Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

Find the largest prime that divides $1\cdot 2\cdot 3+2\cdot 3\cdot 4+\cdots +44\cdot 45\cdot 46$

Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order.

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

Prove the following statement: If a polynomial $f(x)$ with real coefficients takes only nonnegative values, then there exists a positive integer $n$ and polynomials $g_1(x), g_2(x),\cdots, g_n(x)$ such that \[f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2\]

Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$.

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]

The captain and his three sailors get $2009$ golden coins with the same value . The four people decided to divide these coins by the following rules : sailor $1$,sailor $2$,sailor $3$ everyone write down an integer $b_1,b_2,b_3$ , satisfies $b_1\ge b_2\ge b_3$ , and ${b_1+b_2+b_3=2009}$; the captain dosen't know what the numbers the sailors have written . He divides $2009$ coins into $3$ piles , with number of coins: $a_1,a_2,a_3$ , and $a_1\ge a_2\ge a_3$ . For sailor $k$ ($k=1,2,3$) , if $b_k<a_k$ , then he can take $b_k$ coins from the $k$th pile ; if $b_k\ge a_k$ , then he can't take any coins away . At last , the captain own the rest of the coins .If no matter what the numbers the sailors write , the captain can make sure that he always gets $n$ coins . Find the largest possible value of $n$ and prove your conclusion .

How many ways are there to fill in the $ 8$ spots in the picture with letters $A, B, C$ and $D$, using two copies of each letter, such that the spots with identical letters can be connected with a continuous line that stays within the box, without these four lines crossing each other or going through other spots? The lines do not have to be straight. [img]https://cdn.artofproblemsolving.com/attachments/f/f/66c30eaf6fa3b42c5197d0e3a3d59e9160bb8e.png[/img]

Evaluate $$\frac{1}{1!21!} + \frac{1}{3!19!} + \frac{1}{5!16!} + ... + \frac{1}{21!1!}$$

Find all the integers $n \ge 5$ such that the residue of $n$ when divided by each prime number smaller than $\frac{n}{2}$ is odd.

If $60^a = 3$ and $60^b = 5$, then $12^{[(1-a-b)/2(1-b)]}$ is $\text{(A)} \ \sqrt{3} \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt{5} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \sqrt{12}$

Given a square $ABCD$ of side length $6$, the point $E$ is drawn on the line $AB$ such that the distance $EA$ is less than $EB$ and the triangle $\vartriangle BCE$ has the same area as $ABCD$. Compute the shaded area. [img]https://cdn.artofproblemsolving.com/attachments/a/8/5d945a593aee58af3af94f4e8e967eeaeefa6a.png[/img]

The digit in unit place of $1!+2!+\ldots+99!$ is $\textbf{(A)}~3$ $\textbf{(B)}~0$ $\textbf{(C)}~1$ $\textbf{(D)}~7$

Gonçalo writes in a board four of the the following numbers $0, 1, 2, 3, 4$, he can repeat numbers. Nicolas can realize the following operation: change one number of the board, by the remainder(in the division by $5$) of the product of others two numbers of the board. Nicolas wins if all the four numbers are equal, determine if Gonçalo can choose numbers such that Nicolas will never win.

For a positive integer $K$, define a sequence, $\{a_n\}$, as following: $a_1 = K$ and $a_{n+1} =a_n -1$ if $a_n$ is even $a_{n+1} =\frac{a_n - 1}{2}$ if $a_n$ is odd , for all $n \ge 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to $0$.

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposted in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited? $\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$

Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.) For which values of $n$ is it possible to end with only one number remaining on the board? [i](Michael Reitmeir)[/i]

The elevator buttons in Harvard's Science Center form a $3\times 2$ grid of identical buttons, and each button lights up when pressed. One day, a student is in the elevator when all the other lights in the elevator malfunction, so that only the buttons which are lit can be seen, but one cannot see which floors they correspond to. Given that at least one of the buttons is lit, how many distinct arrangements can the student observe? (For example, if only one button is lit, then the student will observe the same arrangement regardless of which button it is.)

Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$

In an $6 \times 6$ grid of lattice points, how many ways are there to choose $ 4$ points that are vertices of a nondegenerate quadrilateral with at least one pair of opposite sides parallel to the sides of the grid?

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]