In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

Two cubes are inscribed in a sphere of radius $R$. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube

It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.

Compute the minimum possible value of $(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2$ For real values $x$

For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.

In a trapezoid $ABCD$, the internal bisector of angle $A$ intersects the base $BC$(or its extension) at the point $E$. Inscribed in the triangle $ABE$ is a circle touching the side $AB$ at $M$ and side $BE$ at the point $P$. Find the angle $DAE$ in degrees, if $AB:MP=2$.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Given positive integers $a,b,$ find the least positive integer $m$ such that among any $m$ distinct integers in the interval $[-a,b]$ there are three pair-wise distinct numbers that their sum is zero. [i]Proposed by Marian Tetiva, Romania[/i]

What is the value of $1234+2341+3412+4123$? $\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$

Let $A$ be an $n\times n$ nonsingular matrix with complex elements, and let $\overline{A}$ be its complex conjugate. Let $B = A\overline{A}+I$, where $I$ is the $n\times n$ identity matrix. (a) Prove or disprove: $A^{-1}BA = \overline{B}$. (b) Prove or disprove: the determinant of $A\overline{A}+I$ is real.

For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?

The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then $m$ is: $\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ \text{one of the integers}~ 4,5,6,7~\text{and one other positive integer}$

Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.

Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$.

Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.

Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial $$q(x) = c_3x^3+c_2x^2+c_1x+c_0$$such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$ $(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11$

Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.

A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases: (a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$. (There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.) (b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens (i.e. each square covered by at least one queen). Remark: If $A$ consists of just one square, then $k = 1$ and the required way is the following: all kings to the left of the square of $A$ make one move to the right.

Given are positive integers $a, b$ satisfying $a \geq 2b$. Does there exist a polynomial $P(x)$ of degree at least $1$ with coefficients from the set $\{0, 1, 2, \ldots, b-1 \}$ such that $P(b) \mid P(a)$?

*. Positive numbers $x_1, x_2, ..., x_{100}$ satisfy the system $$\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\ x_1 + x_2 + ...+ x_{100} < 300 \end{cases}$$ Prove that among these numbers there are three whose sum is greater than $100$.

For what $n$ can the vertices of a regular $n$-gon be connected in a $n$-link closed polyline so that such a polyline does not have three equal links?

Let $AD$ be an internal angle bisector in triangle $\Delta ABC$. An arbitrary point $M$ is chosen on the closed segment $AD$. A parallel to $BC$ through $M$ cuts $AB$ at $N$. Let $AD, CM$ cut circumcircle of $\Delta ABC$ at $K, L$, respectively. Prove that $K,N,L$ are collinear.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Simplify \[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\] where $a, b, c$ are positive real numbers.