The cells of an $n \times n$ table are filled with the numbers $1,2,\dots,n$ for the first row, $n+1,n+2,\dots,2n$ for the second, and so on until $n^2-n,n^2-n+1,\dots,n^2$ for the $n$-th row. Peter picks $n$ numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum $S$ of the numbers he has chosen. Prove that Peter always gets the same number for $S$, no matter how he chooses his $n$ numbers.

The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)

For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!\plus{}1$ and $ (n\plus{}1)!$. Find, without proof, a formula for $ f(n)$.

Consider triangle $\vartriangle ABC$ with $\angle C = 90^o$. Let $P$ be the midpoint of $\overline{AC}$ so that $AP = PC = 1$, and suppose $\angle BAC = \angle CBP$. Compute $AB^2$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$. Let $M$ be the midpoint of the arc $ABC$. The circle with center $M$ and radius $MA$ meets $AD, AB$ at $X, Y$. The point $Z \in XY$ with $Z \neq Y$ satisfies $BY=BZ$. Show that $\angle BZD=\angle BCD$.

All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

Find all the ways of placing the integers $1,2,3,\cdots,16$ in the boxes below, such that each integer appears in exactly one box, and the sum of every pair of neighboring integers is a perfect square. [asy] import graph; real r=10; size(r*cm); picture square1; draw(square1, (0,0)--(0,1)--(1,1)--(1,0)--cycle); add(scale(r/31*cm)*square1,(0,0)); picture square2; draw(square2, (-1,0.5)--(0,0.5)--(0,1)--(1,1)--(1,0)--(0,0)--(0,0.5)); for(int i=1; i<16; ++i) { add(scale(r/31*cm)*square2,(i,0)); }[/asy]

Every cell of $100\times 100$ table is colored black or white. Every cell on table border is black. It is known, that in every $2\times 2$ square there are cells of two colors. Prove, that exist $2\times 2$ square that is colored in chess order.

Given a square $ABCD$ of side $a$, we consider the circle $\omega$, tangent to side $BC$ and to the two semicircles of diameters $AB$ and $CD$. Calculate the radius of circle $\omega$,

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

Alex and Brian take turns shooting free throws until they each shoot twice. Alex and Brian have 80% and 60% chances of making their free throws, respectively. What is the probability that after each free throw they take, Alex has made at least as many free throws as Brian if Brian shoots first?

In an acute triangle $ABC$, its inscribed circle touches the sides $AB,BC$ at the points $C_1,A_1$ respectively. Let $M$ be the midpoint of the side $AC$, $N$ be the midpoint of the arc $ABC$ on the circumcircle of triangle $ABC$, and $P$ be the projection of $M$ on the segment $A_1C_1$. Prove that the points $P,N$ and the incenter $I$ of the triangle $ABC$ lie on the same line.

For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix $$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$ is positive definite.

Petya drew a pentagon $ABCDE$ on the plane. After that, Vasya marked all the points $S$ in a given half-space relative to the plane of the pentagon so that in the pyramid $SABCD$ exactly two side faces are perpendicular to the plane of the base $ABCD$, and the height is $1$. How many points could have Vasya?

For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$. Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.

Is there a triangle with sides of integer lengths such that the length of the shortest side is $ 2007$ and that the largest angle is twice the smallest?

All diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? $\textbf{(A)} \ 49 \qquad \textbf{(B)} \ 65 \qquad \textbf{(C)} \ 70 \qquad \textbf{(D)} \ 96 \qquad \textbf{(E)} \ 128$

One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy counts a total of $28$ heads belonging to the people, horses, and dog. She counts a total of $92$ legs belonging to everyone, and notes that nobody is missing any legs. Upon returning home Wendy gives Alexis a little problem solving practice, "I saw $28$ heads and $92$ legs belonging to people, horses, and dogs. Assuming two legs per person and four for the other animals, how many people did I see?" Alexis scribbles out some algebra and answers correctly. What is her answer?

On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial on the blackboard has $37$ distinct positive roots. [i](8 points)[/i] [i]Alexandr Kuznetsov[/i]

For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.

Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes: - if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table; - if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$. As a function of $k$, which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$?

i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$. ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]

Let $D$ and $E$ be points on side $[AB]$ of a right triangle with $m(\widehat{C})=90^\circ$ such that $|AD|=|AC|$ and $|BE|=|BC|$. Let $F$ be the second intersection point of the circumcircles of triangles $AEC$ and $BDC$. If $|CF|=2$, what is $|ED|$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ 1+\sqrt 2 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 2\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

The sum of 5 consecutive even integers is $ 4$ less than the sum of the first $ 8$ consecutive odd counting numbers. What is the smallest of the even integers? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.