In a football tournament, 15 teams participated, each of which played exactly once with all the others. A win awarded 3 points, a draw 1 point, and a loss 0 points.

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

Answer: 34


How many six-digit numbers can be formed using six different digits? Among these numbers, how many contain exactly four odd digits?

Answer: 33600


In the triangle \( ABC \), on the longest side \( BC \) with length \( b \), point \( M \) is chosen. Find the minimum distance between the centers of the circles circumscribed around triangles \( BAM \) and \( ACM \).

Answer: \frac{b}{2}


The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone.

Answer: \frac{1152}{125} \pi


Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers.

Answer: 2525


Let  $P$  be the portion of the graph of  $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$  located in the first quadrant (not including the  $x$  and  $y$  axes). Let the shortest possible distance between the origin and a point on  $P$  be  $d$ . Find  $\lfloor 1000d \rfloor$ .

*Proposed by **Th3Numb3rThr33***

Answer: 433


Find all three-digit numbers $\overline{abc}$ that satisfy the equation $\overline{abc}(a+b+c)^{3}$.

Answer: 512


There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write 2 more fours. After performing this operation several times, only 3 numbers remain on the blackboard, and it's no longer possible to perform the operation. What is the product of the remaining three numbers?

Answer: 12


Find all values of the positive integer  $k$  that has the property:
There are no positive integers  $a,b$  such that the expression  $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$  is a composite positive number.

Answer: 1


Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted?

Answer: \frac{145}{147}


In the interval $[0, \pi]$, a number $\vartheta$ is randomly selected. The probability that $\sqrt{2} \leq \sqrt{2}\cos\vartheta + \sqrt{2}\sin\vartheta \leq 2$ holds true is $\_\_\_\_\_\_\_\_\cdot$

Answer: \frac{1}{2}


Let triangle  $ABC$  be such that  $AB = AC = 22$  and  $BC = 11$ . Point  $D$  is chosen in the interior of the triangle such that  $AD = 19$  and  $\angle ABD + \angle ACD = 90^o$  . The value of  $BD^2 + CD^2$  can be expressed as  $\frac{a}{b}$  , where  $a$  and  $b$  are relatively prime positive integers. Compute  $100a + b$ .

Answer: 36104


In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees.

Answer: 75


For a positive integer  $K$ , define a sequence,  $\{a_n\}$ , as following:  $a_1 = K$  and $a_{n+1} =a_n -1$  if  $a_n$  is even $a_{n+1} =\frac{a_n - 1}{2}$  if  $a_n$  is odd ,  for all  $n \ge 1$ .
Find the smallest value of  $K$ , which makes  $a_{2005}$  the first term equal to  $0$ .

Answer: 2^{1003}-2


Find all functions  $f:\mathbb{N} \to \mathbb{N}$  so that for any distinct positive integers  $x,y,z$  the value of  $x+y+z$  is a perfect square if and only if  $f(x)+f(y)+f(z)$  is a perfect square.

Answer: f(x)=k^2x


150 people were surveyed and asked: "Do you think teal is more green or blue?" Of them, 90 believe teal is "more green," and 50 believe it's "more blue." Additionally, 40 believe it's both "more green" and "more blue." Another 20 think teal is neither "more green" nor "more blue."

How many of those 150 people believe that teal is "more blue"?

Answer: 80


A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m.

Answer: 13


A  quartic funtion  $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$  touches the line  $ y \equal{} px \plus{} q$  at  $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$  Find the area of the region bounded by these graphs in terms of  $ a,\ \alpha ,\ \beta$ .

Answer: \frac{\beta-\alpha)^5}{30}


Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^2\) and \(13^2\)?

Answer: 165


Use each of the five digits $2, 4, 6, 7$ and $9$ only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product?

Answer: 762


The following  $100$  numbers are written on the board:   $$ 2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1. $$  Alice chooses two numbers  $a,b,$  erases them and writes the number  $\dfrac{ab - 1}{a+b+2}$  on the board. She keeps doing this until a single number remains on the board.

If the sum of all possible numbers she can end up with is  $\dfrac{p}{q}$  where  $p, q$  are coprime, then what
is the value of  $\log_{2}(p+q)$ ?

Answer: 100


Let  $a_1,a_2,a_3,...$  be a sequence of positive real numbers such that:

(i) For all positive integers  $m,n$ , we have  $a_{mn}=a_ma_n$ (ii) There exists a positive real number  $B$  such that for all positive integers  $m,n$  with  $m<n$ , we have  $a_m < Ba_n$ Find all possible values of  $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$

Answer: 0


A group has elements \( g \) and \( h \) satisfying:
- \( ghg = hg^2h \),
- \( g^3 = 1 \),
- \( h^n = 1 \) for some odd \( n \).

Prove \( h = 1 \).

Answer: 1


Find all prime numbers that are equal to the sum of two prime numbers and the difference of two prime numbers.

Answer: 5


A sequence of \( n \) positive integers is considered full if for each \( k > 1 \), \( k \) occurs only if \( k-1 \) occurs before the last occurrence of \( k \). How many full sequences are there for each \( n \)?

Answer: n!


Calculate the value of $V_3$ for the polynomial $f(x) = 2x^6 + 5x^5 + 6x^4 + 23x^3 - 8x^2 + 10x - 3$ at $x = -4$ using the Horner's method.

Answer: -49


In triangle \( \triangle ABC \), it is given that \( \overrightarrow{AB} \cdot \overrightarrow{AC} + 2 \overrightarrow{BA} \cdot \overrightarrow{BC} = 3 \overrightarrow{CA} \cdot \overrightarrow{CB} \).

(1) Let the lengths of \( BC \), \( CA \), and \( AB \) be \( a \), \( b \), and \( c \) respectively. Prove that \( a^2 + 2b^2 = 3c^2 \).

(2) Find the minimum value of \( \cos C \).

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


Lynne chooses four distinct digits from 1 to 9 and arranges them to form the 24 possible four-digit numbers. These 24 numbers are added together giving the result \(N\). For all possible choices of the four distinct digits, what is the largest sum of the distinct prime factors of \(N\)?

Answer: 146


Point  $T$  lies on the bisector of  $\angle B$  of acuteangled  $\triangle ABC$ . Circle  $S$  with diameter  $BT$  intersects  $AB$  and  $BC$  at points  $P$  and  $Q$ . Circle, that goes through point  $A$  and tangent to  $S$  at  $P$  intersects line  $AC$  at  $X$ .  Circle, that goes through point  $C$  and tangent to  $S$  at  $Q$  intersects line  $AC$  at  $Y$ . Prove, that  $TX=TY$

Answer: TX=TY


In a row, there are 99 people - knights and liars (knights always tell the truth, and liars always lie). Each of them said one of two phrases: "To my left, there are twice as many knights as liars" or "To my left, there are as many knights as liars." In reality, there were more knights than liars, and more than 50 people said the first phrase. How many liars said the first phrase?

Answer: 49


Given \( a, b \in \mathbb{R}^+, \) and the equation \( a \sqrt{1 - b^2} - b \sqrt{1 - a^2} = ab \), find the maximum possible value of \( \frac{a}{b} + \frac{b}{a} \).

Answer: \sqrt{5}


Let $ABC$ be an equilateral triangle. Denote the distances from a point $P$ inside the triangle to the vertices as $x$, $y$, and $z$, and the distances to the sides of the triangle as $u$, $v$, and $w$. Prove that

$$
x + y + z \geq 2(u + v + w)
$$

Can there be a case where the equality holds?

Answer: x+y+z\ge2(u+v+w)


Given a point P on the hyperbola $C_1: \frac{x^2}{16} - \frac{y^2}{9} = 1$, a point Q on the circle $C_2: (x - 5)^2 + y^2 = 1$, and a point R on the circle $C_3: (x + 5)^2 + y^2 = 1$, find the maximum value of $|PQ| - |PR|$.

Answer: 10


Suppose the state of Georgia uses a license plate format "LLDLLL", and the state of Nebraska uses a format "LLDDDDD". Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Nebraska issue than Georgia?

Answer: 21902400


How many distinct arrangements of the letters in the word "basics" are there, specifically those beginning with a vowel?

Answer: 120


Find all positive integers  $n$  which satisfy the following tow conditions:
(a)  $n$  has at least four different positive divisors;
(b) for any divisors  $a$  and  $b$  of  $n$  satisfying  $1<a<b<n$ , the number  $b-a$  divides  $n$ .

*(4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)*

Answer: 6,8,12


The volume of a regular tetrahedron \( ABCD \) is 1, and \( O \) is its center. The regular tetrahedron \( A'B'C'D' \) is symmetrical to \( ABCD \) with respect to the point \( O \). Determine the volume of the common part of these two regular tetrahedrons.

Answer: \frac{1}{2}


In a tetrahedron $ABCD$ with volume 12, let $E$, $F$, and $G$ be points on edges $AB$, $BC$, and $AD$ respectively, such that $AE = 2EB$, $BF = FC$, and $AG = 2GD$. A section is made through points $E$, $F$, and $G$ forming cross-section $EFHG$. Given that the distance from point $C$ to this section is 1, find the area of this section.

Answer: 7


Given the function $y=\cos \left( 2x+\dfrac{\pi}{3} \right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\cos 2x$.

Answer: \dfrac{\pi}{6}


(1) If \( n \) is a positive integer, then \(\mathrm{C}_{2 n}^{n}=\frac{(2 n)!}{(n!)^{2}}\) is a positive integer that can be divided by a prime \( p \) where \( n < p \leqslant 2n \), and \(\mathrm{C}_{2 n}^{n}<2^{2 n}\). Prove this.

(2) Let \(\pi(x)\) denote the number of primes \( p \) less than or equal to \( x \), where \( x \) is a positive real number (e.g. \(\pi(10)=4\)). Prove that if \( n \geqslant 3 \) (where \( n \in \mathbb{N} \)), then:

1. \(\pi(2n) < \pi(n) + \frac{2n}{\log_{2} n}\);
2. \(\pi(2^n) < \frac{2^{n+1} \log_{2}(n-1)}{n}\);
3. For all real numbers \( x \geqslant 8 \), \(\pi(x) < \frac{4x \log_{2}(\log_{2} x)}{\log_{2} x}\).

Answer: \pi(x)<\frac{4x\log_2(\log_2x)}{\log_2x}


Given an ellipse M: $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ (where $a>b>0$) whose eccentricity is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and the major axis of the ellipse is 4.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and P$(1, \sqrt {2})$ is a point on ellipse M, find the maximum area of $\triangle PAB$.

Answer: \sqrt {2}


Answer: 16


A line passes through the center of a regular $n$-gon inscribed in a unit circle. Find the sum of the squares of the distances from the vertices of the $n$-gon to this line.

