In how many ways can $n^2$ distinct real numbers be arranged into an $n\times n$ array $(a_{ij })$ such that max$_{j}$ min $_i \,\, a_{ij} $= min$_i$ max$_j \,\, a_{ij}$?

Given a triangle with sides $a, b, c$ that satisfy $\frac{a}{b+c}=\frac{c}{a+b}$. Determine the angles of this triangle, if you know that one of them is equal to $120^0$.

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.

Let $n\geq 2$ and $\mathcal{M}$ be a subset of $S_n$ with at least two elements, and which is closed under composition. Consider a function $f:\mathcal{M}\rightarrow\mathbb{R}$ which satisfies $$|f(\sigma\tau)-f(\sigma)-f(\tau)|\leq 1,$$ for all $\sigma,\tau\in\mathcal{M}$. Prove that $$\max_{\sigma,\tau\in\mathcal{M}}|f(\sigma)-f(\tau)|\leq 2-\frac{2}{|\mathcal{M}|}.$$

Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which \[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34? \]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$

Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3\plus{}b^3\equal{}c^3\plus{}d^3$ and $ a\plus{}b\equal{}c\plus{}d$.

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

a) Given a convex hexagon $ABCDEF$ with all the equal angles. Prove that $$|AB|-|DE| = |EF|-|BC| = |CD|-|FA|$$ b) The opposite problem: Prove that it is possible to construct a convex hexagon with equal angles of six segments $a_1,a_2,...,a_6$, whose lengths satisfy the condition $$a_1-a_4 = a_5-a_2 = a_3-a_6$$

A [i]tile[/i] $T$ is a union of finitely many pairwise disjoint arcs of a unit circle $K$. The [i]size[/i] of $T$, denoted by $|T|$, is the sum of the lengths of the arcs $T$ consists of, divided by $2\pi$. A [i]copy[/i] of $T$ is a tile $T'$ obtained by rotating $T$ about the centre of $K$ through some angle. Given a positive real number $\varepsilon < 1$, does there exist an infinite sequence of tiles $T_1,T_2,\ldots,T_n,\ldots$ satisfying the following two conditions simultaneously: 1) $|T_n| > 1 - \varepsilon$ for all $n$; 2) The union of all $T_n'$ (as $n$ runs through the positive integers) is a proper subset of $K$ for any choice of the copies $T_1'$, $T_2'$, $\ldots$, $T_n', \ldots$? [hide=Note] In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.[/hide]

Given a $2015$-tuple $(a_1,a_2,\ldots,a_{2015})$ in each step we choose two indices $1\le k,l\le 2015$ with $a_k$ even and transform the $2015$-tuple into $(a_1,\ldots,\dfrac{a_k}{2},\ldots,a_l+\dfrac{a_k}{2},\ldots,a_{2015})$. Prove that starting from $(1,2,\ldots,2015)$ in finite number of steps one can reach any permutation of $(1,2,\ldots,2015)$.

Given a circle with diameter $AB$. Points $C$ and $D$ are selected on the circumference of the circle such that the chord $CD$ intersects $AB$ inside the circle, at point $P$. The ratio of the arc length $\overarc {AC}$ to the arc length $\overarc {BD}$ is $4 : 1$ , while the ratio of the arc length $\overarc{AD}$ to the arc length $\overarc {BC}$ is $3 : 2$ . Find $\angle{APC}$ , in degrees.

Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid: $$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$$ b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid: $$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$$

Two streams of water flow through the U-shaped tubes shown. The tube on the left has cross-sectional area $A$, and the speed of the water flowing through it is $v$; the tube on the right has cross-sectional area $A'=1/2A$. If the net force on the tube assembly is zero, what must be the speed $v'$ of the water flowing through the tube on the right? Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit. [asy] // Code by riben size(300); draw(arc((0,0),10,90,270)); draw(arc((0,0),7,90,270)); draw((0,10)--(25,10)); draw((0,-10)--(25,-10)); draw((0,7)--(25,7)); draw((0,-7)--(25,-7)); draw(ellipse((25,8.5),0.5,1.5)); draw(ellipse((25,-8.5),0.5,1.5)); draw((20,8.5)--(7,8.5),EndArrow(size=7)); draw((7,-8.5)--(20,-8.5),EndArrow(size=7)); draw(arc((-22,0),12,90,-90)); draw(arc((-22,0),7,90,-90)); draw((-22,12)--(-42,12)); draw((-22,-12)--(-42,-12)); draw((-22,7)--(-42,7)); draw((-22,-7)--(-42,-7)); draw(ellipse((-42,9.5),1.5,2.5)); draw(ellipse((-42,-9.5),1.5,2.5)); draw((-38,9.5)--(-23,9.5),EndArrow(size=7)); draw((-23,-9.5)--(-38,-9.5),EndArrow(size=7)); [/asy] (A) $1/2v$ (B) $v$ (C) $\sqrt{2}v$ (D) $2v$ (E) $4v$

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

In the acute-angled triangle $ABC$, the segment $AP$ was drawn and the center was marked $O$ of the circumscribed circle. The circumcircle of triangle $ABP$ intersects the line $AC$ for the second time at point $X$, the circumcircle of the triangle $ACP$ intersects the line $AB$ for the second time at the point $Y$. Prove that the lines $XY$ and $PO$ are perpendicular if and only if $P$ is the foor of the bisector of the triangle $ABC$.

In rectangle $ABCD$, diagonal $AC$ is met by the angle bisector from $B$ at $B'$ and the angle bisector from $D$ at $D'$. Diagonal $BD$ is met by the angle bisector from $A$ at $A'$ and the angle bisector from $C$ at $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ the area of rectangle $ABCD$. What is the ratio of the longer side and shorter side of rectangle $ABCD$?

$n$ people sit in a circle. Each of them is either a liar (always lies) or a truthteller (always tells the truth). Every person knows exactly who speaks the truth and who lies. In their turn, each person says 'the person two seats to my left is a truthteller'. It is known that there's at least one liar and at least one truthteller in the circle. [list=a] [*] Is it possible that $n=2017?$ [*] Is it possible that $n=5778?$ [/list]

Through point $P$, which lies on one of the sides of the triangle $ABC$, draw a line dividing its area in half.

Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$

(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one? (b) The same question for a $7 \times 7$ board. (A Shapovalov_

Simon plays a game on an $n\times n$ grid of cells. Initially, each cell is filled with an integer. Every minute, Simon picks a cell satisfying the following: [list] [*] The magnitude of the integer in the chosen cell is less than $n^{n^n}$ [*] The sum of all the integers in the neighboring cells (sharing one side with the chosen cell) is non-zero [/list] Simon then adds each integer in a neighboring cell to the chosen cell. Show that Simon will eventually not be able to make any valid moves.

In the beginning, number $1$ has been written to point $(0,0)$ and $0$ has been written to any other point of integral coordinates. After every second, all numbers are replaced with the sum of the numbers in four neighbouring points at the previous second. Find the sum of numbers in all points of integral coordinates after $n$ seconds.