0,0)--(1,0),dashed);
label("$\theta$", (0.2, 0.2/3), fontsize(8));
label("$r$", (0.409575,0.28679), NW, fontsize(8));
[/asy]

[i]Problem proposed by Ahaan Rungta[/i]

Answer: 12


Exercise 6. The words of the language ababaa are sequences of characters "a" or "b" such that every letter "b" is followed by an "a". Determine the number of 6-letter words in the language ababaa.

Answer: 13


9.3. Vasya must write one digit on each face of several dice so that any ordered combination of three digits from 000 to 999 inclusive can be obtained by selecting some three different dice and placing them with the appropriate sides up in the correct order. At the same time, the digits 6 and 9 do not transform into each other when rotated 180 degrees. What is the minimum number of dice Vasya should use?

Answer: 5


Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions:
(1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$;
(2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that
$a,b\in A_r\cap A_s\cap A_t$;
(3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$.
Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.

Answer: 11


4. 199 A car departs from point $O$ along a straight highway, maintaining a constant speed $v$. At the same time the car starts, a person on a bicycle departs from a point that is a distance $a$ from $O$ and a distance $b$ from the highway, aiming to deliver a letter to the car's driver. What is the minimum speed the cyclist must travel at to achieve his goal?

Answer: \dfrac{v b}{a}


On an $n$ × $n$ board, the set of all squares that are located on or below the main diagonal of the board is called the$n-ladder$. For example, the following figure shows a $3-ladder$:
[asy]
draw((0,0)--(0,3));
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,0)--(1,3));
draw((0,2)--(2,2));
draw((2,0)--(2,2));
draw((0,3)--(1,3));
draw((3,0)--(3,1));
[/asy]
In how many ways can a $99-ladder$ be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?

Answer: 2^{98}


3. In a school from I to VIII grade, there is only one class per grade. In grades I-IV, there are a total of 130 students. In grade V, there are 7 more students than in grade II, in grade VI, there are 5 fewer students than in grade I, in grade VII, there are 10 more students than in grade IV, and in grade VIII, there are 4 fewer students than in grade I. How many students are there in grades I-VIII in total?

Answer: 268


10.8. On a circle of length 2013, 2013 points are marked, dividing it into equal arcs. A chip is placed at each marked point. We define the distance between two points as the length of the shorter arc between them. For what largest $n$ can the chips be rearranged so that there is again one chip at each marked point, and the distance between any two chips that were initially no more than $n$ apart has increased?

( D. Khramov)

Answer: 670


6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is 2, and

the lateral edge is 6.

Answer: \dfrac{15\sqrt{2}}{4}


## Task B-4.1.

If in each term of the binomial expansion $(x-y)^{n}, n \geqslant 2, x, y \neq 0$, we substitute $x=672 y$, the sum of the third and fourth terms will be equal to 0. Determine the 2018th term in the expansion of this binomial.

Answer: -1356096 y^{2018}


7. In square $ABCD$ with side length $4$, $E$ is the midpoint of $AB$, and $F$ is a moving point on side $BC$. If $\triangle ADE$ and $\triangle DCF$ are folded along $DE$ and $DF$ respectively, so that $A$ and $C$ coincide at $A'$, then the maximum distance from point $A'$ to the plane $DEF$ is $\qquad$.

Answer: \dfrac{4\sqrt{5}}{5}


(IMO 2016 P5) The equation

$$
(x-1) \times \ldots \times(x-2016)=(x-1) \times \ldots \times(x-2016)
$$

is written on the board. We want to erase some of the linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors to erase to achieve this goal.

Answer: 2016


1. The sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=1, a_{2}=3, \text { and } \\
a_{n+2}=\left|a_{n+1}\right|-a_{n}
\end{array}\left(n \in \mathbf{N}_{+}\right) .
$$

Let $\left\{a_{n}\right\}$'s sum of the first $n$ terms be $S_{n}$. Then $S_{100}=$

Answer: 89


Let \( a, b, x, \) and \( y \) be four integers such that \( a x + b y = 1 \). Show that the application \(\varphi: \mathbb{Z}^{2} \rightarrow \mathbb{Z} / a b \mathbb{Z}\) defined by \(\varphi(u, v) = u b y + v a x \) is a group homomorphism with kernel \(\operatorname{Ker}(\varphi) = a \mathbb{Z} \times b \mathbb{Z} \).