Answer: \frac{n}{2}


On a number line, there are three points A, B, and C which represent the numbers -24, -10, and 10, respectively. Two electronic ants, named Alpha and Beta, start moving towards each other from points A and C, respectively. Alpha moves at a speed of 4 units per second, while Beta moves at a speed of 6 units per second.

(1) At which point on the number line do Alpha and Beta meet?
(2) After how many seconds will the sum of Alpha's distances to points A, B, and C be 40 units? If at that moment Alpha turns back, will Alpha and Beta meet again on the number line? If they can meet, find the meeting point; if they cannot, explain why.

Answer: -44


An acute angle at the base of a trapezoid inscribed in a circle with a radius of 13 is $30^{\circ}$. The leg (non-parallel side) of the trapezoid is 10. Find the median (the midsegment) of the trapezoid.

Answer: 12


Among all fractions of the form \(\frac{m}{n}\), where \(m\) and \(n\) are four-digit numbers with the same sum of digits, the largest one was chosen. Find it and write your answer as an improper fraction (do not round or simplify it!).

Answer: \frac{9900}{1089}


Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,$ and $BC=6$. What is $AC$?
$\textbf{(A)}\; 3+\sqrt{5} \qquad\textbf{(B)}\; 6 \qquad\textbf{(C)}\; \dfrac{9}{2}\sqrt{2} \qquad\textbf{(D)}\; 8-\sqrt{2} \qquad\textbf{(E)}\; 7$

Answer: \textbf{(B)}\;6


When $(a-b)^n,n\ge2,ab\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals:

Answer: 2k+1


How many triangles can be formed using the vertices of a regular hexacontagon (a 60-sided polygon), avoiding the use of any three consecutive vertices in forming these triangles?

Answer: 34160


Let  $m$  and  $n$  be positive integers.  Mr. Fat has a set  $S$  containing every rectangular tile with integer side lengths and area of a power of  $2$ .  Mr. Fat also has a rectangle  $R$  with dimensions  $2^m \times 2^n$  and a  $1 \times 1$  square removed from one of the corners.  Mr. Fat wants to choose  $m + n$  rectangles from  $S$ , with respective areas  $2^0, 2^1, \ldots, 2^{m + n - 1}$ , and then tile  $R$  with the chosen rectangles.  Prove that this can be done in at most  $(m + n)!$  ways.

*Palmer Mebane.*

Answer: (+n)!


We have a set of  ${221}$  real numbers whose sum is  ${110721}$ . It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to  ${2004}$ .

Answer: 2004


There are two ways of choosing six different numbers from the list \( 1,2,3,4,5,6,7,8,9 \) so that the product of the six numbers is a perfect square. Suppose that these two perfect squares are \( m^{2} \) and \( n^{2} \), with \( m \) and \( n \) positive integers and \( m \neq n \). What is the value of \( m+n \)?

Answer: 108


Let \( P, Q, R, S \) be four points inside the tetrahedron \( A-BCD \), where \( Q, R, S, P \) are the midpoints of \( PA, QB, RC, SD \) respectively. If \( V_{P-ABC} \) denotes the volume of the tetrahedron \( P-ABC \), and similarly for the others, then what is the ratio \( V_{P-ABC} : V_{P-BCD} : V_{P-DAB} : V_{P-DAC} = \) ?

Answer: 8:1:4:2


The circle \(\gamma_{1}\) centered at \(O_{1}\) intersects the circle \(\gamma_{2}\) centered at \(O_{2}\) at two points \(P\) and \(Q\). The tangent to \(\gamma_{2}\) at \(P\) intersects \(\gamma_{1}\) at the point \(A\) and the tangent to \(\gamma_{1}\) at \(P\) intersects \(\gamma_{2}\) at the point \(B\), where \(A\) and \(B\) are distinct from \(P\). Suppose \(PQ \cdot O_{1}O_{2} = PO_{1} \cdot PO_{2}\) and \(\angle APB\) is acute. Determine the size of \(\angle APB\) in degrees.

Answer: 30


Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey?

Answer: 11


Let $M=123456789101112\dots5253$ be the $96$-digit number that is formed by writing integers from $1$ to $53$ in order. What is the remainder when $M$ is divided by $55$?

Answer: 53


Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the service in this online store.

Answer: 0.64


Among all the simple fractions with a numerator and denominator that are two-digit numbers, find the smallest fraction greater than $\frac{3}{4}$. Provide its numerator in the answer.

Answer: 73


Let \( (a_1, a_2, \dots, a_{12}) \) be a list of the first 12 positive integers such that for each \( 2 \le i \le 12 \), either \( a_i+1 \) or \( a_i-1 \) or both appear somewhere before \( a_i \) in the list. Determine the number of such lists.

Answer: 2048


Compute the product of the sums of the squares and the cubes of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 1 = 0,\] given that all roots are real and nonnegative.

Answer: 13754


A set  $ X$  of positive integers is called *nice* if for each pair  $ a$ ,  $ b\in X$  exactly one of the numbers  $ a \plus{} b$  and  $ |a \minus{} b|$  belongs to  $ X$  (the numbers  $ a$  and  $ b$  may be equal). Determine the number of nice sets containing the number 2008.

*Author: Fedor Petrov*

Answer: 8


Find the largest value of the expression   $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$  , where  $p,R, r$  is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.

Answer: \frac{5\sqrt{3}}{2}


Given an integer \( n>1 \), let \( a_{1}, a_{2}, \cdots, a_{n} \) be distinct non-negative real numbers. Define the sets
\[ A=\left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}, \]
\[ B=\left\{a_{i} a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}. \]
Find the minimum value of \( \frac{|A|}{|B|} \), where \( |X| \) denotes the number of elements in the set \( X \).

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


In the diagram, all rows, columns, and diagonals have the sum 12. Find the sum of the four corner numbers.

Answer: 16


A motorcyclist traveled from point $A$ to point $B$, which is 120 km away from $A$. On his way back, he set off at the same speed but had to stop for 10 minutes an hour after starting. After this stop, he continued his journey to $A$ with an increased speed of 6 km/h. What was the motorcyclist's initial speed, given that the return trip took the same amount of time as the trip from $A$ to $B$?

Answer: 48


Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to the volume of Bonnie's prism. Find the ratio of the total length of Bonnie's wire to Roark's wire. Express your answer as a common fraction.

Answer: \frac{1}{16}


In how many ways can you form 5 quartets from 5 violinists, 5 violists, 5 cellists, and 5 pianists?

Answer: (5!)^3


The largest four-digit number whose digits add to 17 is 9800. The 5th largest four-digit number whose digits have a sum of 17 is:

Answer: 9611


Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immediately return and meet again at point C. Given that A walks 60 meters per minute and B walks 80 meters per minute, how far apart are points A and B?

Answer: 1680


The trinomial \( a x^{2} + b x + c \) for all integers \( x \) is an exact fourth power. Prove that then \( a = b = 0 \).

Answer: 00


Denote by  $\mathbb{Q}^+$  the set of positive rational numbers. A function  $f : \mathbb{Q}^+ \to \mathbb{Q}$  satisfies

•  $f(p) = 1$  for all primes  $p$ , and

•  $f(ab) = af(b) + bf(a)$  for all  $ a,b \in \mathbb{Q}^+ $ .

For which positive integers  $n$  does the equation  $nf(c) = c$  have at least one solution  $c$  in  $\mathbb{Q}^+$ ?

Answer: n


Although I am certain that my clock is 5 minutes fast, it is actually 10 minutes slow. On the other hand, my friend's clock is really 5 minutes fast, even though he thinks it is correct. We scheduled a meeting for 10 o'clock and plan to arrive on time. Who will arrive first? After how much time will the other arrive?

Answer: 20


A set A, consisting of certain natural numbers from 1 to 49, is called dense if it contains more than 40 numbers and does not contain any six consecutive numbers. How many dense sets exist?

Answer: 495


Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes?

Answer: 26


Michael has 16 white socks, 3 blue socks, and 6 red socks in a drawer. He randomly takes socks from the drawer (one at a time) until he has a pair of the same color. What is the largest number of socks he could possibly withdraw in this fashion?

Answer: 4


Blind boxes are a new type of product. Merchants package different styles of products from the same series in boxes with the same appearance, so that consumers do not know which style of product they are buying. A merchant has designed three types of dolls, $A$, $B$, and $C$, in the same series, and sells them in blind boxes. It is known that the production ratio of the three types of dolls $A$, $B$, and $C$ is $6:3:1$. Using frequency to estimate probability, calculate the probability that a consumer randomly buys $4$ blind boxes at once and finds all three types of dolls inside.

Answer: 0.216


The distance between the centers of non-intersecting circles is $a$. Prove that the intersection points of the common external tangents with the common internal tangents lie on one circle, and find its radius.

Answer: \frac{}{2}


Determine the maximum value of  $m^{2}+n^{2}$ , where  $m$  and  $n$  are integers satisfying  $m,n\in \{1,2,...,1981\}$  and  $(n^{2}-mn-m^{2})^{2}=1.$

Answer: 3524578


Suppose we have  $10$  balls and  $10$  colors. For each ball, we (independently) color it one of the  $10$  colors, then group the balls together by color at the end. If  $S$  is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of  $S$  written as a decimal.

*Proposed by Michael Kural*

Answer: 55


A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) with a base \( A B C \) and lateral edges \( A A_{1}, B B_{1}, C C_{1} \) is inscribed in a sphere of radius 6. The segment \( C D \) is the diameter of this sphere. Find the volume of the prism if \( A D = 4 \sqrt{6} \).

Answer: 48\sqrt{15}


There are four piles of stones: one with 6 stones, two with 8 stones, and one with 9 stones. Five players numbered 1, 2, 3, 4, and 5 take turns, in the order of their numbers, choosing one of the piles and dividing it into two smaller piles. The loser is the player who cannot do this. State the number of the player who loses.

Answer: 3


A large $5 \times 5 \times 5$ cube is formed using 125 small $1 \times 1 \times 1$ cubes. There are three central columns, each passing through the small cube at the very centre of the large cube: one from top to bottom, one from front to back, and one from left to right. All of the small cubes that make up these three columns are removed. What is the surface area of the resulting solid?

Answer: 192


The alphabet contains \( n > 1 \) letters; a word is any finite sequence of letters in which any two neighboring letters are different. A word is called good if it is not possible to delete all letters except for four such that the remaining sequence is of the form \( a a b b \), where \( a \) and \( b \) are different letters. Find the maximum possible number of letters in a good word.

(D. Khramiov)

Answer: 2n+1


In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $a = b\cos C + c\sin B$.
(1) Find angle $B$.
(2) If $b = 4$, find the maximum area of triangle $ABC$.

