The vertices of a regular $2012$-gon are labelled with the numbers $1$ through $2012$ in some order. Call a vertex a peak if its label is larger than the label of its two neighbours, and a valley if its label is smaller than the label of its two neighbours. Show that the total number of peaks is equal to the total number of valleys.

Suppose that the automorphism group of the finite undirected graph $ X\equal{}(P, E)$ is isomorphic to the quaternion group (of order $ 8$). Prove that the adjacency matrix of $ X$ has an eigenvalue of multiplicity at least $ 4$. ($ P\equal{} \{ 1,2,\ldots, n \}$ is the set of vertices of the graph $ X$. The set of edges $ E$ is a subset of the set of all unordered pairs of elements of $ P$. The group of automorphisms of $ X$ consists of those permutations of $ P$ that map edges to edges. The adjacency matrix $ M\equal{}[m_{ij}]$ is the $ n \times n$ matrix defined by $ m_{ij}\equal{}1$ if $ \{ i,j \} \in E$ and $ m_{i,j}\equal{}0$ otherwise.) [i]L. Babai[/i]

In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.

[b]5.[/b] Determine the functions $G$ defined on the set of all non-zero real numbers the values of which are regular matrices of order $2$, and the functions $f$ mapping the two-dimensional real vector space $E_2$ into itself, such that for any vector $y \in E_2$ and for any regular matrix $X$ of order $2$, $f(X_y)= G(det X)Xf(y)$ ($det X $ denotes the determinant of $X$).[b](A. 5)[/b]

Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).

In a $50\times 50$ checkered square, each cell is colored in one of the 100 given colors so that all colors are used and there does not exist a monochromatic domino. Galia wants to repaint all the cells of one of the colors in a different color (from the given 100 colors) so that a monochromatic domino still won't exist. Is it true that Galia will surely be able to do this [i]Proposed by G. Sharafutdinova[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x+y)\leq f(x^2+y)$$ for all $x,y$.

Let $ABC$ be a triangle with incenter $I$. Let $P$ and $Q$ be points such that $IP\perp AC$, $IQ\perp AB$, and $IA\perp PQ$. Assume that $BP$ and $CQ$ intersect at the point $R\neq A$ on the circumcircle of $ABC$ such that $AR\parallel BC$. Given that $\angle B-\angle C=36^\circ$, the value of $\cos A$ can be expressed in the form $\frac{m-\sqrt n}{p}$ for positive integers $m,n,p$ and where $n$ is not divisible by the square of any prime. Find the value of $100m+10n+p$. [i]Proposed by Michael Ren[/i]

In the isosceles triangle $ABC$ ($AC = BC$), $AB =\sqrt3$ and the altitude $CD =\sqrt2$. Let $E$ and $F$ be the midpoints of $BC$ and $DB$, respectively and $G$ be the intersection of $AE$ and $CF$. Prove that $D$ belongs to the angle bisector of $\angle AGF$.

If we define $\otimes(a,b,c)$ by \begin{align*} \otimes(a,b,c) = \frac{\max(a,b,c)- \min(a,b,c)}{a+b+c-\min(a,b,c)-\max(a,b,c)}, \end{align*} compute $\otimes(\otimes(7,1,3),\otimes(-3,-4,2),1)$.

Given a convex quadrilateral $ABCD$, find the locus of the points $P$ inside the quadrilateral such that \[S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}\] (where $S_X$ denotes the area of triangle $X$).

A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$. [i]Proposed by Michał Pilipczuk[/i]

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?