Answer: \operatorname{Ker}(\varphi) = a \mathbb{Z} \times b \mathbb{Z}


5. In a Cartesian coordinate system, points with both coordinates being integers are called integer points, and the set of all integer points is denoted as $X$. Find the largest real number $\alpha$, such that there exists a function $f: X \rightarrow \mathbf{N}^{*}$, satisfying: (1) There exists a positive real number $c$, such that for each positive integer $n$, the number of integer points $A$ satisfying $f(A)=n$ is greater than $c n^{\alpha}$; (2) For any integer point $\triangle A B C$ (i.e., $A, B, C$ are all integer points), $f(A)>|B C|, f(B)>|A C|$, $f(C)>|A B|$ at least one of these conditions holds.

Answer: 1


A circle of radius $1+\sqrt{2}$ is circumscribed around an isosceles right triangle. Find the radius of the circle that touches the legs of this triangle and internally touches the circle circumscribed around it.

Answer: 2


[ Proportional means in a right triangle ] [ Isosceles, inscribed, and circumscribed trapezoids

In a right triangle $A B C$, point $D$ is the midpoint of the height dropped to the hypotenuse $A B$. Lines symmetric to $A B$ with respect to $A D$ and $B D$ intersect at point $F$. Find the ratio of the areas of triangles $A B F$ and $A B C$.

#

Answer: \dfrac{4}{3}


3.1 Two ants are running along a circle towards each other at constant speeds. While one of them runs 9 laps, the other runs 6. At the point where the ants meet, a red dot appears. How many red dots will there be on the circle?

![](https://cdn.mathpix.com/cropped/2024_05_06_6a9b962ccc5e03d35f9eg-05.jpg?height=449&width=531&top_left_y=495&top_left_x=822)

#

Answer: 5


4. Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Answer: 5412


In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.

[asy] fill((0.5, 4.5)--(1.5,4.5)--(1.5,2.5)--(0.5,2.5)--cycle,lightgray); fill((1.5,3.5)--(2.5,3.5)--(2.5,1.5)--(1.5,1.5)--cycle,lightgray); label("$8$", (1, 0)); label("$C$", (2, 0)); label("$8$", (3, 0)); label("$8$", (0, 1)); label("$C$", (1, 1)); label("$M$", (2, 1)); label("$C$", (3, 1)); label("$8$", (4, 1)); label("$C$", (0, 2)); label("$M$", (1, 2)); label("$A$", (2, 2)); label("$M$", (3, 2)); label("$C$", (4, 2)); label("$8$", (0, 3)); label("$C$", (1, 3)); label("$M$", (2, 3)); label("$C$", (3, 3)); label("$8$", (4, 3)); label("$8$", (1, 4)); label("$C$", (2, 4)); label("$8$", (3, 4));[/asy]

Find the number of different paths.

Answer: 24


A square with side length \(x\) is inscribed in a right triangle with sides of length \(3\), \(4\), and \(5\) so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \(y\) is inscribed in another right triangle with sides of length \(3\), \(4\), and \(5\) so that one side of the square lies on the hypotenuse of the triangle. Find the value of \(\dfrac{x}{y}\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 72


Sides \(\overline{AB}\) and \(\overline{AC}\) of equilateral triangle \(ABC\) are tangent to a circle at points \(B\) and \(C\) respectively. Find the fraction of the area of \(\triangle ABC\) that lies outside the circle. The original answer is in the form \(\frac{k}{m}-\frac{n\sqrt{3}\pi}{p}\). Please give the value of \(k + m + n + p\).

Answer: 38


A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, $(2, -2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x, y)$ to each of $(x, y + 1)$, $(x + 1, y + 1)$, $(x + 1, y)$, $(x + 1, y - 1)$, $(x, y - 1)$, $(x - 1, y - 1)$, $(x - 1, y)$, or $(x - 1, y + 1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or at one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Answer: 39


How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property. Provide your answer as an integer.