Answer: 4\sqrt{2} + 4


Compute the sum of all positive integers  $n$  such that  $n^n$  has 325 positive integer divisors. (For example,  $4^4=256$  has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)

Answer: 93


A circular disk is divided by $2n$ equally spaced radii($n>0$) and one secant line. The maximum number of non-overlapping
areas into which the disk can be divided is
$\textbf{(A) }2n+1\qquad \textbf{(B) }2n+2\qquad \textbf{(C) }3n-1\qquad \textbf{(D) }3n\qquad  \textbf{(E) }3n+1$

Answer: \textbf{(E)}3n+1


Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated once each (meaning two pairs of identical letters), and digits can be repeated?

Answer: 390000


Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______.

Answer: 288


Let $a_{1}, a_{2}, \cdots, a_{6}$ be any permutation of $\{1,2, \cdots, 6\}$. If the sum of any three consecutive numbers is not divisible by 3, how many such permutations exist?

Answer: 96


Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation
\[x(x-y)+y(y-z)+z(z-x) = 1?\]
$\textbf{(A)} \: x>y$ and $y=z$
$\textbf{(B)} \: x=y-1$ and $y=z-1$
$\textbf{(C)} \: x=z+1$ and $y=x+1$
$\textbf{(D)} \: x=z$ and $y-1=x$
$\textbf{(E)} \: x+y+z=1$

Answer: \textbf{(D)}


Find the number of ordered pairs $(x,y)$ of real numbers such that
\[16^{x^2 + y} + 16^{x + y^2} = 1.\]

Answer: 1


Given a circle $\mathrm{e}O: x^{2}+y^{2}=r^{2}$ and a fixed line $l: x=a$. Let $M$ be a moving point on $l$. $A_{1}$ and $A_{2}$ are the two intersection points of $\mathrm{e}O$ and the x-axis. Lines $M A_{1}$ and $M A_{2}$ intersect the circle $\mathrm{e}O$ again at points $P$ and $Q$, respectively. Prove that the line PQ passes through a fixed point.

Answer: (\frac{r^2}{},0)


Let  $n$  be a positive integer. Each of the numbers  $1,2,3,\ldots,100$  is painted with one of  $n$  colors in such a way that two distinct numbers with a sum divisible by  $4$  are painted with different colors. Determine the smallest value of  $n$  for which such a situation is possible.

Answer: 25


Initially, the numbers $1, 2, 3, \cdots, 2013$ are written on a blackboard. In each step, you can erase two numbers $a$ and $b$ from the blackboard and write $a + b$. What is the minimum number of steps required to ensure that the sum of any remaining numbers on the blackboard is not equal to 2014?

Answer: 503


Points \( A, B, C \) in the plane satisfy \( \overline{AB} = 2002 \) and \( \overline{AC} = 9999 \). The circles with diameters \( AB \) and \( AC \) intersect at \( A \) and \( D \). If \( \overline{AD} = 37 \), what is the shortest distance from point \( A \) to line \( BC \)?

Answer: 37


Our next object up for bid is an arithmetic progression of primes. For example, the primes 3,5, and 7 form an arithmetic progression of length 3. What is the largest possible length of an arithmetic progression formed of positive primes less than 1,000,000? Be prepared to justify your answer.

Answer: 12


Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$?

Answer: 572


Let  $ABC$  be a triangle with  $\angle A = 60^o$ . Line  $\ell$  intersects segments  $AB$  and  $AC$  and splits triangle  $ABC$  into an equilateral triangle and a quadrilateral. Let  $X$  and  $Y$  be on  $\ell$  such that lines  $BX$  and  $CY$  are perpendicular to ℓ. Given that  $AB = 20$  and  $AC = 22$ , compute  $XY$  .

Answer: 21


Suppose that each of the 5 persons knows a piece of information, each piece is different, about a certain event. Each time person  $A$  calls person  $B$ ,  $A$  gives  $B$  all the information that  $A$  knows at that moment about the event, while  $B$  does not say to  $A$  anything that he knew.

(a) What is the minimum number of calls are necessary so that everyone knows about the event?
(b) How many calls are necessary if there were  $n$  persons?

Answer: 2n-2


Team A and Team B each has 7 players who will compete in the predetermined order of a Go competition. The first players from each team compete first, and the loser is eliminated while the winner continues to compete against the next player from the losing team, and so on, until all members of one team are eliminated. How many possible sequences of matches are there until one team is completely eliminated?

Answer: 3432


A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly.

Answer: 1001001


In how many ways can the integers from  $1$  to  $n$  be ordered subject to the condition that, except for the first integer on the left, every integer differs by  $1$  from some integer to the left of it?

Answer: 2^{n-1}


Given a hyperbola with its left and right foci being $F_1$ and $F_2$ respectively, and the length of chord $AB$ on the left branch passing through $F_1$ is 5. If $2a=8$, calculate the perimeter of $\triangle ABF_2$.

Answer: 26


Among six different quadratic trinomials, which differ by permutations of coefficients, what is the maximum number that may have no roots?

Answer: 6


Among all triangles with the sum of the medians equal to 3, find the triangle with the maximum sum of the heights.

Answer: 3


The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means "less than";
the symbol $>$ means "greater than."
The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that:

Answer: $-1<x<7$


In triangle \( \triangle ABC \), \( AB = 1 \), \( AC = 2 \), and \( \cos B = 2 \sin C \). What is the length of \( BC \)?

Answer: \frac{\sqrt{2}+\sqrt{14}}{2}


The quadratic function \( f(x) = ax^2 + bx + c \), where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \), satisfies the following conditions:
1. \( f(-1) = 0 \);
2. When \( x \in \mathbf{R} \), \( f(x) \geq x \);
3. When \( x \in (0, 2) \), \( f(x) \leq \frac{(x+1)^2}{4} \).

Given \( x_1, x_2, x_3 \in (0, 2) \) such that \( \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} = 3 \), find the minimum value of \( f(x_1) f(x_2) f(x_3) \).

Answer: 1


Adia writes a list in increasing order of the integers between 1 and 100, inclusive, that cannot be written as the product of two consecutive positive integers. What is the 40th integer in her list?

Answer: 46


Answer: \text{B}


Let $n > 1$ be an integer. If $x_{1}, \ldots, x_{n}$ are the roots of the polynomial $X^{n} + a_{1} x^{n-1} + \cdots + a_{n-1}$ and $y_{1}, \ldots, y_{n-1}$ are the roots of the derived polynomial $n x^{n-1} + (n-1) a_{1} X^{n-2} + \cdots + 2 a_{n-2} X + a_{n-1}$, show that:

$$
\frac{1}{n} \sum_{k=1}^{n} x_{k}^{2}>\frac{1}{n-1} \sum_{k=1}^{n} y_{k}^{2}
$$

Answer: \frac{1}{n}\sum_{k=1}^nx_k^2>\frac{1}{n-1}\sum_{k=1}^{n-1}y_k^2


The number of integers \(N\) from 1 to 1990 for which \(\frac{N^{2}+7}{N+4}\) is not a reduced fraction is:

Answer: 86


Let $A$, $B$, and $C$ be three points on a line. Let $\Gamma_{1}$ and $\Gamma_{2}$ be the circles with diameters $[AB]$ and $[AC]$, respectively. Let $\omega_{0}$ be the circle with diameter $[BC]$. Then, let $\omega_{n}$ be the circle tangent to $\Gamma_{1}$, $\Gamma_{2}$, and $\omega_{n-1}$ that is not $\omega_{n-2}$. Let $r_{n}$ be the radius of $\omega_{n}$ and let $d_{n}$ be the distance from the center of $\omega_{n}$ to the line $(AC)$. Show that $r_{n}=2n \cdot d_{n}$.

Answer: r_{n}=2n\cdotd_{n}


The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5,1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?

Answer: 4 \sqrt{5}


Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:

First method:

a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:

\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.48 \text{ (billion rubles)} \]

b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{J P Y} = \frac{478.48}{1388.01} \approx 34.47\% \]

c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:

\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]

Second method:

a) Determine the share of the euro in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{E U R} = \frac{41.89}{1388.01} \approx 3.02\% \]

b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{J P Y} = 100 - 3.02 - 0.2 - 22.31 - 39.98 - 0.02 = 34.47\% \]

c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:

\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]

Answer: -12.6


Given that the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfy \( \frac{z_{3}-z_{1}}{z_{2}-z_{1}} = a \mathrm{i} \) where \( a \) is a non-zero real number (\( a \in \mathbf{R}, a \neq 0 \)), find the angle between the vectors \( \overrightarrow{Z_{1} Z_{2}} \) and \( \overrightarrow{Z_{1} Z_{3}} \).

Answer: \frac{\pi}{2}


An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions.

Answer: 2000


A rook traversed an \$n\$×\$n\$ chessboard, visiting each square exactly once, with each move being exactly one square. The squares are numbered from 1 to \$n^2\$ in the order of the rook's traversal. Let \$ M \$ be the maximum difference between the numbers of adjacent (by side) squares. What is the smallest possible value of \$ \mathrm{M} \$?

Answer: 2n-1


Given that the function \( f(x) \) is a decreasing function on \( \mathbf{R} \) and is an odd function. If \( m \) and \( n \) satisfy the system of inequalities \(\left\{\begin{array}{l}f(m)+f(n-2) \leqslant 0, \\ f(m-n-1) \leqslant 0,\end{array}\right.\) then the range of \( 5m-n \) is \(\qquad\).

Answer: [7,+\infty)


A solitaire game is played on an \( m \times n \) rectangular board, using \( mn \) markers that are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one can take away one marker with its black side up, but must then turn over all markers that are in squares having an edge in common with the square of the removed marker. Determine all pairs \( (m, n) \) of positive integers such that all markers can be removed from the board.

Answer: 1


Let  $ABCD$  be a square with side length  $2$ , and let a semicircle with flat side  $CD$  be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle?

Answer: 4 - 2\sqrt{3}


Let $N$ be the number of consecutive $0$'s at the right end of the decimal representation of the product $1!2!3!4!\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$.

Answer: 124


The numbers from 1 to 9 are divided into three groups of three numbers, and then the numbers in each group are multiplied. $A$ is the largest of the three products. What is the smallest possible value of $A$?

Answer: 72


How many ways are there to arrange the numbers \(\{1,2,3,4,5,6,7,8\}\) in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.

Answer: 36


Points \( K \) and \( N \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) respectively, such that \( AK = BK \) and \( AN = 2NC \).

In what ratio does the segment \( KN \) divide the median \( AM \) of triangle \( ABC \)?

Answer: 4:3


Let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be the three interior angles of $\triangle \mathrm{ABC}$. Find the maximum value of the real number $\lambda$ such that $\frac{1}{\sin A}+\frac{1}{\sin B} \geq \frac{\lambda}{3+2 \cos C}$ always holds.

Answer: 8


In the number \(2 \_ 0 \_ 1 \_ 6 \_ 0 \_ 2 \_\), each of the 6 stars needs to be replaced with any of the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) (digits can be repeated) so that the resulting 12-digit number is divisible by 18. How many ways can this be done?

