Calculate $ \begin{pmatrix}1&0&0& \ldots &0\\\binom{1}{0} &\binom{1}{1} &0& \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \binom{n}{0} &\binom{n}{1} & \binom{n}{2} & \ldots & \binom{n}{n}\end{pmatrix}^{-1} . $

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]

Find all reals $a$ such that the sequence $\{x(n)\}$, $n=0,1,2, \ldots$ that satisfy: $x(0)=1996$ and $x_{n+1} = \frac{a}{1+x(n)^2}$ for any natural number $n$ has a limit as n goes to infinity.

A cyclic quadrilateral $ABCD$ is given. Let $I{}$ and $J{}$ be the centers of circles inscribed in the triangles $ABC$ and $ADC$. It turns out that the points $B, I, J, D$ lie on the same circle. Prove that the quadrilateral $ABCD$ is tangential.

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

For how many integers is the number $x^4-51x^2+50$ negative? $ \textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }14\qquad\textbf{(E) }16\qquad $

Determine the number of sequences of points $(x_1, y_1),(x_2, y_2), \dots ,(x_{4570}, y_{4570})$ on the plane satisfying the following two properties: $\text{(i)}$ $\{x_1,x_2,\dots,x_{4570}\}=\{1,2,\dots,2014\}$ and $\{y_1,y_2,\dots,y_{4570}\}=\{1,2,\dots,2557\}$ $\text{(ii)} $ For each $i = 1, 2,\dots , 4569$, exactly one of $x_i = x_{i+1}$ and $y_i = y_{i+1}$ holds.

Let be four $ n\times n $ real matrices $ A,B,C,D $ having the property that $ C+D\sqrt{-1} $ is the inverse of $ A+B\sqrt{-1} . $ Show that $ \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A. $ [i]Vasile Pop[/i]

Let $f : Z^2 \to C$ be a function such that $f(x+11, y) = f(x, y+11) = f(x, y)$, and $f(x, y)f(z,w) = f(xz - yw,xw + yz)$. How many possible values can $f(1, 1)$ have?

Tom throws a football to Wes, who is a distance $l$ away. Tom can control the time of flight $t$ of the ball by choosing any speed up to $v_{\text{max}}$ and any launch angle between $0^\circ$ and $90^\circ$. Ignore air resistance and assume Tom and Wes are at the same height. Which of the following statements is [b]incorrect[/b]? $ \textbf{(A)}$ If $v_{\text{max}} < \sqrt{gl}$, the ball cannot reach Wes at all. $ \\ $ $ \textbf{(B)}$ Assuming the ball can reach Wes, as $v_{\text{max}}$ increases with $l$ held fixed, the minimum value of $t$ decreases. $ \\ $ $ \textbf{(C)}$ Assuming the ball can reach Wes, as $v_{\text{max}}$ increases with $l$ held fixed, the maximum value of $t$ increases. $ \\ $ $ \textbf{(D)}$ Assuming the ball can reach Wes, as $l$ increases with $v_{\text{max}}$ held fixed, the minimum value of $t$ increases. $ \\ $ $ \textbf{(E)}$ Assuming the ball can reach Wes, as $l$ increases with $v_{\text{max}}$ held fixed, the maximum value of $t$ increases.

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

In a quadrilateral $ABCD$, we have $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$ and $\angle BCD = 70^{\circ}$ . Let $ M, N$ be the mid-points of $AB$ and $CD$ respectively. Suppose $P$ is a point on the segment $M N$ such that $\frac{AM}{CN} = \frac{MP}{PN}$ and $AP = CP$ . Find $\angle AP C$.

Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$? [size=85][color=#0000FF][Mod edit: Question fixed][/color][/size]

Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.

A polynomial $w$ of degree $n > 1$ has $n$ distinct zeros $x_1,x_2,...,x_n$. Prove that: $$\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.$$

Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins): At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture. Is there a winning strategy available for Sasha?

There are $2008$ trinomials $x^2-a_kx+b_k$ where $a_k$ and $b_k$ are all different numbers from set $(1,2,...,4016)$. These trinomials has not common real roots. We mark all real roots on the $Ox$-axis. Prove, that distance between some two marked points is $\leq \frac{1}{250}$

A tennis tournament has $n>2$ players and any two players play one game against each other (ties are not allowed). After the game these players can be arranged in a circle, such that for any three players $A,B,C$, if $A,B$ are adjacent on the circle, then at least one of $A,B$ won against $C$. Find all possible values for $n$.

Suppose that $a$ and $b$ are real numbers such that the line $y = ax + b$ intersects the graph of $y = x^2$ at two distinct points $A$ and $B$. If the coordinates of the midpoint of $AB$ are $(5, 101)$, compute $a + b$.

Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess. Prove that we can choose three members of the club who play three different games amongst each other.

Find the total number of different integer values the function \[ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] \] takes for real numbers $x$ with $0 \leq x \leq 100$.

A computer can do $10,000$ additions per second. How many additions can it do in one hour? $\text{(A)}\ 6\text{ million} \qquad \text{(B)}\ 36\text{ million} \qquad \text{(C)}\ 60\text{ million} \qquad \text{(D)}\ 216\text{ million} \qquad \text{(E)}\ 360\text{ million}$

An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

Let $c$ be a positive integer. Consider the sequence $x_1,x_2,\ldots$ which satisfies $x_1=c$ and, for $n\ge 2$, \[x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1\] where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine an expression for $x_n$ in terms of $n$ and $c$. [i]Huang Yumin[/i]

Prove that for each natural number $k$ there exists a natural number $n(k)$, such that for each $m\geq n(k)$ and each set $M$ of $m$ points in the plane, there can be chosen $k$ triangles, so that each has an angle greater than $120^\circ$.