Answer: 226


The vertices \(V\) of a centrally symmetric hexagon in the complex plane are given by
\[ V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}. \]
For each \(j\), \(1\leq j\leq 12\), an element \(z_j\) is chosen from \(V\) at random, independently of the other choices. Let \(P={\prod}_{j=1}^{12}z_j\) be the product of the \(12\) numbers selected. Find the probability that \(P=-1\). The original answer is in the form \(\frac{k}{m^n}\), please provide the value of \(k + m + n\).

Answer: 233


A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. Find the probability that $3$ draws are required. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 6


Example 5 A $98 \times 98$ chessboard is displayed on a computer screen, with its squares colored like a chessboard. You are allowed to select any rectangle with the mouse (the sides of the rectangle must lie on the grid lines), and then click the mouse button, which will change the color of each square in the rectangle (black to white, white to black). Find the minimum number of operations required to make all the squares on the chessboard the same color.
(1998, United States of America Mathematical Olympiad)

Answer: 98


Eliane wants to choose her schedule for swimming. She wants to attend two classes per week, one in the morning and one in the afternoon, not on the same day, nor on consecutive days. In the morning, there are swimming classes from Monday to Saturday, at $9 \mathrm{~h}$, $10 \mathrm{~h}$, and $11 \mathrm{~h}$, and in the afternoon, from Monday to Friday, at $17 \mathrm{~h}$ and $18 \mathrm{~h}$. In how many distinct ways can Eliane choose her schedule?

Answer: 96


From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition:
$f(i) \geq i-1$   $i=1,...,n$
one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies
$f(i) \leq i+1$  $i=1,...,n$
Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.

Answer: 6


When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck $n$ trees and lose 1 energy, where $n$ is her current energy, or rest (i.e. buck 0 trees) and gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed?

[i]Anderson Wang.[/i]

[hide="Clarifications"][list=1][*]The problem asks for the maximum *total* number of trees she can buck in 60 minutes, not the maximum number she can buck on the 61st minute.
[*]She does not have an energy cap. In particular, her energy may go above 100 if, for instance, she chooses to rest during the first minute.[/list][/hide]

Answer: 4293


In triangle $ABC$, side $AB=15$ and $AC=10$, $AD$ is the angle bisector of angle $A$. A line is drawn from point $D$, parallel to $AB$, until it intersects $AC$ at point $E$. Find $AE$, $EC$, and $DE$.

#

Answer: 6


Emelyanov L.A.

The median $A A_{0}$ of triangle $A B C$ is laid off from point $A_{0}$ perpendicular to side $B C$ to the exterior of the triangle. Denote the second end of the constructed segment by $A_{1}$. Similarly, points $B_{1}$ and $C_{1}$ are constructed. Find the angles of triangle $A_{1} B_{1} C_{1}$, if the angles of triangle $A B C$ are $30^{\circ}, 30^{\circ}$, and $120^{\circ}$.

Answer: 60^\circ


5. Given a tetrahedron $ABCD$ such that $AB=\sqrt{3}, AD=$ $BC=\sqrt{10}, AC=CD=BD=\sqrt{7}$. The volume of the tetrahedron is $\qquad$.

Answer: \sqrt{2}


## Task B-2.3.

In a triangle with side lengths $a, b, c$ and opposite angles $\alpha, \beta, \gamma$ respectively, the following equality holds:

$$
a^{3}+b^{3}+c^{3}=a b(a+b)-b c(b+c)+a c(a+c)
$$

If $\cos \beta=\frac{24}{25}$, determine the ratio in which the height from the vertex of angle $\alpha$ divides the opposite side.

Answer: \dfrac{49}{576}


Example 2 As shown in Figure 2, the three edges of the cube are $AB$, $BC$, and $CD$, and $AD$ is the body diagonal. Points $P$, $Q$, and $R$ are on $AB$, $BC$, and $CD$ respectively, with $AP=5$, $PB=15$, $BQ=15$, and $CR=10$. What is the area of the polygon formed by the intersection of the plane $PQR$ extended in all directions with the cube?