Answer: 26244


A village of knights and liars is arranged on a grid in the shape of a $9 \times 9$ square, with each cell containing one person - either a knight or a liar. Knights always tell the truth, while liars always lie. Cells are considered neighbors if they share a side or a corner. Each inhabitant stated: "Among my neighbors, there is an odd number of liars." Is the total number of liars in the village even or odd?

Answer: odd


A secret base is surrounded by a transparent winding fence in the shape of a non-convex polygon, and there is a swamp outside. A straight power line with 36 poles runs through the swamp, with some poles outside the base and some inside. (The power line does not pass through the vertices of the fence.) A spy walks around the base outside along the fence so that the fence is always to his right. Each time he ends up on the power line, he counts how many poles are on his left (he can see them all). By the time the spy has walked around the entire fence, he has counted a total of 2015 poles. How many poles are inside the base?

Answer: 1


Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by
$\textbf{(A)}\ p\%\qquad \textbf{(B)}\ \frac{p}{1+p}\%\qquad \textbf{(C)}\ \frac{100}{p}\%\qquad \textbf{(D)}\ \frac{p}{100+p}\%\qquad \textbf{(E)}\ \frac{100p}{100+p}\%$

Answer: \textbf{(E)}\\frac{100p}{100+p}


One morning at 9:00 AM, a pedestrian named Fedya left the village of Fedino and headed towards the village of Novoverandovo. At the same time, a cyclist named Vera left Novoverandovo heading towards Fedya. It is known that by the time they met, Fedya had covered one-third of the distance between the villages. However, if Fedya had left an hour earlier, he would have covered half the distance by the time they met. At what time did Fedya and Vera meet? Assume that the speeds of Vera and Fedya are constant.

Answer: 11:00


Let  $Z$  denote the set of points in  $\mathbb{R}^{n}$  whose coordinates are  $0$  or  $1.$  (Thus  $Z$  has  $2^{n}$  elements, which are the vertices of a unit hypercube in  $\mathbb{R}^{n}$ .) Given a vector subspace  $V$  of  $\mathbb{R}^{n},$  let  $Z(V)$  denote the number of members of  $Z$  that lie in  $V.$  Let  $k$  be given,  $0\le k\le n.$  Find the maximum, over all vector subspaces  $V\subseteq\mathbb{R}^{n}$  of dimension  $k,$  of the number of points in  $V\cap Z.$

Answer: 2^k


Ahmad walks up a flight of eight steps, going up either one or two steps at a time. There is a hole on the sixth step, so he cannot use this step. In how many different ways can Ahmad reach the top step?

Answer: 8


Given four positive constants \( a, b, c, \) and \( d \), find the minimum value of \( c x + d y^{2} \) when the real numbers \( x \) and \( y \) satisfy \( a x^{2} + b y^{2} = 1 \).

Answer: -\frac{}{\sqrt{}}


A sequence is defined as follows: $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$.  Given that $a_{28}= 6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum_{k=1}^{28}a_k$ is divided by 1000.

Answer: 834


Given that a school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible.

Answer: 90


Given that point \( P \) lies on the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of \( P \).

Answer: -\frac{64}{5}


A hotelier wanted to equip the dining room with new chairs. In the catalog, he chose a type of chair. When placing the order, he learned from the manufacturer that as part of a discount event, every fourth chair is offered at half price and that, therefore, he could save the equivalent cost of seven and a half chairs from the original plan. The hotelier calculated that for the originally planned amount, he could purchase nine more chairs than he intended.

How many chairs did the hotelier originally want to buy?

(L. Simünek)

Hint: First solve the problem without the information that, for the originally planned amount, nine more chairs could be bought.

Answer: 63


8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.

a) How many four-digit numbers are liked by both Andrey and Tanya?

b) Find the total sum of the digits of all such four-digit numbers.

Answer: 14580


Equilateral triangles \( ABF \) and \( BCG \) are constructed outside regular pentagon \( ABCDE \). Compute \( \angle FEG \).

Answer: 48


Find all quadruplets of positive integers  $(m,n,k,l)$  such that  $3^m = 2^k + 7^n$  and  $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$ .

Answer: (2,1,1,1)


Given a rectangle \(ABCD\), different points are chosen on two of its sides: five points on \(AB\) and six points on \(BC\). How many different triangles can be formed with vertices at the chosen points?

Answer: 135


Coordinate System and Parametric Equation
Given the ellipse $(C)$: $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\triangle PAB$.

Answer: 6(\sqrt{2} + 1)


There are 167 distinct cards with the numbers \(1, 3, 11, 3^{2}, 11^{2}, \ldots, 3^{83}, 11^{83}\) (each card has exactly one number, each number appears exactly once). In how many ways can 3 cards be chosen such that the product of the numbers on the chosen cards is the square of an integer divisible by 33?

Answer: 139523


In the convex quadrilateral $ABCD$, it is known that $\angle CBD = 58^\circ$, $\angle ABD = 44^\circ$, and $\angle ADC = 78^\circ$. Find the angle $\angle CAD$.

Answer: 58


Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$

Answer: 407


Does there exist a function \( s: \mathbb{Q} \rightarrow \{-1,1\} \) such that if \( x \) and \( y \) are distinct rational numbers satisfying \( xy = 1 \) or \( x + y \in \{0, 1\} \), then \( s(x) s(y) = -1 \)? Justify your answer.

Answer: Yes


In a certain city, license plate numbers consist of 6 digits (ranging from 0 to 9), but it is required that any two plates differ by at least 2 digits (for example, license plates 038471 and 030471 cannot both be used). Determine the maximum number of different license plates that can be issued and provide a proof.

Answer: 100000


A regular 400-gon is divided into parallelograms. Prove that among these parallelograms there are at least 100 rectangles.

Answer: 100


If $\sin \alpha + \sin \gamma = 2 \sin \beta$, prove that:
$$
\tan \frac{\alpha + \beta}{2} + \tan \frac{\beta + \gamma}{2} = 2 \tan \frac{\gamma + \alpha}{2}.
$$

Answer: \tan\frac{\alpha+\beta}{2}+\tan\frac{\beta+\gamma}{2}=2\tan\frac{\gamma+\alpha}{2}


In the arithmetic sequence \(\left\{a_{n}\right\}\), \(a_{2}=5\) and \(a_{6}=21\). Let the sum of the first \(n\) terms of the sequence \(\left\{\frac{1}{a_{n}}\right\}\) be denoted as \(S_{n}\). If the inequality \(S_{2 n+1}-S_{n} \leqslant \frac{m}{15}\) holds for any \(n \in \mathbf{N}_{+}\), then the minimum value of the positive integer \(m\) is \(\qquad\).

Answer: 5


What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7?

Answer: 49999


How many six-digit numbers with all distinct digits can be formed? Among these numbers, how many have exactly four odd digits?

Answer: 33600


In the language of the "Tekimar" tribe, there are only 7 letters: A, E, I, K, M, R, T, but the alphabetical order of these letters is unknown. A word is any sequence of the seven different letters of the alphabet, and no other words exist in the language. The tribe's chief wrote out all the existing words in alphabetical order and noted that the word "Metrika" holds position 3634 in this list. What position does the word "Materik" hold in this list?

Answer: 3745


In the acute triangle  $ABC$ , the sum of the distances from the vertices  $B$  and  $C$  to of the orthocenter  $H$  is equal to  $4r,$  where  $r$  is the radius of the circle inscribed in this triangle. Find the perimeter of triangle  $ABC$  if it is known that  $BC=a$ .

(Gryhoriy Filippovskyi)

Answer: 3a


Square \(ABCD\) has a side length of \(1\). Points \(P\) and \(Q\) are located on segments \(AB\) and \(AD\) respectively. If the perimeter of \(\triangle APQ\) is \(2\), find \(\angle PCQ\).

Answer: 45


A rectangular floor that is $12$ feet wide and $19$ feet long is tiled with rectangular tiles that are $1$ foot by $2$ feet. Find the number of tiles a bug visits when walking from one corner to the diagonal opposite corner.

Answer: 30


On the coordinate plane (\( x; y \)), a circle with radius 4 and center at the origin is drawn. A line given by the equation \( y = 4 - (2 - \sqrt{3}) x \) intersects the circle at points \( A \) and \( B \). Find the sum of the length of segment \( A B \) and the length of the shorter arc \( A B \).

Answer: 4\sqrt{2 - \sqrt{3}} + \frac{2\pi}{3}


Find the value of $m$ for which the quadratic equation $3x^2 + mx + 36 = 0$ has exactly one solution in $x$.

Answer: 12\sqrt{3}


In the village of Matitika, five friends live along a straight road in the following order: Alya, Bella, Valya, Galya, and Dilya. Each of them calculated the sum of distances (in meters) from her house to the houses of the others. Bella reported the number 700, Valya reported 600, Galya reported 650. How many meters are there between Bella's and Galya's houses?

Answer: 150


Let \( x, y, z, A, B, C \in \mathbb{R} \), with
\[ A + B + C = \pi. \]

Define \( G_n = x^n \sin nA + y^n \sin nB + z^n \sin nC \) for \( n \in \mathbb{Z}_{+} \).

Given \( G_1 = G_2 = 0 \), find the value of \( G_n \).

Answer: 0


Let \( K \) be the midpoint of edge \( BC \), and \( M \) be the center of face \( DCDC_1 \). Since points \( A \) and \( K \) lie in the plane of the bottom face, the line \( AK \) intersects the extension of edge \( DC \) at some point \( O \). Triangles \( ABK \) and \( KCO \) are congruent, hence \( CO = AB = DC \) and \( DO = 2 \cdot DC \). Since points \( M \) and \( O \) lie in the plane of face \( DCDC_1 \), the line \( MO \) intersects edges \( CC_1 \) and \( D_1D \) at some points \( L \) and \( N \). Thus, the section is a quadrilateral \( AKLN \), which is a trapezoid because \( AN \parallel KL \). Given the side length of the cube is 1, it is easy to find by similarity of triangles that \( LC = D_1N = \frac{1}{3} \) and \( DN = \frac{2}{3} \).

Using the Pythagorean theorem, we find the sides of trapezoid \( AN = \sqrt{1 + \frac{4}{9}} = \frac{\sqrt{13}}{3} \), \( AK = \sqrt{1 + \frac{1}{4}} = \frac{\sqrt{5}}{2} \), \( KL = \sqrt{\frac{1}{4} + \frac{1}{9}} = \frac{\sqrt{13}}{6} \), \( NL = \frac{\sqrt{10}}{3} \).

In trapezoid \( AKNL \), draw the line \( KF \parallel LN \) and the height of the trapezoid \( KP \). Since \( KL = \frac{1}{2}AN \), it follows that \( AF = FN \) and \( LK \).

