Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

Find all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that for all positive integers $m,n$ and $a$ we have a) $f(f(m)f(n))=mn$ and b) $f(2024a+1)=2024a+1$.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

A round-robin tournament has six competititors. Each round between two players is equally likely to result in a win for a given player, a loss for that player, or a tie. The results of the tournament are \textit{nice} if for all triples of distinct players $(A, B, C)$, 1. If $A$ beat $B$ and $B$ beat $C$, then $A$ also beat $C$; 2. If $A$ and $B$ tied, then either $C$ beat both $A$ and $B$, or $C$ lost to both $A$ and $B$. The probability that the results of the tournament are $\textit{nice}$ is $p = \tfrac{m}{n}$, for coprime positive integers $m$ and $n$. Find $m$. [i]Proposed by Michael Tang[/i]

Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X, Y$ vary on $\omega$, the line $XY$ passes through a fixed point. [i]Proposed by Aaron Thomas, UK[/i]

Prove that if $A{}$ and $B{}$ are $n\times n$ matrices with complex entries which satisfy \[A=AB-BA+A^2B-2ABA+BA^2+A^2BA-ABA^2,\]then $\det(A)=0$.

$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.

Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?

You are given a large collection of identical heavy balls and lightweight rods. When two balls are placed at the ends of one rod and interact through their mutual gravitational attraction (as is shown on the left), the compressive force in the rod is $F$. Next, three balls and three rods are placed at the vertexes and edges of an equilateral triangle (as is shown on the right). What is the compressive force in each rod in the latter case? [asy] size(300); real x=-25; draw((x,-8)--(x,8),linewidth(6)); filldraw(Circle((x,8),2.5),grey); filldraw(Circle((x,-8),2.5),grey); draw((0,-8)--(0,8)--(8*sqrt(3),0)--cycle,linewidth(6)); filldraw(Circle((0,8),2.5),grey); filldraw(Circle((0,-8),2.5),grey); filldraw(Circle((8*sqrt(3),0),2.5),grey); [/asy] (A) $\frac{1}{\sqrt{3}}F$ (B) $\frac{\sqrt{3}}{2}F$ (C) $F$ (D) $\sqrt{3}F$ (E) $2F$

A convex $n$-gon and its $n-3$ diagonals which have no common point inside the polygon form a [i]subdivision graph[/i]. Show that if and only if $3|n$, there exists a [i]subdivision graph [/i]that can be drawn in one closed stroke. (i.e. start from a certain vertex, get through every edges and diagonals exactly one time, finally back to the starting vertex.)

Find the smallest positive integer \( k \geq 2 \) for which there exists a polynomial \( f(x) \) of degree \( k \) with integer coefficients and a leading coefficient of \( 1 \) that satisfies the following condition: (Condition) For any two integers \( m \) and \( n \), if \( f(m) - f(n) \) is a multiple of \( 31 \), then \( m - n \) is a multiple of \( 31 \).

Jordan has an infinite geometric series of positive reals whose sum is equal to $2\sqrt2 + 2$. It turns out that if Jordan squares each term of his geometric series and adds up the resulting numbers, he get a sum equal to $4$. If Jordan decides to take the fourth power of each term of his original geometric series and add up the resulting numbers, what sum will he get?

There are red and blue balls in an urn : $1024$ in total. In one round, we do the following: we draw the balls from the urn two by two. After all balls have been drawn, we put a new ball back into the urn for each pair of drawn balls: the colour of the new ball depends on that of the drawn pair. For two red balls drawn, we put back a red ball. For two blue balls, we put back a blue ball. For a red and a blue ball, we put back a black ball. For a red and a black ball, we put back a red ball. For a blue and a black ball, we put back a blue ball. Finally, for two black balls we put back a black ball. Then the next round begins. After $10$ rounds, a single ball remains in the urn, which is red. What is the maximal number of blue balls that might have been in the urn at the very beginning?

