Let $a$ and $b$ be positive integers with $\gcd(a, b)=1$. Show that every integer greater than $ab-a-b$ can be expressed in the form $ax+by$, where $x, y \in \mathbb{N}_{0}$.

Prove that there are no positive integers $a_1, a_2, \ldots , a_{2021}$ (not necessarily distinct) such that for $k = 1, 2, 3, \ldots , 2021$ the number of elements in the set $$A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}$$ be exactly $a_k$.

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

In a dance, a group $S$ of $1994$ students stand in a big circle. Each student claps the hands of each of his two neighbours a number of times. For each student $x,$ let $f(x)$ be the total number of times $x$ claps the hands of his neighbours. As an example, suppose there are $3$ students $A, B$ and $C$. A claps hand with $B$ two times, $B$ claps hand with $C$ three times and $C$ claps hand with $A$ five times. Then $f(A) = 7, f(B) = 5$ and $f(C) = 8.$ (i) Prove that $\{f(x) | x \in S\}\ne\{n | n$ is an integer, $2 \le n \le 1995\}$. (ii) Find an example in which $\{f(x) | x \in S\} = \{n | n$ is an integer, $n \ne 3, 2 \le n \le 1996\}$

Let $\mathcal A = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a [i]regular 2014-simplex[/i], meaning the $2015$ vertices of $\mathcal A$ lie in $2014$-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by $10^6$. [i]Proposed by Robin Park[/i]

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have?

[b]p1.[/b] (See the diagram below.) $ABCD$ is a square. Points $G$, $H$, $I$, and $J$ are chosen in the interior of $ABCD$ so that: (i) $H$ is on $\overline{AG}$, $I$ is on $\overline{BH}$, $J$ is on $\overline{CI}$, and $G$ is on $\overline{DJ}$ (ii) $\vartriangle ABH \sim \vartriangle BCI \sim \vartriangle CDJ \sim \vartriangle DAG$ and (iii) the radii of the inscribed circles of $\vartriangle ABH$, $\vartriangle BCI$, $\vartriangle CDJ$, $\vartriangle DAK$, and $GHIJ$ are all the same. What is the ratio of $\overline{AB}$ to $\overline{GH}$? [img]https://cdn.artofproblemsolving.com/attachments/f/b/47e8b9c1288874bc48462605ecd06ddf0f251d.png[/img] [b]p2.[/b] The three solutions $r_1$, $r_2$, and $r_3$ of the equation $$x^3 + x^2 - 2x - 1 = 0$$ can be written in the form $2 \cos (k_1 \pi)$, $2 \cos (k_2 \pi)$, and $2 \cos (k_3 \pi)$ where $0 \le k_1 < k_2 < k_3 \le 1$. What is the ordered triple $(k_1, k_2, k_3)$? [b]p3.[/b] $P$ is a convex polyhedron, all of whose faces are either triangles or decagons ($10$-sided polygon), though not necessarily regular. Furthermore, at each vertex of $P$ exactly three faces meet. If $P$ has $20$ triangular faces, how many decagonal faces does P have? [b]p4.[/b] $P_1$ is a parabola whose line of symmetry is parallel to the $x$-axis, has $(0, 1)$ as its vertex, and passes through $(2, 2)$. $P_2$ is a parabola whose line of symmetry is parallel to the $y$-axis, has $(1, 0)$ as its vertex, and passes through $(2, 2)$. Find all four points of intersection between $P_1$ and $P_2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The number $12$ is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers $2$ or $3$ (a division is possible only if the result is an integer) . Prove that the number that will be written on the board in exactly one hour will not be equal to $54$.

As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$. [asy] size(200); pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C); draw(F--C--A--B--C^^A--D^^B--E); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("84", centroid(H, C, E), fontsize(9.5)); label("35", centroid(H, B, D), fontsize(9.5)); label("30", centroid(H, F, B), fontsize(9.5)); label("40", centroid(H, A, F), fontsize(9.5));[/asy]

Compute the unique five-digit positive integer $\underline{abcde}$ such that $a \neq 0, c \neq 0,$ and $$\underline{abcde}=(\underline{ab}+\underline{cde})^2.$$

Shikaku and his son Shikamaru must climb a staircase that has $2022$ steps; the steps are listed $1$, $2$, $...$ , $2022$ and the floor is considered step $0$. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of $7$ steps, that is, if they are in the steps $m$ and$ n$, then it must always be true that $|m-n| \le 7$. For the game they establish the following rules: a) They move alternately in turns. b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously. c) If a player has just moved to the $n$-th step, then on the next turn the other player cannot be moved to any of the steps $n-1$, $n$ or $n + 1$, except when it is for reach the last step. d) Whoever reaches the last step (listed with $2022$) wins. Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.

Prove that if $ a,x,y$ are $ p$-adic integers different from $ 0$ and $ p | x, pa | xy$, then \[ \frac 1y \frac{(1\plus{}x)^y\minus{}1}{x} \equiv \frac{\log (1\plus{}x)}{x} \;\;\;\; ( \textrm{mod} \; a\ ) \\\\ .\] [i]L. Redei[/i]

For any real number $ x$ prove that: \[ x\in \mathbb{Z}\Leftrightarrow \lfloor x\rfloor \plus{}\lfloor 2x\rfloor\plus{}\lfloor 3x\rfloor\plus{}...\plus{}\lfloor nx\rfloor\equal{}\frac{n(\lfloor x\rfloor\plus{}\lfloor nx\rfloor)}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]

How many zeros are at the end of the product \[25\times 25\times 25\times 25\times 25\times 25\times 25\times 8\times 8\times 8?\] $\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 12$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ satisfies, $$2+f(x)f(y)\leq xy+2f(x+y+1).$$

The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.

Let $ABCD$ be a convex, non-cyclic quadrilateral with $E$ the intersection of its diagonals. Given $\angle ABD+\angle DAC = \angle CBD+\angle DCA$, $AB = 10$, $BC = 15$, $AE = 7$, and $EC = 13$, find $BD$.

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a [i]clique[/i] if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its [i]size[/i]. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room. [i]Author: Vasily Astakhov, Russia[/i]

In a scientific conference, all participants can speak in total $2 \cdot n$ languages ($n \geq 2$). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer $k$, $1 \leq k \leq n-1$ there are at most $k-1$ languages such that each of these languages is spoken by at most $k$ participants. Show that we can choose a group from $2 \cdot n$ participants which in total can speak $2 \cdot n$ languages and each language is spoken by exactly two participants from this group.

Find the smallest natural number that is the sum of $9$ consecutive natural numbers, is the sum of $10$ consecutive natural numbers and is also the sum of $11$ consecutive natural numbers.

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.

One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $a_1,a_2,\dotsc$ be a sequence of real numbers from the interval $[0,1]$. Prove that there is a sequence $1\leqslant n_1<n_2<\dotsc$ of positive integers such that $$A=\lim_{\substack{i,j\to \infty \\ i\neq j}} a_{n_i+n_j}$$exists, i.e., for every real number $\epsilon >0$, there is a constant $N_{\epsilon}$ that $|a_{n_i+n_j}-A|<\epsilon$ is satisfied for any pair of distinct indices $i,j>N_{\epsilon}$.