By the law of cosines, we have \( KF^2 = (AK)^2 + (AF)^2 - 2 \cdot AK \cdot AF \cdot \cos \hat{A} \), thus \( \cos \hat{A} = \frac{3}{\sqrt{65}} \) hence \( \sin \hat{A} = \sqrt{\frac{56}{65}} = \frac{2\sqrt{14}}{\sqrt{65}} \). Consequently, \( KP = \sqrt{\frac{14}{13}} \) and the area \(S\) of the trapezoid \( \text{trapezoid } = \frac{1}{2} KP \cdot \frac{\sqrt{13}}{2} = \frac{\sqrt{14}}{4} \).

Answer: \frac{\sqrt{14}}{4}


Let \( n > m \geq 1 \) be natural numbers such that the groups of the last three digits in the decimal representation of \( 1978^m \) and \( 1978^n \) coincide. Find the ordered pair \( (m, n) \) of such \( m \) and \( n \) for which \( m+n \) is minimal.

Answer: (3,103)


Find the functions \( f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*} \) that satisfy, for all \( x, y > 0 \),

\[ f(2 x f(3 y)) + f\left(27 y^{3} f(2 x)\right) = 6 x y + 54 x y^{3} \]

Answer: f(x)=x


Lynne is tiling her long and narrow rectangular front hall. The hall is 2 tiles wide and 13 tiles long. She is going to use exactly 11 black tiles and exactly 15 white tiles. Determine the number of distinct ways of tiling the hall so that no two black tiles are adjacent (that is, share an edge).

Answer: 486


In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are then placed on top of the grid. Five of the circles have a diameter equal to a side of a small square (1 cm), and one large circle has a diameter equal to the length of four sides of a small square (4 cm). The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?

Answer: 41.25


Consider a chess board, with the numbers  $1$  through  $64$  placed in the squares as in the diagram below.

\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24
\hline
25 & 26 & 27 & 28 & 29 & 30 & 31 & 32
\hline
33 & 34 & 35 & 36 & 37 & 38 & 39 & 40
\hline
41 & 42 & 43 & 44 & 45 & 46 & 47 & 48
\hline
49 & 50 & 51 & 52 & 53 & 54 & 55 & 56
\hline
57 & 58 & 59 & 60 & 61 & 62 & 63 & 64
\hline
\end{tabular}\]

Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain?

Note. For any  $2\times3$  or  $3\times2$  rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.

Answer: 1056


A convex polygon is divided into parallelograms. A vertex of the polygon that belongs to only one parallelogram is called a good vertex. Prove that there are at least three good vertices.

Answer: 3


Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$?

Answer: 72


A sphere inscribed in a pyramid \( SABC \) touches the faces \( SAB, SBC, SCA \) at points \( D, E, F \) respectively.

Find all possible values of the sum of the angles \( SDA, SEB \), and \( SFC \).

Answer: 360


Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit.

Answer: 8281


If \( N(t) = 100 \times 18^{t} \) and \( P = N(0) \), find \( P \).

A fox ate \( P \) grapes in 5 days, each day eating 6 more than on the previous day. If he ate \( Q \) grapes on the first day, find \( Q \).

If \( Q \% \) of \( \frac{25}{32} \) is \( \frac{1}{Q} \% \) of \( R \), find \( R \).

If one root of the equation \( 3 x^{2} - a x + R = 0 \) is \( \frac{50}{9} \) and the other root is \( S \), find \( S \).

Answer: 3


Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and the product of its first $n$ terms is $T_n$, where the first term $a_1 > 1$, and $a_{2014}a_{2015} - 1 > 0$, $\frac{a_{2014} - 1}{a_{2015} - 1} < 0$, find the largest natural number $n$ such that $T_n > 1$.

Answer: 4028


Answer: 4


In the Cartesian coordinate plane $(xOy)$, a moving point $P$ passes through the chords $PA$ and $PB$ of the circles $C_{1}$: $x^{2}+y^{2}+2x+2y+1=0$ and $C_{2}$: $x^{2}+y^{2}-4x-6y+9=0$ respectively, where $A$ and $B$ are the points of intersection. If $|PA|=|PB|$, find the minimum value of $|OP|$.

Answer: \frac{4}{5}


There are  $20$  geese numbered  $1-20$  standing in a line. The even numbered geese are standing at the front in the order  $2,4,\dots,20,$  where  $2$  is at the front of the line. Then the odd numbered geese are standing behind them in the order,  $1,3,5,\dots ,19,$  where  $19$  is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered  $1,2,\dots,20$  (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?

*Author: Ray Li*

Answer: 55


Given the parabola $y = -2x^2 + 4x + m$.
1. For what value of $m$ does the parabola intersect the x-axis at exactly one point?
2. If two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the parabola have $x$-coordinates satisfying $x_1 > x_2 > 2$, compare the values of $y_1$ and $y_2$.

Answer: -2


In the Cartesian coordinate system, given the set of points $I=\{(x, y) \mid x$ and $y$ are integers, and $0 \leq x \leq 5,0 \leq y \leq 5\}$, find the number of distinct squares that can be formed with vertices from the set $I$.

Answer: 105


Given an equilateral triangle $ABC$ with side length 4. Points $D$, $E$, and $F$ are located on sides $BC$, $CA$, and $AB$ respectively, with $|AE| = |BF| = |CD| = 1$. Construct lines $AD$, $BE$, and $CF$ which intersect to form triangle $\triangle RQS$. Point $P$ moves within triangle $\triangle RQS$ and on its edges. The distances from point $P$ to the sides of $\triangle ABC$ are denoted as $x$, $y$, and $z$ respectively.

1. Prove that when $P$ is at a vertex of $\triangle RQS$, the product $xyz$ achieves its minimum value.
2. Determine the minimum value of the product $xyz$.

Answer: \frac{648\sqrt{3}}{2197}


For which values of the parameter \( \boldsymbol{a} \) does the equation \( |f(x) - 5| = p(x) \), where \( f(x) = \left| \frac{x^2 - 10x + 25}{x - 5} - \frac{x^2 - 3x}{3 - x} \right| \) and \( p(x) = a \), have three solutions? If there are multiple values for the parameter, specify their product.

Answer: 20


A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.

Answer: 5/2


Prime number  $p>3$  is congruent to  $2$  modulo  $3$ . Let  $a_k = k^2 + k +1$  for  $k=1, 2, \ldots, p-1$ . Prove that product  $a_1a_2\ldots a_{p-1}$  is congruent to  $3$  modulo  $p$ .

Answer: 3


Let the circle \( O: x^{2} + y^{2} = 5 \) intersect the parabola \( C: y^{2} = 2px \) (where \( p > 0 \)) at the point \( A(x_{0}, 2) \). Let \( AB \) be a diameter of the circle \( O \), and a line passing through \( B \) intersects the parabola \( C \) at two distinct points \( D \) and \( E \). Find the product of the slopes of the lines \( AD \) and \( AE \).

Answer: 2


Answer: \frac{45}{2}


Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements.

Answer: 240


Let \( S = \{1, 2, \cdots, 2025\} \). A subset of \( S \) with 51 elements is called a "good subset" if the sum of its elements is divisible by 5. Find the number of good subsets of \( S \).

Answer: \frac{1}{5}\binom{2025}{51}


Solve the system of equations:
\[
\left\{
\begin{array}{l}
x^{2} y + x y^{2} + 3x + 3y + 24 = 0, \\
x^{3} y - x y^{3} + 3x^{2} - 3y^{2} - 48 = 0
\end{array}
\right.
\]

Answer: (-3,-1)


Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If  $\tfrac{A}{B} = \tfrac{m}{n}$  where  $m,n$  are relatively prime positive integers, determine  $100m+n$ .

*Proposed by Lewis Chen*

Answer: 103


Given the sequence $\left\{r_{n}\right\}$ such that
$$
r_{1}=2, \quad r_{n}=r_{1} r_{2} \cdots r_{n-1}+1,
$$

let $n$ be a given positive integer, and $a_{1}, a_{2}, \cdots, a_{n}$ be $n$ positive integers such that $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}<1$. Prove that:
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}} \leq \frac{1}{r_{1}}+\frac{1}{r_{2}}+\cdots+\frac{1}{r_{n}}.
$$

Answer: \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\le\frac{1}{r_{1}}+\frac{1}{r_{2}}+\cdots+\frac{1}{r_{n}}


A cat broke into a wine store at night. It jumped onto a shelf where wine bottles were arranged in a long row - the first third of bottles were priced at 160 CZK each, the next third were priced at 130 CZK each, and the last third at 100 CZK each. First, the cat knocked over a bottle priced at 160 CZK, which was at the very beginning of the row, and then continued knocking over one bottle after another without skipping any. It knocked down a total of 25 bottles before it stopped. In the morning, the owner regretted that the cat did not start its rampage from the other end of the shelf. If it had broken the same number of bottles from the other end, the damage would have been 660 CZK less. How many bottles were originally on the shelf?

Answer: 36


Let \( \triangle ABC \) be an acute triangle with circumcenter \( O \) such that \( AB = 4 \), \( AC = 5 \), and \( BC = 6 \). Let \( D \) be the foot of the altitude from \( A \) to \( BC \), and \( E \) be the intersection of \( AO \) with \( BC \). Suppose that \( X \) is on \( BC \) between \( D \) and \( E \) such that there is a point \( Y \) on \( AD \) satisfying \( XY \parallel AO \) and \( YO \perp AX \). Determine the length of \( BX \).

Answer: \frac{96}{41}


Given that the three interior angles $A, B, C$ of $\triangle ABC$ form an arithmetic sequence, and the corresponding sides are $a, b, c$, and $a, c, \frac{4}{\sqrt{3}} b$ form a geometric sequence, find $S_{\triangle ABC}$ such that $a^2 = \quad$.

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


Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.

Answer: 1/8


Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of the caught fish himself. How many fish did Vasya catch?

Answer: 10


There is a deck of 52 cards, with 13 cards of each suit. Ivan draws one card at a time from the deck without returning the drawn cards. Each time before drawing a card, Ivan guesses a suit. Prove that if Ivan always guesses a suit that has no fewer cards remaining in the deck than any other suit, the guessed suit will match the suit of the drawn card at least 13 times.

Answer: 13


Find all such odd natural numbers $n>1$ that for any relatively prime divisors $a$ and $b$ of the number $n$, the number $a+b-1$ is also a divisor of $n$.

Answer: n


Given the vector $\boldsymbol{m}=(\sin B, 1-\cos B)$, which forms an angle of $\frac{\pi}{3}$ with the vector $\boldsymbol{n}=(2,0)$, where $A, B, C$ are the internal angles of $\triangle ABC$, find the value of the angle $B$.

