Prove that for every non-negative integer $m$ there exists a polynomial w with integer coefficients such that $2^m$ is the greatest common divisor of the numbers $$ a_n = 3^n + w(n), n = 0, 1, 2, ....$$

An acute triangle \(ABC\) has circumcircle \(\Gamma\) and circumcentre \(O\). The incentres of \(AOB\) and \(AOC\) are \(I_b\) and \(I_c\) respectively. Let \(M\) be the the point on \(\Gamma\) such that \(MB = MC\) and \(M\) lies on the same side of \(BC\) as \(A\). Prove that the points \(M\), \(A\), \(I_b\), and \(I_c\) are concyclic.

The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set $A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$. Compute the sum $A+B$ in terms of $x$.

[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\ [b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.

Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.)

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

The game tic-tac is played on a $3$ by $3$ square grid between players $X$ and $O$. They take turns, and on their turn a player writes their symbol onto one empty space of the grid. A player wins if they fill a row or column with three copies of their symbol; a player filling a main diagonal does [i]not[/i] end the game in a win for that player. If the grid is filled without determining the winner, the game is a draw. Assuming player $X$ goes first and the players draw the game, how many possibilities are there for the final state of the grid? $\text{(A) }24\qquad\text{(B) }33\qquad\text{(C) }36\qquad\text{(D) }45\qquad\text{(E) }126$

In a park there are 23 trees $t_0,t_1,\dots,t_{22}$ in a circle and 22 birds $b_1,n_2,\dots,b_{22}.$ Initially, each bird is in a tree. Every minute, the bird $b_i, 1\leqslant i\leqslant 22$ flies from the tree $t_j{}$ to the tree $t_{i+j}$ in clockwise order, indices taken modulo 23. Prove that there exists a moment when at least 6 trees are empty.

Let $\triangle ABC$ have $\angle A=20^{\circ}$ and $\angle C=40^{\circ}$. We've constructed the angle bisector $AL$ ($L\in BC$) and the external angle bisector $CN$ ($N\in AB$). Find $\angle CLN$.

In a general triangle $ ADE$ (as shown) lines $ \overline{EB}$ and $ \overline{EC}$ are drawn. Which of the following angle relations is true? [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair B = (1,0); pair C = (2,0); pair D = (3,0); pair E = (1.25,1.75); draw(A--D--E--cycle); draw(E--B); draw(E--C); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,SE); label("$E$",E,N); label("$y$",E,3SW + 3S); label("$w$",E,7S + E); label("$b$",E,3SE + 4S + E); label("$x$",A,NE); label("$z$",B,NW); label("$m$",B,NE); label("$n$",C,NW); label("$c$",C,NE); label("$a$",D,NW+W);[/asy] $ \textbf{(A)}\ x \plus{} z \equal{} a \plus{} b\qquad \textbf{(B)}\ y \plus{} z \equal{} a \plus{} b\qquad \textbf{(C)}\ m \plus{} x \equal{} w \plus{} n\qquad \\ \textbf{(D)}\ x \plus{} z \plus{} n \equal{} w \plus{} c \plus{} m\qquad \textbf{(E)}\ x \plus{} y \plus{} n \equal{} a \plus{} b \plus{} m$

The point $D$ on the altitude $AA_1$ of an acute triangle $ABC$ is such that $\angle BDC=90^\circ$; $H$ is the orthocentre of $ABC$. A circle with diameter $AH$ is constructed. Prove that the tangent drawn from $B$ to this circle is equal to $BD$.

If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .

The lengths of the diagonals of a rhombus are, in inches, two consecutive integers. The area of the rhombus is $210$ sq. in. Find its perimeter, in inches.

The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.

Let $ABCD$ be a circumscribed quadrilateral and $T=AC\cap BD$. Let $I_1$, $I_2$, $I_3$, $I_4$ the incenters of $\Delta TAB$, $\Delta TBC$, $TCD$, $TDA$, respectively, and $J_1$, $J_2$, $J_3$, $J_4$ the incenters of $\Delta ABC$, $\Delta BCD$, $\Delta CDA$, $\Delta DAB$. Show that $I_1I_2I_3I_4$ is a cyclic quadrilateral and its center is $J_1J_3\cap J_2J_4$

For a positive integer $n$, there is a school with $2n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is no more than $n$, find the maximum number of well-formed set. Here, an empty set and a set with one student is regarded as well-formed as well.

Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.

There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

[b]p1.[/b] Find the sum of all $3$-digit positive integers $\overline{abc}$ that satisfy $$\overline{abc} = {n \choose a}+{n \choose b}+ {n \choose c}$$ for some $n \le 10$. [b]p2.[/b] Feng and Trung play a game. Feng chooses an integer $p$ from $1$ to $90$, and Trung tries to guess it. In each round, Trung asks Feng two yes-or-no questions about $p$. Feng must answer one question truthfully and one question untruthfully. After $15$ rounds, Trung concludes there are n possible values for $p$. What is the least possible value of $n$, assuming Feng chooses the best strategy to prevent Trung from guessing correctly? [b]p3.[/b] A hypercube $H_n$ is an $n$-dimensional analogue of a cube. Its vertices are all the points $(x_1, .., x_n)$ that satisfy $x_i = 0$ or $1$ for all $1 \le i \le n$ and its edges are all segments that connect two adjacent vertices. (Two vertices are adjacent if their coordinates differ at exactly one $x_i$ . For example, $(0,0,0,0)$ and $(0,0,0,1)$ are adjacent on $H_4$.) Let $\phi (H_n)$ be the number of cubes formed by the edges and vertices of $H_n$. Find $\phi (H_4) + \phi (H_5)$. [b]p4.[/b] Denote the legs of a right triangle as $a$ and $b$, the radius of the circumscribed circle as $R$ and the radius of the inscribed circle as $r$. Find $\frac{a+b}{R+r}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $\triangle ABC$, $AB = AC$. Its circumcircle, $\Gamma$, has a radius of 2. Circle $\Omega$ has a radius of 1 and is tangent to $\Gamma$, $\overline{AB}$, and $\overline{AC}$. The area of $\triangle ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$, where $b$ is squarefree and $\gcd (a, c) = 1$. Compute $a + b + c$. [i]Proposed by Aaron Lin[/i]

At the start of the Mighty Mathematicians Football Team's first game of the season, their coach noticed that the jersey numbers of the 22 players on the field were all the numbers from 1 to 22. At halftime, the coach substituted her goal-keeper, with jersey number 1, for a reserve player. No other substitutions were made by either team at or before halftime. The coach noticed that after the substitution, no two players on the field had the same jersey number and that the sums of the jersey numbers of each of the teams were exactly equal. Determine * the greatest possible jersey number of the reserve player, * the smallest possible (positive) jersey number of the reserve player.