Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.

There are $n \ge 7$ points in the plane, no $3$ of which are collinear. At least $7$ pairs of points are joined by line segments. For every aforementioned line segment $s$, let $t(s)$ be the number of triangles for which the segment $s$ is a side. Prove that there exist different line segments $s_1, s_2, s_3,$ and $s_4$ such that \[t(s_1) = t(s_2) = t(s_3) = t(s_4)\] holds. Proposed by [i]Viktor Simjanoski[/i]

A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$, \[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\] Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$ Evaluate \[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]

A positive integer $ K$ is given. Define the sequence $ (a_n)$ by $ a_1\equal{}1$ and $ a_n$ is the $ n$-th natural number greater than $ a_{n\minus{}1}$ which is congruent to $ n$ modulo $ K$. $ (a)$ Find an explicit formula for $ a_n$. $ (b)$ What is the result if $ K\equal{}2?$

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation: $$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$ then $ z_1^6=z_2^6 $

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that $O$ is the centre of the circumscribed circle. [u]other formulation:[/u] $P$ is a convex polygon and $X$ is an interior point such that for every pair of vertices $A, B$, the triangle $XAB$ is isosceles. Prove that all the vertices of $P$ lie on a circle with center $X$.

(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \] (b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]

Determine the number of $2021$-tuples of positive integers such that the number $3$ is an element of the tuple and consecutive elements of the tuple differ by at most $1$. [i]Walther Janous (Austria)[/i]

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

Numbers $1,\ldots,2025$ are written in a circle in increasing order. For every three consecutive numbers $i,j,k$ we consider the polynomial $(x-i)(x-j)(x-k)$. Let $s(x)$ be the sum of all $2025$ these polynomials. Prove that $s(x)$ has an integral root. [i]A. Voidelevich[/i]

A square, a regular pentagon, and a regular hexagon are all inscribed in the same circle. The $15$ vertices of these polygons divide the circle into at most $15$ arcs. Let $M$ be the degree measure of the longest of these arcs. Find the minimum possible value for $M$.

Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.

Suppose that $G$ is a finite group generated by the two elements $g$ and $h,$ where the order of $g$ is odd. Show that every element of $G$ can be written in the form \[g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}\] with $1\le r\le |G|$ and $m_n,n_1,m_2,n_2,\dots,m_r,n_r\in\{1,-1\}.$ (Here $|G|$ is the number of elements of $G.$)

$\textbf{(Lucas Numbers)}$ The Lucas numbers are defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \ge 0$. There are $N$ integers $1 \le n \le 2016$ such that $L_n$ contains the digit $1$. Estimate $N$. An estimate of $E$ earns $\left\lfloor 20 - 2|N-E| \right\rfloor$ or $0$ points, whichever is greater.

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$. Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$. [i]Proposed by Greece[/i]

$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that \[ p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1 \] Show that $p'(x) = nq(x)$.

3. Eight farmers have a checkered $8 \times 8$ field. There is a fence along the boundary of the field. The entire field is completely covered with berries (there is a berry in every point of the field, except the points of the fence). The farmers divided the field along the grid lines in 8 plots of equal area (every plot is a polygon), however they did not demarcate their boundaries. Each farmer takes care of berries only inside his own plot (not on its boundaries). A farmer will notice a loss only if at least two berries disappeared inside his plot. There is a crow which knows all of the above, except the location of boundaries of plots. Can the crow carry off 9 berries from the field so that for sure no farmer will notice this? Tatiana Kazitsyna

At one school, $85$ percent of the students are taking mathematics courses, $55$ percent of the students are taking history courses, and $7$ percent of the students are taking neither mathematics nor history courses. Find the percent of the students who are taking both mathematics and history courses.

Let $p$ be a prime number. Show that $7^p+3p-4$ is not a perfect square.

Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.

A real-valued function $ f$ satisfies for all reals $ x$ and $ y$ the equality \[ f (xy) \equal{} f (x)y \plus{} x f (y). \] Prove that this function satisfies for all reals $ x$ and $ y \ne 0$ the equality \[ f\left(\frac{x}{y}\right)\equal{}\frac{f (x)y \minus{} x f (y)}{y^2} \]

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying \[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \] for all integers $x$. [i]Proposed by Ankan Bhattacharya[/i]