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


At least  $ n - 1$  numbers are removed from the set  $\{1, 2, \ldots, 2n - 1\}$  according to the following rules:
(i) If  $ a$  is removed, so is  $ 2a$ ;
(ii) If  $ a$  and  $ b$  are removed, so is  $ a \plus{} b$ .
Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.

Answer: n^2


Determine the least odd number  $a > 5$  satisfying the following conditions: There are positive integers  $m_1,m_2, n_1, n_2$  such that  $a=m_1^2+n_1^2$ ,  $a^2=m_2^2+n_2^2$ , and  $m_1-n_1=m_2-n_2.$

Answer: 261


In an equilateral triangle \(ABC\), a point \(P\) is chosen such that \(AP = 10\), \(BP = 8\), and \(CP = 6\). Find the area of this triangle.

Answer: 36 + 25\sqrt{3}


Two numbers \( x \) and \( y \) satisfy the equation \( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.

Answer: -5


Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

Answer: 486


A permutation of \(\{1, 2, \ldots, 7\}\) is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation \((3, 4, 2, 1, 6, 5, 7)\) can be partitioned correctly into the blocks \([3, 4, 2, 1]\) and \([6, 5, 7]\), since when these blocks are sorted, the permutation becomes \((1, 2, 3, 4, 5, 6, 7)\). Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.

Answer: \frac{151}{105}


What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$?

Answer: 16.\overline{6}\%


Given a triangle  $ABC$  with angles  $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let  $H$  be its orthocentre, and  $O$  be its circumcenter. Let  $F$  be the midpoint of side  $AB$ , and  $Q$  be the foot of the perpendicular from  $B$  onto  $AC$ . Denote by  $X$  the intersection point of the lines  $FH$  and  $QO$ . Suppose the ratio of the length of  $FX$  and the circumradius of the triangle is given by  $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of  $1000a + 100b + 10c + d$ .

Answer: 1132


A child eats a jar of jam in six minutes, while Karlson eats it twice as fast. How long will it take for them to eat the jar of jam together?

Answer: 2\,


Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\_\_\_\_\_\_$.

Answer: \frac{1}{4}


Given \( a \geq b \geq c \geq d > 0 \) and \( a^{2} + b^{2} + c^{2} + d^{2} = \frac{(a + b + c + d)^{2}}{3} \), find the maximum value of \(\frac{a + c}{b + d}\).

Answer: \frac{7+2\sqrt{6}}{5}


Suppose that  $ABCD$  is a rectangle with sides of length  $12$  and  $18$ . Let  $S$  be the region of points contained in  $ABCD$  which are closer to the center of the rectangle than to any of its vertices. Find the area of  $S$ .

Answer: 54


Calculate the lengths of the arcs of curves defined by the equations in polar coordinates.

$$
\rho=5 e^{5 \varphi / 12},-\frac{\pi}{2} \leq \varphi \leq \frac{\pi}{2}
$$

Answer: 26\sinh(\frac{5\pi}{24})


A three-digit number is composed of three different non-zero digits in base ten. When divided by the sum of these three digits, the smallest quotient value is what?

Answer: 10.5


For a finite sequence $B = (b_1, b_2, \dots, b_m)$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + T_2 + \dots + T_m}{m},\]
where $T_k = b_1 + b_2 + \dots + b_k$ for $1 \leq k \leq m$.

If the Cesaro sum of the 50-term sequence $(b_1, b_2, \dots, b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$?

Answer: 492


$M$  is the midpoint of the side  $AB$  in an equilateral triangle  $\triangle ABC.$  The point  $D$  on the side  $BC$  is such that  $BD : DC = 3 : 1.$  On the line passing through  $C$  and parallel to  $MD$  there is a point  $T$  inside the triangle  $\triangle ABC$  such that  $\angle CTA = 150.$  Find the  $\angle MT D.$ *(K. Ivanov )*

Answer: 120


Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer.

Answer: 5600


If \( p \geq 5 \) is a prime number, let \( q \) be the smallest prime number such that \( q > p \) and let \( n \) be the number of positive divisors of \( p+q \) (including 1 and \( p+q \)).

a) Show that no matter which prime number \( p \) is chosen, the number \( n \) is greater than or equal to 4.

b) Find the smallest possible value \( m \) that \( n \) can take among all possible choices of \( p \):

b1) Give an example of a prime number \( p \) such that the value \( m \) is reached.

b2) Show that there is no prime number \( p \) for which the value of \( n \) is less than \( m \).

Answer: 6


Let \( f(x) = x^{3} - 3x \). Compute the number of positive divisors of
\[ \left\lfloor f\left(f\left(f\left(f\left(f\left(f\left(f\left(f\left(\frac{5}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right\rfloor \]
where \( f \) is applied 8 times.

Answer: 6562


Given a sequence of distinct positive integers $(i\_1, i\_2, \ldots, i\_n)$ where $n$ is a positive integer greater than or equal to 2, if $i\_p > i\_q$ when $p > q$, then $i\_p$ and $i\_q$ are called a "good order" of the sequence. The number of "good orders" in a sequence is called the "good order number" of that sequence. For example, the sequence (1, 3, 4, 2) has "good orders" "1, 3", "1, 4", "1, 2", "3, 4", so its "good order number" is 4. If the "good order number" of the sequence of distinct positive integers $(a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7)$ is 3, then the "good order number" of $(a\_7, a\_6, a\_5, a\_4, a\_3, a\_2, a\_1)$ is $\_\_\_\_\_\_\_.$

Answer: 18


If the sequence \(\{a_n\}\) satisfies \(a_1 = \frac{2}{3}\) and \(a_{n+1} - a_n = \sqrt{\frac{2}{3} \left(a_{n+1} + a_n\right)}\), then find the value of \(a_{2015}\).

Answer: 1354080


In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

Answer: 112


Let \( T \) be a right triangle with sides having lengths 3, 4, and 5. A point \( P \) is called awesome if \( P \) is the center of a parallelogram whose vertices all lie on the boundary of \( T \). What is the area of the set of awesome points?

Answer: 3/2


Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if
the average value of the polynomial on each circle centered at the origin is $0$.
The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$.
Find the dimension of $V$.

Answer: 2020050


How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$?
$\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Answer: \textbf{(C)}2


Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$.
Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF\cdot XG$?

Answer: 17


A right cylinder is given with a height of  $20$  and a circular base of radius  $5$ . A vertical planar cut is made into this base of radius  $5$ . A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the pieces is  $100\sqrt2$ . Then the volume of the larger piece can be written as  $a + b\pi$ , where  $a, b$  are positive integers. Find  $a + b$ .

Answer: 625


The base of a pyramid is a triangle with sides of lengths \(a\), \(a\), and \(b\). All lateral edges are inclined at an angle of \(60^{\circ}\) to the plane of the base. Determine the volume of the pyramid.

Answer: \frac{^2b\sqrt{3}}{12}


Vanya thought of a seven-digit number and then subtracted the sum of all its digits except one. He got 9875352. What number would Vanya get if he subtracted the sum of all the digits except the second one from the left?

Answer: 9875357


(The full score for this question is 8 points) There are 4 red cards labeled with the numbers 1, 2, 3, 4, and 2 blue cards labeled with the numbers 1, 2. Four different cards are drawn from these 6 cards.
(1) If it is required that at least one blue card is drawn, how many different ways are there to draw the cards?
(2) If the sum of the numbers on the four drawn cards equals 10, and they are arranged in a row, how many different arrangements are there?

Answer: 96


Given the ellipse $E$: $\frac{x^{2}}{2}+y^{2}=1$ with its right focus $F$, two perpendicular lines passing through $F$ intersect with $E$ at points $A$, $C$ and $B$, $D$.

1. Can the quadrilateral $ABCD$ form a parallelogram? Please explain the reason.
2. Find the minimum value of $|AC|+|BD|$.

Answer: \frac{8 \sqrt{2}}{3}


Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
$(\textbf{A})\: 1.6\qquad(\textbf{B}) \: 1.8\qquad(\textbf{C}) \: 2.0\qquad(\textbf{D}) \: 2.2\qquad(\textbf{E}) \: 2.4$

Answer: (\textbf{D})\:2.2


For a positive integer \( n \) and a real number \( x \) such that \( 0 \leqslant x < n \), define

\[ f(n, x) = (1 - \{x\}) \cdot \binom{n}{\lfloor x \rfloor} + \{x\} \cdot \binom{n}{\lfloor x \rfloor + 1} \]

where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to the real number \( x \), and \( \{x\} = x - \lfloor x \rfloor \). If integers \( m, n \geq 2 \) satisfy

\[ f\left(m, \frac{1}{n}\right) + f\left(m, \frac{2}{n}\right) + \cdots + f\left(m, \frac{mn - 1}{n}\right) = 123, \]

find the value of

\[ f\left(n, \frac{1}{m}\right) + f\left(n, \frac{2}{m}\right) + \cdots + f\left(n, \frac{mn - 1}{m}\right). \]

Answer: 74


Three fair, six-sided dice are rolled. What is the probability that the sum of the three numbers showing is less than 16?

Answer: \frac{103}{108}


Fill the numbers $1,2,\cdots,36$ into a $6 \times 6$ grid, placing one number in each cell, such that the numbers in each row are in increasing order from left to right. What is the minimum possible sum of the six numbers in the third column?

Answer: 108


The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$.

Answer: 90


A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

Answer: 54\sqrt{3}-18\pi


Given a convex quadrilateral \(ABCD\) with \(\angle C = 57^{\circ}\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\), find the measure of angle \(D\) in degrees.

Answer: 168


How many solutions are there to the equation
\[ m^{4}+8n^{2}+425=n^{4}+42m^{2}, \]
where \( m \) and \( n \) are integers?

Answer: 16


The high-speed train "Sapsan," approaching a railway station at a speed of \( v = 216 \) km/h, emits a warning sound signal lasting \( \Delta t = 5 \) seconds when it is half a kilometer away from the station. What will be the duration of the signal \( \Delta t_{1} \) from the perspective of passengers standing on the platform? The speed of sound in the air is \( c = 340 \) m/s.

Answer: 4.12


Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and 1 form the side lengths of an obtuse triangle.

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


Let \( ABCD \) be a parallelogram. Let \( E \) be the midpoint of \( AB \) and \( F \) be the midpoint of \( CD \). Points \( P \) and \( Q \) are on segments \( EF \) and \( CF \), respectively, such that \( A, P \), and \( Q \) are collinear. Given that \( EP = 5 \), \( PF = 3 \), and \( QF = 12 \), find \( CQ \).

Answer: 8


A circle \( C \) touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that the radius of \( C \) must be larger than the radius of the middle of the three circles.