Answer: 525


1. We understand a palindrome as a natural number that reads the same forwards and backwards, for example, 16 261. Find the largest four-digit palindrome whose square is also a palindrome.

Answer: 2002


The people in an infinitely long line are numbered  $1,2,3,\dots$ . Then, each person says either ``Karl" or ``Lark" independently and at random. Let  $S$  be the set of all positive integers  $i$  such that people  $i$ ,  $i+1$ , and  $i+2$  all say ``Karl," and define  $X = \sum_{i \in S} 2^{-i}$ . Then the expected value of  $X^2$  is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Ankit Bisain*

Answer: 1028


7. The area of quadrilateral $\mathrm{ABCD}$ is $48 \mathrm{~cm}^{2}$ and point $\mathrm{D}$ lies on segment $\overline{\mathrm{BE}}$ such that $|E D|:|D B|=1: 2$. What is the area of triangle $\triangle \mathrm{ABC}$?

![](https://cdn.mathpix.com/cropped/2024_06_03_a89fbb865052a7f27847g-08.jpg?height=485&width=580&top_left_y=791&top_left_x=561)

Answer: 72


6. As shown in the figure, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $M, N$ are the midpoints of edges $A_{1} B_{1}, B B_{1}$ respectively. A section is made through points $D, M, N$ of the cube, and it is known that the section is a polygon $\Gamma$. Then, the cosine value of the interior angle of $\Gamma$ at vertex $D$ is $\qquad$

Answer: \dfrac{4}{13}


2.45 A monotonically increasing sequence of integers, if its first term is odd, the second term is even, the third term is odd, the fourth term is even, $\cdots$, and so on, is called an alternating sequence. The empty set is also considered an alternating sequence. Let $A(n)$ denote the number of alternating sequences, where the integers are all taken from the set $\{1,2, \cdots, n\}$. Clearly, $A(1)=2, A(2)=3$. Find $A(20)$, and prove your conclusion.

Answer: 17711


In right triangle \(ABC\), angle \(ACB\) is a right angle. Let \(E\) be the point where the angle bisector of \(\angle ABC\) intersects side \(AC\). Point \(D\) is the midpoint of side \(AB\), and \(O\) is the point of intersection of segments \(BE\) and \(CD\). A perpendicular line is drawn from point \(O\) to \(BO\) meeting side \(BC\) at point \(F\). It is given that \(FC = b\) and \(OC = \frac{3b}{2}\). Find the area of triangle \(ABC\).

Answer: \dfrac{8\sqrt{11}}{5} b^2


Example 8 A coach can travel from Nanjing to Shanghai on the Shanghai-Nanjing Expressway in $4 \mathrm{~h}$. It is known that a coach departs from Nanjing to Shanghai every half hour, and at the same time, a coach also departs from Shanghai to Nanjing. If a passenger takes a bus from Nanjing to Shanghai, how many coaches heading from Shanghai to Nanjing can he encounter during his entire journey? (Including the vehicles arriving at the Nanjing station and the vehicles leaving the Shanghai station.)

Answer: 17


In a psychology experiment, an image of a cat or an image of a dog is flashed briefly onto a screen and then Anna is asked to guess whether the image showed a cat or a dog. This process is repeated a large number of times with an equal number of images of cats and images of dogs shown. If Anna is correct $95 \%$ of the time when she guesses "dog" and $90 \%$ of the time when she guesses "cat", determine the ratio of the number of times she guessed "dog" to the number of times she guessed "cat".

Answer: \dfrac{8}{9}


8. A circle with center $O$ is inscribed in a rectangular trapezoid with bases $B C$ and $A D$, where $B C: A D=2: 3$. The lines $D O$ and $B C$ intersect at point $E$. In what ratio does the tangent to the circle, drawn from point $E$, divide the side $A D$ of the trapezoid, counting from point A.

Answer: \dfrac{2}{13}


Problem 2. Find the side length of the smallest equilateral triangle in which three disks with radii 2,3 and 4 without common inner points can be placed.

