A sequence $\{a_n\}$ satisfies $$a_0=0, a_1=a_2=1, a_{3n} = a_n, a_{3n+1}=a_{3n+2} = a_n+1$$ for all $n \geq 1$. Compute $a_{2021}$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 6)[/i]

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer. Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$

Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

Given an octagon with the equal angles. The lengths of all the sides are integers. Prove that the opposite sides are equal in pairs. [u]alternate wording[/u] Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

Let $\vartriangle ABC$ be a right triangle such that $AB = 3$, $BC = 4$, $AC = 5$. Let point $D$ be on $AC$ such that the incircles of $\vartriangle ABD$ and $\vartriangle BCD$ are mutually tangent. Find the length of $BD$.

There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$.

A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?

An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that \[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\] The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

A given truncated pyramid has triangular bases. The areas of the bases are $B_1$ and $B_2$ and the area of the surface is $S$. Prove that if there exists a plane parallel to the bases whose intersection divides the pyramid to two truncated pyramids in which may be inscribed by spheres then $$S=(\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2$$ [i]G. Gantchev[/i]

A pirate ship spots, $10$ nautical miles to the east, an oblivious caravel sailing $60$ south of west at a steady $12 \text{ nm/hour}$. What is the minimum speed that the pirate ship must maintain at to be able to catch the caravel?

On sides $BC$, $CA$ and $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are taken, respectively, so that the radii of the circles inscribed in triangles $A_1BC_1$, $AB_1C_1$ and $A_1B_1C$ are equal to each other and equal to $r$. The radius of the circle inscribed in triangle $A_1B_1C_1$ is equal to $r_1$. Find the radius of the circle inscribed in triangle $ABC$.

Real numbers $x$ and $y$ satisfy \begin{align*} x^2 + y^2 &= 2023 \\ (x-2)(y-2) &= 3. \end{align*} Find the largest possible value of $|x-y|$. [i]Proposed by Howard Halim[/i]

Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)

The roots of the polynomial $x^4 - 4ix^3 +3x^2 -14ix - 44$ form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?

Given a parallelogram $ABCD$. A circle passing through $A$ meets the line segments $AB, AC$ and $AD$ at inner points $M,K,N$, respectively. Prove that \[|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|\]

Find the second smallest positive integer $n$ such that when $n$ is divided by $5,$ the remainder is $3,$ and when $n$ is divided by $7,$ the remainder is $4.$

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. [i]Bulgaria[/i]

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.

A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.

Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$

$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.

(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$. (b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers. Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$