Answer: R>r_2


The numbers \( x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n} \) satisfy the condition \( x_{1}^{2}+\ldots+x_{n}^{2}+y_{1}^{2}+\ldots+y_{n}^{2} \leq 2 \). Find the maximum value of the expression

$$
A=\left(2\left(x_{1}+\ldots+x_{n}\right)-y_{1}-\ldots-y_{n}\right) \cdot \left(x_{1}+\ldots+x_{n}+2\left(y_{1}+\ldots+y_{n}\right)\right)
$$

Answer: 5n


In a regular triangular pyramid, the side of the base is equal to \(a\), and the angle between the slant height (apothem) and a lateral face is \(\frac{\pi}{4}\). Find the height of the pyramid.

Answer: \frac{\sqrt{6}}{6}


Find the smallest prime number that can not be written in the form  $\left| 2^a-3^b \right|$  with non-negative integers  $a,b$ .

Answer: 41


In an isosceles trapezoid with bases 1 and 4, two circles are inscribed such that each circle touches the other circle, the two non-parallel sides, and one of the bases. Find the area of the trapezoid.

Answer: \frac{15\sqrt{2}}{2}


A triangle has side lengths 7, 8, and 9.  There are exactly two lines that simultaneously bisect the perimeter and area of the triangle.  Let $\theta$ be the acute angle between these two lines.  Find $\tan \theta.$

[asy]
unitsize(0.5 cm);

pair A, B, C, P, Q, R, S, X;

B = (0,0);
C = (8,0);
A = intersectionpoint(arc(B,7,0,180),arc(C,9,0,180));
P = interp(A,B,(12 - 3*sqrt(2))/2/7);
Q = interp(A,C,(12 + 3*sqrt(2))/2/9);
R = interp(C,A,6/9);
S = interp(C,B,6/8);
X = extension(P,Q,R,S);

draw(A--B--C--cycle);
draw(interp(P,Q,-0.2)--interp(P,Q,1.2),red);
draw(interp(R,S,-0.2)--interp(R,S,1.2),blue);

label("$\theta$", X + (0.8,0.4));
[/asy]

Answer: 3 \sqrt{5} + 2 \sqrt{10}


Alice, Bob, and Charlie roll a 4, 5, and 6-sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls?

Answer: \frac{13}{30}


In triangle \( \triangle ABC \), the sides opposite to the angles \( \angle A \), \( \angle B \), and \( \angle C \) are denoted as \( a \), \( b \), and \( c \) respectively. If \( b^{2}=a^{2}+c^{2}-ac \), and \( c-a \) is equal to the height \( h \) from vertex \( A \) to side \( AC \), then find \( \sin \frac{C-A}{2} \).

Answer: \frac{1}{2}


Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(\operatorname{tg}\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$

Answer: 1


The domain of the function \( f(x) \) is \(\mathbf{R}\) and it satisfies the following conditions: for \( x \in [0, 1) \), \( f(x) = 2^x - x \), and for any real number \( x \), \( f(x) + f(x+1) = 1 \). Let \( a = \log_2 3 \). Find the value of the expression \( f(a) + f(2a) + f(3a) \).

Answer: \frac{17}{16}


If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$?

Answer: 20


In triangle \(ABC\) with side lengths \(a, b, c\) satisfying the condition \(a + b + c = 1\), prove that:
$$
5\left(a^{2}+b^{2}+c^{2}\right) + 18 abc \geq \frac{7}{3}.
$$

Answer: \frac{7}{3}


In \(\triangle ABC\), \(BC = a\), \(CA = b\), \(AB = c\). If \(2a^{2} + b^{2} + c^{2} = 4\), then the maximum area of \(\triangle ABC\) is ______.

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


In $\triangle ABC$, $AB = 33 \text{ cm}$, $AC = 21 \text{ cm}$ and $BC = x \text{ cm}$, where $x$ is an integer. $D$ is a point on segment $AB$ and $E$ is a point on segment $AC$ such that $AD = DE = EC = y \text{ cm}$, where $y$ is also an integer. Find $x$.

Answer: 30


The sum of the plane angles of a trihedral angle is $180^\circ$. Find the sum of the cosines of the dihedral angles of this trihedral angle.

Answer: 1


In how many ways can two disjoint subsets be selected from a set with $n$ elements?

Answer: 3^n


$201$  positive integers are written on a line, such that both the first one and the last one are equal to  $19999$ . Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.

Answer: 19800


A certain chemical reaction requires a catalyst to accelerate the reaction, but using too much of this catalyst affects the purity of the product. If the amount of this catalyst added is between 500g and 1500g, and the 0.618 method is used to arrange the experiment, then the amount of catalyst added for the second time is \_\_\_\_\_\_ g.

Answer: 882


How many positive integers less than 10,000 have at most two different digits?

Answer: 927


Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously?

Answer: 35


For natural number  $n$ , define the function  $f(n)$  to be the number you get by  $f(n)$  adding the digits of the number  $n$ . For example,  $f(16)=7$ ,  $f(f(78))=6$ , and  $f(f(f(5978)))=2$ .
Find the least natural number  $n$  such that  $f(f(f(n)))$  is not a one-digit number.


A team of mowers had to mow two meadows, one twice the size of the other. The team mowed the larger meadow for half a day. Afterward, they split in half: one half stayed on the larger meadow and finished mowing it by evening, while the other half mowed the smaller meadow but did not finish it. How many mowers were in the team if it is known that one mower completed the remaining work the next day?

Answer: 8


For a natural number \( n \), let \( S_{n} \) denote the least common multiple of all the numbers \( 1,2, \ldots, n \). Does there exist a natural number \( m \) such that \( S_{m+1}=4 S_{m} \)?

Answer: No


The function \( g \), defined on the set of integers, satisfies the following conditions:

1) \( g(1) > 1 \)
2) \( g(x+y) + x g(y) + y g(x) = g(x) g(y) + x + y + xy \) for any \( x, y \in \mathbb{Z} \);
3) \( 3 g(x) = g(x+1) + 2x - 1 \) for any \( x \in \mathbb{Z} \).

Find \( g(5) \).

Answer: 248


Solve the following system of equations in integer numbers:
$$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

Answer: (1, 0, -1)


Consider non-negative real numbers  $a, b, c$  satisfying the condition  $a^2 + b^2 + c^2 = 2$  . Find the maximum value of the following expression   $$ P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c $$

Answer: 3


In trapezoid \(ABCD\), the side \(AB\) is perpendicular to both \(AD\) and \(BC\), and \(|AB|=\sqrt{|AD| \cdot |BC|}\). Let \(E\) be the point where the non-parallel sides intersect, \(O\) be the point where the diagonals intersect, and \(M\) be the midpoint of \(AB\). Find \(\angle EOM\).

Answer: 90


Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy's treasure and halving the number of pirates. On the 48th day of sailing, the pirates reached the deserted island, and Billy buried all his treasure at the marked spot—exactly 1000 doubloons. How many pirates set off with Billy to the deserted island?

Answer: 30


In $\triangle ABC$, $\cos \frac{C}{2} = \frac{2\sqrt{5}}{5}$, $\overrightarrow{A H} \cdot \overrightarrow{BC} = 0$, and $\overrightarrow{AB} \cdot (\overrightarrow{CA} + \overrightarrow{CB}) = 0$.

Find the eccentricity of the hyperbola passing through point $C$ and having $A$ and $H$ as its foci.

Answer: 2


Given that $F$ is the focus of the parabola $y^{2}=4x$, and a perpendicular line to the directrix is drawn from a point $M$ on the parabola, with the foot of the perpendicular being $N$. If $|MF|= \frac{4}{3}$, then $\angle NMF=$     .

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


Let \(2n\) real numbers \(a_1, a_2, \cdots, a_{2n}\) satisfy the condition \(\sum_{i=1}^{2n-1} (a_{i+1} - a_i)^2 = 1\). Find the maximum value of \(\left(a_{n+1} + a_{n+2} + \cdots + a_{2n}\right) - \left(a_1 + a_2 + \cdots + a_n\right)\).

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


Let  $ABC$  be an acute triangle with orthocenter  $H$ . Let  $M$ ,  $N$  and  $R$  be the midpoints of  $AB$ ,  $BC$  an  $AH$ , respectively. If  $A\hat{B}C=70^\large\circ$ , compute  $M\hat{N}R$ .

Answer: 20


For a positive real number $x > 1,$ the Riemann zeta function $\zeta(x)$ is defined by
\[\zeta(x) = \sum_{n = 1}^\infty \frac{1}{n^x}.\]Compute
\[\sum_{k = 2}^\infty \{\zeta(2k - 1)\}.\]Note: For a real number $x,$ $\{x\}$ denotes the fractional part of $x.$

Answer: \frac{1}{4}


For which integers \( n \in \{1, 2, \ldots, 15\} \) is \( n^n + 1 \) a prime number?

Answer: 1,2,4


Given that \( P \) is a moving point on the parabola \( y^2 = 2x \), and points \( B \) and \( C \) are on the \( y \)-axis, the circle \((x-1)^2 + y^2 = 1\) is the incircle of \( \triangle PBC \). Then, the minimum value of \( S_{\triangle PBC} \) is ______.

Answer: 8


What is the probability that in a randomly chosen arrangement of the numbers and letters in "HMMT2005," one can read either "HMMT" or "2005" from left to right?

Answer: 23/144


In a circle, mutually perpendicular diameters $AB$ and $CD$ are drawn, and a point $M$ is taken arbitrarily on the arc $AC$. Find $MB + MD$ if $MA + MC = a$.

Answer: (1+\sqrt{2})


The angular measure of an arc is $110^{\circ}$. Find the angle between the chord and the extension of the radius drawn to the end of the arc.

Answer: 145


We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have?

Answer: 32


Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.

Author: Nikola Petrovi?

Answer: 2n(n - 1) + 1


Let $PQRS$ be a convex quadrilateral, and let $H_P,$ $H_Q,$ $H_R,$ $H_S$ denote the centroids of triangles $QRS,$ $PRS,$ $PQS,$ and $PQR,$ respectively. Calculate $\frac{[H_P H_Q H_R H_S]}{[PQRS]}.$

Answer: \frac{1}{9}


Let a  $9$ -digit number be balanced if it has all numerals  $1$  to  $9$ . Let  $S$  be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of  $k$  such that any two subsequences of  $S$  which has consecutive  $k$  numerals are different from each other.

Answer: 17


The positive numbers  $a, b, c$  are such that  $a^2<16bc, b^2<16ca$  and  $c^2<16ab$ . Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]

Answer: ^2+b^2+^2<2(++ca)


Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the value of $a$ when $a=$____, such that the minimum value of the algebraic expression $\frac{{2{a^2}+1}}{{ab}}-2$ is ____.

Answer: 2\sqrt{3}


A total of 2015 tickets, numbered 1, 2, 3, 4, ..., 2014, 2015, are placed in an empty bag. Alfie removes ticket \( a \) from the bag. Bernice then removes ticket \( b \) from the bag. Finally, Charlie removes ticket \( c \) from the bag. They notice that \( a < b < c \) and \( a + b + c = 2018 \). In how many ways could this happen?