Answer: 11\sqrt{3}


$\left[\begin{array}{c}\text { Quadrilateral Pyramid }\end{array}\right]$

In the pyramid $S A B C D$, the base is a quadrilateral $A B C D$, where sides $A D$ and $B C$ are parallel, side $A B$ is 4, side $B C$ is 8, and angle $A B C$ is $60^{\circ}$. The edge $S B$ is $8 \sqrt{2}$. Find the volume of the pyramid, given that two planes can be drawn through the lines $A D$ and $B C$, not coinciding with the base of the pyramid, and intersecting the pyramid in equal quadrilaterals.

Answer: \dfrac{160\sqrt{3}}{3}


25. As shown in the figure, the sum of the numbers on the opposite faces of the cube is 7. If it is stipulated that the outside of side 2 is the front, and the cube is first flipped backward 15 times, then flipped to the right 30 times, each time flipping $90^{\circ}$, then the number on the top of the cube at this time is $\qquad$ .

Answer: 2


Problem 10.1. Find all values of the real parameter $a$ for which the equation $x^{3}-3 x^{2}+\left(a^{2}+2\right) x-a^{2}=0$ has three distinct roots $x_{1}$, $x_{2}$ and $x_{3}$ such that $\sin \left(\frac{2 \pi}{3} x_{1}\right), \sin \left(\frac{2 \pi}{3} x_{2}\right)$ and $\sin \left(\frac{2 \pi}{3} x_{3}\right)$ form (in some order) an aritmetic progression.

Answer: 0


320 Suppose there are $2 k+1$ consecutive natural numbers,
where the sum of the squares of the $k$ larger natural numbers is equal to the sum of the squares of the remaining $k+1$ smaller natural numbers (such as $5^{2}=4^{2}+3^{2}$ $\left.(k=1), 14^{2}+13^{2}=12^{2}+11^{2}+10^{2}(k=2)\right)$. For convenience, such an equation is called a "modified way". Clearly, the natural numbers $3, 4, 5, 10, 11, 12, 13, 14$ are all numbers in the modified way. The question is: Is the four-digit number 2012 a number in the modified way?

Answer: 2012


Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.

Answer: 1998001


3. [30] Let $M$ be a $2014 \times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F \in \mathcal{F}(M)$ such that $\operatorname{det}(M+F) \neq 0$.

Answer: (2013!!)^2


Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers.  Find $m-n.$ AIME 1995 Problem 1.png
Please give the answer directly without any intermediate steps.

Answer: 255


Column 1. A tour group is selecting visit locations from $A, B, C, D, E$ under the following constraints:
(1) If visiting $A$, must also visit $B$,
(2) At least one of $D, E$ must be visited,
(3) Only one of $B, C$ can be visited,
(4) Both $C, D$ must be visited or neither,
(5) If visiting $E$, both $A, D$ must be visited.
Please explain the reasoning, which places can the tour group visit at most?

Answer: 2


4. (7 points) When copying a problem, Lin Lin mistakenly wrote a repeating decimal as 0.123456. If the digits are correct but the dots indicating the repeating section were omitted, there are $\qquad$ possible original decimals.

Answer: 6


40. A TV station plans to broadcast 10 episodes of a TV series in 5 days (in order), with the possibility of not broadcasting on some days. There are $\qquad$ ways to broadcast the series.

Answer: 1001


In the parallelogram \(ABCD\), side \(AB\) is equal to 6, and the height dropped to the base \(AD\) is 3. The angle bisector of \(\angle BAD\) intersects side \(BC\) at point \(M\), with \(MC = 4\). Point \(N\) is the intersection of the bisector \(AM\) and diagonal \(BD\). Find the area of triangle \(BNM\).

Answer: \dfrac{27}{8}


1. One day, Alice found herself in a magical forest where two amazing twins, Tra-la-la and Tru-la-la, lived. Tra-la-la always lied on Mondays, Tuesdays, and Wednesdays, but told the truth all other days of the week. Tru-la-la always lied on Thursdays, Fridays, and Saturdays, but told the truth all other days of the week. One day, Alice met both brothers (it was impossible to tell who was who just by looking), who declared:

First: “I lie on Saturdays”

Second: “I will lie tomorrow”

First: “And I also lie on Sundays”

Determine which day of the week this meeting could have taken place.