The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

In all triangles $ABC$ does it hold that: $$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

Consider 2024 distinct prime numbers $p_1, p_2, \dots, p_{2024}$ such that \[p_1+p_2+\dots+p_{1012}=p_{1013}+p_{1014}+\dots+p_{2024}.\] Let $A=p_1p_2\dots p_{1012}$ and $B=p_{1013}p_{1014}\dots p_{2024}$. Prove that $|A-B|\geq 4$.

Humberto and Luciano use the break between classes to have fun with the following game: Humberto writes a list of distinct positive integers on a green sheet of paper and hands it to Luciano. Luciano then writes on a board all the possible sums, without repetitions, of one or more different numbers written on the green sheet. For example, if Humberto writes $1$, $3$ and $4$ on the green sheet, Luciano will write $1$, $3$, $4$, $5$, $7$ and $8$ on the board. (a) Let $n$ be a positive integer. Determine all positive integers $k$ such that Humberto can write a list of $n$ numbers on the green sheet in order to guarantee that Luciano will write exactly $k$ numbers on the board. (b) Luciano now decides to write a list of $m$ distinct positive integers on a yellow sheet of paper. Determine the smallest positive integer $m$ such that it is possible for Luciano to write this list so that, for any list that Humberto writes on the green sheet, with a maximum of $2023$ numbers, not all the numbers on the yellow sheet will be written on the board.

In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?

Let $a_1, a_2, a_3, \ldots$ be an infinite sequence of positive integers such that $a_1=4$, $a_2=12$, and for all positive integers $n$, \[a_{n+2}=\gcd\left(a_{n+1}^2-4,a_n^2+3a_n \right).\] Find, with proof, a formula for $a_n$ in terms of $n$.

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{2n-1}^{2}+F_{2n+1}^{2}+1=3F_{2n-1}F_{2n+1}$ for all $n \ge 1$.

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 + |(a - b)(b - c)(c - a)|.$$

[b]p10 [/b]Kathy has two positive real numbers, $a$ and $b$. She mistakenly writes $$\log (a + b) = \log (a) + \log( b),$$ but miraculously, she finds that for her combination of $a$ and $b$, the equality holds. If $a = 2022b$, then $b = \frac{p}{q}$ , for positive integers $p, q$ where $gcd(p, q) = 1$. Find $p + q$. [b]p11[/b] Points $X$ and $Y$ lie on sides $AB$ and $BC$ of triangle $ABC$, respectively. Ray $\overrightarrow{XY}$ is extended to point $Z$ such that $A, C$, and $Z$ are collinear, in that order. If triangle$ ABZ$ is isosceles and triangle $CYZ$ is equilateral, then the possible values of $\angle ZXB$ lie in the interval $I = (a^o, b^o)$, such that $0 \le a, b \le 360$ and such that $a$ is as large as possible and $b$ is as small as possible. Find $a + b$. [b]p12[/b] Let $S = \{(a, b) | 0 \le a, b \le 3, a$ and $b$ are integers $\}$. In other words, $S$ is the set of points in the coordinate plane with integer coordinates between $0$ and $3$, inclusive. Prair selects four distinct points in $S$, for each selected point, she draws lines with slope $1$ and slope $-1$ passing through that point. Given that each point in $S$ lies on at least one line Prair drew, how many ways could she have selected those four points?

All $n+3$ offices of University of Somewhere are numbered with numbers $0,1,2, \ldots ,$ $n+1,$ $n+2$ for some $n \in \mathbb{N}$. One day, Professor $D$ came up with a polynomial with real coefficients and power $n$. Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the $i$th office, for $i$ $\in$ $\{0,1, \ldots, n+1 \}$ he wrote $2^i$ and on the $(n+2)$th office he wrote $2^{n+2}$ $-n-3$. [list=a] [*] Prove that Professor D made a calculation error [*] Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them. [/list]

Let $m, n$ be positive integers with $m<n$ and consider an $n\times n$ board from which its upper left $ m\times m$ part has been removed. An example of such board for $n=5$ and $m=2$ is shown below. Determine for which pairs $(m, n)$ this board can be tiled with $3\times 1$ tiles. Each tile can be positioned either horizontally or vertically so that it covers exactly three squares of the board. The tiles should not overlap and should not cover squares outside of the board.

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection? $\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$