Answer: 338352


In any tetrahedron, do the heights intersect at a single point?

Answer: No


Fedya was 7 kopecks short of buying a portion of ice cream, and Masha was only 1 kopeck short. However, even when they combined all their money, it was still not enough to buy one portion of ice cream. How much did one portion of ice cream cost?

Answer: 7


Sofia and Viktor are playing the following game on a  $2022 \times 2022$  board:
 - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
 - Then Viktor without seeing the table, chooses a positive integer  $n$ ;
 - After that Viktor looks at the table covered with dominoes, chooses and fixes  $n$  of them;
 - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before.
If she achieves that, she wins, otherwise Viktor wins. What is the minimum number  $n$  for which Viktor can always win, no matter the starting covering of dominoes.

*Proposed by Viktor Simjanoski*

Answer: 1011^2


Answer: \textbf{(B)}\40


Point \( A \) lies on the side \( LM \) of triangle \( KLM \) with an angle of \( 120^\circ \) at vertex \( K \). The circles inscribed in triangles \( AKL \) and \( AKM \) have centers at \( F \) and \( O \) respectively. Find the radius of the circle circumscribed around triangle \( FKO \) if \( AF = 3 \) and \( AO = 6 \).

Answer: \sqrt{15}


A subset of a student group is called an *ideal company* if
1) in this subset, all girls are liked by all young men,
2) no one can be added to this subset without violating condition  $1$ .
In a certain group,  $9$  female students and  $15$  students study. Warden of the group made a list of all kinds of ideal companies in this group. What is the largest number of companies on this list?

Answer: 512


For each positive integer \( n \), define \( s(n) \) to equal the sum of the digits of \( n \). For example, \( s(2023)=2+0+2+3 \). The number of integers \( n \) with \( 100 \leq n \leq 999 \) and \( 7 \leq s(n) \leq 11 \) is \( S \). What is the integer formed by the rightmost two digits of \( S \)?

Answer: 24


The rules of a solitaire card game are as follows: Seven pairs of distinct cards are placed into a backpack. The player randomly draws cards from the backpack one at a time and returns them unless a pair is drawn, in which case the pair is set aside. Every turn, the player draws exactly three cards (or fewer if fewer than three cards remain). If the three cards drawn are all distinct (each one not forming part of a pair with the others), the game ends. Otherwise, the play continues until the backpack is empty. What is the probability that the backpack is emptied?

Answer: \frac{27}{5005}


Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \epsilon$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \epsilon - \delta \epsilon^2.\]
Determine the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 17


Determine the number of solutions of the simultaneous equations  $ x^2 \plus{} y^3 \equal{} 29$  and  $ \log_3 x \cdot \log_2 y \equal{} 1.$

Answer: 2


Given a convex quadrilateral \(ABCD\). It is known that \(\angle C = 57^\circ\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\). How many degrees is angle \(D\)?

Answer: 168


Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}$

Answer: \textbf{(B)}\\frac{2+\sqrt{3}}{3}


On side \( AB \) of triangle \( ABC \), point \( K \) is marked, and on side \( AC \), point \( M \) is marked. Segments \( BM \) and \( CK \) intersect at point \( P \). It turns out that angles \( APB \), \( BPC \), and \( CPA \) are each equal to \( 120^\circ \). Additionally, the area of quadrilateral \( AKPM \) is equal to the area of triangle \( BPC \). Find angle \( BAC \).

Answer: 60


A hemisphere is placed on a sphere of radius \(100 \text{ cm}\). The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in cm).

Answer: 300


If you toss a fair coin \( n+1 \) times and I toss it \( n \) times, what is the probability that you get more heads?

Answer: \frac{1}{2}


If the function \( f(x)=\left(x^{2}-a x+2 a\right) \ln (x+1) \) passes through all four quadrants, then the range of the real number \( a \) is ______.

Answer: (-\frac{1}{3},0)


Compute the number of functions \( f: \{1, 2, \ldots, 9\} \rightarrow \{1, 2, \ldots, 9\} \) which satisfy \( f(f(f(f(f(x))))) = x \) for each \( x \in \{1, 2, \ldots, 9\} \).

Answer: 3025


A circle intersects the sides $AC$ and $CB$ of an isosceles triangle $ACB$ at points $P$ and $Q$ respectively, and is circumscribed around triangle $ABQ$. The segments $AQ$ and $BP$ intersect at point $D$ such that $AQ: AD = 4:3$. Find the area of triangle $DQB$ if the area of triangle $PQC$ is 3.

Answer: 9/2


Let  $ f (n) $  be a function that fulfills the following properties: $\bullet$  For each natural   $ n $ ,   $ f (n) $  is an integer greater than or equal to  $ 0 $ . $\bullet$   $f (n) = 2010 $ , if  $ n $  ends in  $ 7 $ . For example,   $ f (137) = 2010 $ . $\bullet$  If  $ a $  is a divisor of  $ b $ , then:   $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $ .

Find  $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $  and justify your answer.

Answer: 2010


A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?

Answer: $4(\pi - \sqrt{3})$


Let
\[\mathbf{A} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Compute $\mathbf{A}^{2018}.$

Answer: \begin{pmatrix} \frac{1}{2} & 0 & -\frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{pmatrix}


Let  $ABCD$  be a trapezoid such that  $AB$  is parallel to  $CD$ , and let  $E$  be the midpoint of its side  $BC$ . Suppose we can inscribe a circle into the quadrilateral  $ABED$ , and that we can inscribe a circle into the quadrilateral  $AECD$ . Denote  $|AB|=a$ ,  $|BC|=b$ ,  $|CD|=c$ ,  $|DA|=d$ . Prove that \[a+c=\frac{b}{3}+d;\]  \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]

Answer: \frac{1}{}+\frac{1}{}=\frac{3}{b}


Given \( n \) planes in space (\( n \geq 5 \)) such that any three of them intersect at exactly one point and there is no point in space where more than three of the planes intersect,

Prove that among the regions into which the planes divide the space, there are at least \( \frac{2n-3}{4} \) tetrahedra.

Answer: \frac{2n-3}{4}


A box contains four balls, each with a unique number from 1 to 4, and all balls are identical in shape and size.
(1) If two balls are randomly drawn from the box, what is the probability that the sum of their numbers is greater than 5?
(2) If one ball is drawn from the box, its number is recorded as $a$, and then the ball is put back. Another ball is drawn, and its number is recorded as $b$. What is the probability that $|a-b| \geq 2$?

Answer: \frac{3}{8}


$SM$ is the angle bisector in triangle $SQT$. Point $O$ on side $ST$ is such that the angle $OQT$ is equal to the sum of the angles $QTS$ and $QST$. Prove that $OM$ is the angle bisector of the angle $QOT$.

Answer: OMistheanglebisectorof\angleQOT


From the digits $1$, $2$, $3$, $4$, form a four-digit number with the first digit being $1$, and having exactly two identical digits in the number. How many such four-digit numbers are there?

Answer: 36


Given \( n \) is a positive integer. For a sequence of non-negative real numbers
\[
0 \leqslant a_{1}, a_{2}, \cdots, a_{n} \leqslant 1,
\]
prove that
\[
\frac{1-a_{1} a_{2} \cdots a_{n}}{n} \leqslant \frac{1}{1+a_{1}+a_{2}+\cdots+a_{n}}.
\]

Answer: \frac{1-a_{1}a_{2}\cdotsa_{n}}{n}\leslant\frac{1}{1+a_{1}+a_{2}+\cdots+a_{n}}


An exchange point conducts two types of operations:

1) Give 2 euros - receive 3 dollars and a candy as a gift;
2) Give 5 dollars - receive 3 euros and a candy as a gift.

When the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not acquire any euros, but he received 50 candies. How many dollars did this "gift" cost Buratino?

Answer: 10


Determine if there are other composite numbers smaller than 20 that are also abundant besides number 12. If so, list them. If not, confirm 12 remains the smallest abundant number.

Answer: 12


Let  $ABC$  be a triangle with  $\angle A=\angle C=30^{\circ}.$  Points  $D,E,F$  are chosen on the sides  $AB,BC,CA$  respectively so that  $\angle BFD=\angle BFE=60^{\circ}.$  Let  $p$  and  $p_1$  be the perimeters of the triangles  $ABC$  and  $DEF$ , respectively. Prove that  $p\le 2p_1.$

Answer: p\le2p_1


The 2-digit integers from 19 to 92 are written consecutively to form the integer $N=192021\cdots9192$. Suppose that $3^k$ is the highest power of 3 that is a factor of $N$. What is $k$?
$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 2\quad \text{(D) } 3\quad \text{(E) more than } 3$

Answer: (B)1


Jo and Blair take turns counting numbers, starting with Jo who says "1". Blair follows by saying the next number in the sequence, which is normally the last number said by Jo plus one. However, every third turn, the speaker will say the next number plus two instead of one. Express the $53^{\text{rd}}$ number said in this new sequence.

Answer: 71


The lengths of the sides of an acute triangle form an arithmetic progression with a common difference of 5 cm. Find the largest number with the property that the length of the longest side of any triangle of this type is greater than this number.

Answer: 25\,


An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$  What is the largest possible value of $p$?

Answer: 130


Given the value of $\sin \alpha$, what is the maximum number of possible values for a) $\sin \frac{\alpha}{2}$, b) $\sin \frac{\alpha}{3}$?

Answer: 3


There are 9 swallows sitting on a wire at regular intervals. The distance between the first and the last swallow is $720 \mathrm{~cm}$.

- What is the distance between neighboring swallows?
- How many swallows would be sitting on the wire if three more swallows sat between every two existing ones?

Answer: 33


A positive integer \( n \) is said to be 'good' if \( n^2 - 1 \) can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers.

Answer: 104


Denote by \(\mathbb{N}\) the set of all positive integers. Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that for all positive integers \(m\) and \(n\), the integer \(f(m) + f(n) - mn\) is nonzero and divides \(mf(m) + nf(n)\).

Answer: f(n)=n^2


Given an acute triangle  $ABC$  with altituties AD and BE. O circumcinter of  $ABC$ .If o lies on the segment DE then find the value of  $sinAsinBcosC$

Answer: \frac{1}{2}


Rotate an equilateral triangle with side length $2$ around one of its sides to form a solid of revolution. The surface area of this solid is ______.

Answer: 4\sqrt{3}\pi


Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:

$\textbf{(A)}\ \text{(1) and (4)}\qquad \textbf{(B)}\ \text{(1) and (3)}\qquad \textbf{(C)}\ \text{(1) and (2)}\qquad \textbf{(D)}\ \text{(2) and (4)}\qquad \textbf{(E)}\ \text{(2) and (3)}$

Answer: \textbf{(A)}