Answer: Wednesday


13.19 A person tosses a coin, scoring $a$ points for heads and $b$ points for tails $(a, b$ are positive integers, $a>b$). The scores are accumulated each time. He finds that no matter how many times he tosses the coin, there are 35 scores that cannot be recorded, for example, 58 is one of them. Determine the values of $a$ and $b$.
(32nd Putnam Mathematical Competition, 1971)

Answer: 11


5. Find all values of $a$, for each of which the system

$$
\left\{\begin{array}{l}
|y|+|y-x| \leqslant a-|x-1| \\
(y-4)(y+3) \geqslant(4-x)(3+x)
\end{array}\right.
$$

has exactly two solutions.

Answer: 7


Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n\geq0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point (0,0) together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n.$ Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square. Please provide the value of the function when $n = 10$. The answer should be an integer.

Answer: 51


Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1\leq i\leq m$ and $1\leq j\leq n.$ There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1\leq i\leq m-1$ and $1\leq j\leq n-1.$ If a coin occupies the square $(i,j)$ with $i\leq m-1$ and $j\leq n-1$ and the squares $(i+1,j),(i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1).$ How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves? The original answer is in the format \binom{a}{b}, please provide the value of a+b when $m=1$ and $n=1$.

Answer: 0


For all $n\geq1$, let $$a_n=\sum_{k=1}^{n-1}\frac{\sin\left(\frac{(2k-1)\pi}{2n}\right)}{\cos^2\left(\frac{(k-1)\pi}{2n}\right)\cos^2\left(\frac{k\pi}{2n}\right)}.$$ Determine $$\lim_{n\to\infty}\frac{a_n}{n^3}.$$ The original answer is in the format $\frac{m}{\pi^{n}}$. Please provide the value of m + n.

Answer: 11


Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1.$ Iteratively, for $n=0,1,2,\ldots$, a point $x_{n+1}$ is chosen uniformly from the interval $[0,x_n].$ Let Z be the smallest value of $n$ for which $x_n<\delta.$ Find the expected value of Z, as a function of $\delta$. Please provide the value when $\delta = 10$. The answer should be an integer.

Answer: 0


Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n.$ Evaluate $M(2020). The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 4041


Determine which positive integers n have the following property: For all integers m that are relatively prime to n, there exists a permutation $\pi:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that $\pi(\pi(k))\equiv mk(\bmod n)$ for all $k\in\{1,2,\ldots,n\}$. Provide the sum of all such positive integers n.

Answer: 3


A $2 \times 3$ rectangle has vertices at $(0,0),(2,0),(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the $x$-axis is now back on the $x$-axis.) Find the area of the region above the $x$-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the form of \(\frac{m\pi}{n} + k\), please provide the value of \(m + n + k\).

Answer: 15


A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form $\frac{a\sqrt{b}-c}{d}$, please give the value of a + b + c + d.

Answer: 14


For any pair $(x, y)$ of real numbers, a sequence $\left(a_{n}(x, y)\right)_{n \geq 0}$ is defined as follows:  $$\begin{aligned} a_{0}(x, y) & =x \\ a_{n+1}(x, y) & =\frac{\left(a_{n}(x, y)\right)^{2}+y^{2}}{2}, \quad \text { for all } n \geq 0. \end{aligned}$$  Find the area of the region $\left\{(x, y) \mid\left(a_{n}(x, y)\right)_{n \geq 0}\right.$ converges $\}$. The original answer is in the format $m+n\pi$, please provide the value of m+n.

Answer: 5


Two real numbers x and y are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x / y$ is even? The original answer is in the form $(m-\pi)/n$, please provide the value of m+n.

Answer: 9


Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Please provide the value of $p + q + pq$.

Answer: 11


2. Given a quadrilateral $A B C D$. It is known that $B C=4, B D=9, A C=5, A D=3$. It is also known that at least one of the two unknown sides of the quadrilateral $A B C D$ is also an integer. Find what it can be equal to. In your answer, list all suitable values in ascending order separated by a semicolon.

