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.

Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?

Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

Given a triangle $ABC$, there’s a point $X$ on the side $AB$ such that $2BX = BA + BC$. Let $Y$ be the point symmetric to the incenter $I$ of triangle $ABC$, with respect to point $X$. Prove that $YI_B\perp AB$ where $I_B$ is the $B$-excenter of triangle $ABC$.

A weird calculator has a numerical display and only two buttons, $\boxed{D\sharp}$ and $\boxed{D\flat}$. The first button doubles the displayed number and then adds $1$. The second button doubles the displayed number and then subtracts $1$. For example, if the display is showing $5$, then pressing the $\boxed{D\sharp}$ produces $11$. If the display shows $5$ and we press $\boxed{D\flat}$, we get $9$. If the display shows $5$ and we press the sequence $\boxed{D\sharp}$, $\boxed{D\flat}$, $\boxed{D\sharp}$, $\boxed{D\sharp}$, we get a display of $87$. [list=i] [*] Suppose the initial displayed number is $1$. Give a sequence of exactly eight button presses that will result in a display of $313$. [*] Suppose the initial displayed number is $1$, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced. [/list]

Show that a square can be inscribed in any regular polygon.

Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and \[ a_k = (n-k+1) \cdot c_{k-1}, \quad b_k = \binom nk - c_k - a_k, \quad \text{and} \quad c_k = \frac{b_{k-1}}{k} \] for each integer $1 \leq k \leq n$. $ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$. Proposed by [i]Jonathan Du[/i]

In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$. [i]Proposed by Evin Liang[/i] [hide=Solution][i]Solution[/i]. $\boxed{23}$ Use coordinates with $A = (0,12)$, $B = (5,0)$, and $C = (-9,0)$. Then $M = \left(\dfrac{5}{2},6\right)$ and $E = (0,0)$. By shoelace, the area of $GAME$ is $\boxed{23}$.[/hide]

Divide the set of twelve numbers $A = \{3, 4, 5, ...,13, 14\}$ into two sets $ B$ and $C$ 'of six numbers each according to this condition: for any two different numbers with $ B$ their sum does not belong to $ B$ and for any two different numbers from $C$, the sum does not belong to $C$.

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.