Prove that for all sufficiently large positive integers $d{}$, at least $99\%$ of the polynomials of the form \[\sum_{i\leqslant d}\sum_{j\leqslant d}\pm x^iy^j\]are irreducible over the integers.

Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$? Adam Ardeishar and Michael Ren

Bethan is playing a game on an $n\times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves Bethan realises she cannot make any more moves. Determine the minimum value of $M$. United Kingdom, Sam Bealing

Let $n{}$ be a positive integer, and let $\mathcal{C}$ be a collection of subsets of $\{1,2,\ldots,2^n\}$ satisfying both of the following conditions: Every $(2^n-1)$-element subset of $\{1,2,\ldots,2^n\}$ is a member of $\mathcal{C}$, and Every non-empty member $C$ of $\mathcal{C}$ contains an element $c$ such that $C\setminus\{c\}$ is again a member of $\mathcal{C}$. Determine the smallest size $\mathcal{C}$ may have. Serbia, Pavle Martinovic ́

Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other. Russia, Sergei Avgustinovich and Dmitry Khramtsov

A ternary sequence is one whose terms all lie in the set $\{0, 1, 2\}$. Let $w$ be a length $n$ ternary sequence $(a_1,\ldots,a_n)$. Prove that $w$ can be extended leftwards and rightwards to a length $m=6n$ ternary sequence \[(d_1,\ldots,d_m) = (b_1,\ldots,b_p,a_1,\ldots,a_n,c_1,\ldots,c_q), \quad p,q\geqslant 0,\]containing no length $t > 2n$ palindromic subsequence. (A sequence is called palindromic if it reads the same rightwards and leftwards. A length $t$ subsequence of $(d_1,\ldots,d_m)$ is a sequence of the form $(d_{i_1},\ldots,d_{i_t})$, where $1\leqslant i_1<\cdots<i_t \leqslant m$.)

The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. Iran, Shayan Talaei

Let $ABC$ be an acute scalene triangle, and let $A_1, B_1, C_1$ be the feet of the altitudes from $A, B, C$. Let $A_2$ be the intersection of the tangents to the circle $ABC$ at $B, C$ and define $B_2, C_2$ similarly. Let $A_2A_1$ intersect the circle $A_2B_2C_2$ again at $A_3$ and define $B_3, C_3$ similarly. Show that the circles $AA_1A_3, BB_1B_3$, and $CC_1C_3$ all have two common points, $X_1$ and $X_2$ which both lie on the Euler line of the triangle $ABC$. United Kingdom, Joe Benton

In the triangle $ABC$ with circumcircle $\Gamma$, the incircle $\omega$ touches sides $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $K\neq D$. Line $AK$ meets $\Gamma$ at $L\neq A$. Rays $KI$ and $IL$ meet the circumcircle of triangle $BIC$ at $Q\neq I$ and $P\neq I$, respectively. The circumcircles of triangles $KFB$ and $KEC$ meet $EF$ at $R\neq F$ and $S\neq E$, respectively. Prove that $PQRS$ is cyclic. India, Anant Mugdal

Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct. (For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.) Poland, Wojciech Nadara

For a positive integer $n$, let $\varphi(n)$ and $d(n)$ denote the value of the Euler phi function at $n$ and the number of positive divisors of $n$, respectively. Prove that there are infinitely many positive integers $n$ such that $\varphi(n)$ and $d(n)$ are both perfect squares. Finland, Olli Järviniemi