Answer: 6;7;8


Arrange in ascending order those numbers that contain each of the digits $1,2, \ldots, 9$ exactly once. What number stands in the 100,000th position?

Answer: 358926471


Behind a thin lens with a focal length \( F \) and a diameter \( D \), a flat screen is placed perpendicularly to its optical axis at its focal point. A point light source is placed on the main optical axis at a distance \( d > F \) from the lens. Determine the diameter of the light spot on the screen.

Answer: \dfrac{D F}{d}


1. Given are fifty natural numbers, of which half do not exceed 50, and the other half are greater than 50 but less than 100. The difference between any two of the given numbers is not 0 or 50. Find the sum of these numbers.

Answer: 2525


Let the two branches of the hyperbola $xy=1$ be $C_{1}$ and $C_{2}$ (as shown in the figure), and the three vertices of the equilateral triangle $PQR$ lie on this hyperbola.
(1) Prove that $P, Q, R$ cannot all lie on the same branch of the hyperbola;
(2) Suppose $P(-1,-1)$ is on $C_{2}$, and $Q, R$ are on $C_{1}$. Find: the coordinates of the vertices $Q, R$.

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


11. (6 points) A car rally has two equal-distance courses. The first course starts on a flat road, 26 kilometers from the midpoint it begins to climb; 4 kilometers after passing the midpoint, it is all downhill; The second course also starts on a flat road, 4 kilometers from the midpoint it begins to descend, 26 kilometers after passing the midpoint, it is all uphill. It is known that a certain race car used the same amount of time on these two courses; the speed at the start of the second course is $\frac{5}{6}$ of the speed at the start of the first course; and when encountering an uphill, the speed decreases by $25 \%$, and when encountering a downhill, the speed increases by $25 \%$. Therefore, the distance of each course is $\qquad$ kilometers.

Answer: 92


7. Given $x, y, z \in \mathbf{R}_{+}$, and $x+y+z=6$. Then the maximum value of $x+\sqrt{x y}+\sqrt[3]{x y z}$ is $\qquad$ .

Answer: 8


3. Let $A B C D$ be a quadrilateral for which $\angle B A C=\angle A C B=20^{\circ}, \angle D C A=30^{\circ}$ and $\angle C A D=40^{\circ}$. Determine the measure of the angle $\angle C B D$.

Answer: 80


2. In a beach soccer tournament, 17 teams participate, and each team plays against each other exactly once. Teams are awarded 3 points for a win in regular time, 2 points for a win in extra time, and 1 point for a win on penalties. The losing team does not receive any points. What is the maximum number of teams that can accumulate exactly 5 points?

Answer: 11


Answer: 2


8. (10 points) Cars A and B start from locations $A$ and $B$ respectively at the same time, heading towards each other. They meet after 3 hours. Car A then turns around and heads back to $A$, while Car B continues on. After Car A reaches $A$ and turns around to head towards $B$, it meets Car B again after half an hour. How long does it take for Car B to travel from $A$ to $B$? $\qquad$ hours.

Answer: \dfrac{36}{5}


At the ends of the segment, two ones are written. In the middle between them, their sum - the number 2 - is written. Then, in the middle between each pair of adjacent written numbers, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written?

#

Answer: 1972


B2. Graphically determine the intersection of the graphs of functions $f$ and $g$ with the rules $f(x)=3^{x}-1$ and $g(x)=x^{2}-2 x$. Verify the solution by calculation.

Answer: (0, 0)


9. If $a \in A$, and $a-1 \notin A, a+1 \notin A$, then $a$ is called an isolated element of set $A$. Therefore, the number of four-element subsets of set $M=\{1,2, \cdots, 9\}$ that have no isolated elements is $\qquad$ .

Answer: 21


Answer: 4


Positive integers $x_{1}, \ldots, x_{m}$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_{1}, \ldots, F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$?

(Here $F_{1}, \ldots, F_{2018}$ are the first 2018 Fibonacci numbers: $F_{1}=F_{2}=1, F_{k+1}=F_{k}+F_{k-1}$ for $k>1$.)

Answer: 1009