Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$.) Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and $3\times 1$ without overlapping.[img]https://pp.userapi.com/c850332/v850332712/115884/DKxvALE-sAc.jpg[/img]

Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?

The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.

On a circle $\omega_1$, four points $A$, $C$, $B$, $D$ lie in that order. Prove that $CD^2 = AC \cdot BC + AD \cdot BD$ if and only if at least one of $C$ and $D$ is the midpoint of arc $AB$.

Let $ n $ be a natural number. Consider all permutations $ ({{a} _ {1}}, \ \ldots, \ {{a} _ {2n}}) $ of the first $ 2n $ natural numbers such that the numbers $ | {{a} _ {i +1}} - {{a} _ {i}} |, \ i = 1, \ \ldots, \ 2n-1, $ are pairwise different. Prove that $ {{a} _ {1}} - {{a} _ {2n}} = n $ if and only if $ 1 \le {{a} _ {2k}} \le n $ for all $ k = 1, \ \ldots, \ n $.

Let $ f$ be a continuous, nonconstant, real function, and assume the existence of an $ F$ such that $ f(x\plus{}y)\equal{}F[f(x),f(y)]$ for all real $ x$ and $ y$. Prove that $ f$ is strictly monotone.

Rhombuses $ABCD$ and $A_1B_1C_1D_1$ are equal. Side $BC$ intersects sides $B_1C_1$ and $C_1D_1$ at points $M$ and $N$ respectively. Side $AD$ intersects sides $A_1B_1$ and $A_1D_1$ at points $Q$ and $P$ respectively. Let $O$ be the intersection point of lines $MP$ and $QN$. Find $\angle A_1B_1C_1$ , if $\angle QOP = \frac12 \angle B_1C_1D_1$.

Let $ABCDE$ be a convex pentagon such that \begin{align*} &AB+BC+CD+DE+EA=65 \text{ and} \\ &AC+CE+EB+BD+DA=72. \end{align*} Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of $ABCDE.$

In coordinate plane, a variable point $M$, starting from the origin $O(0, 0)$, moves on the line $l$ with slope $k$, where $k$ is an irrational number. [b](i)[/b] Prove that point $O(0, 0)$ is the only rational point (namely, the coordinates of which are both rationals) on the line $l.$ [b](ii)[/b] Prove that for any number $\varepsilon > 0$, there exist integers $m, n$ such that the distance between $l$ and the point $(m, n)$ is less than $\varepsilon.$

In how many ways can we construct a square with dimensions $3\times 3$ using $3$ white, $3$ green and $3$ red squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .

Let $D$ be an arbitrary point on the side $BC$ of the non-isosceles acute triangle $ABC$. The circle with center $D$ and radius $DA$ intersects the rays $AB^\to$ (after $B$) and $AC^\to$ (after $C$) at $M$ and $N$. Prove that the orthocenter of triangle $AMN$ lies on a fixed line, independent of the choice of $D$.

Prove that if the sum of $x^5,y^5$ and $z^5$, where $x,y$ and $z$ are integer numbers, is divisible by $25$ then the sum of some two of them is divisible by $25$.

For all positive integers $k$, define $f(k)=k^2+k+1$. Compute the largest positive integer $n$ such that \[2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.\][i]Proposed by David Altizio[/i]

Compute the largest root of $x^4-x^3-5x^2+2x+6$.

Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

If circular arcs $ AC$ and $ BC$ have centers at $ B$ and $ A$, respectively, then there exists a circle tangent to both $ \stackrel{\frown}{AC}$ and $ \stackrel{\frown}{BC}$, and to $ \overline{AB}$. If the length of $ \stackrel{\frown}{BC}$ is $ 12$, then the circumference of the circle is [asy]unitsize(4cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,.375); pair A=(-.5,0); pair B=(.5,0); pair C=shift(-.5,0)*dir(60); draw(Arc(A,1,0,60)); draw(Arc(B,1,120,180)); draw(A--B); draw(Circle(O,.375)); dot(A); dot(B); dot(C); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N);[/asy]$ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 28$

Construct a parallelogram $ABCD$, if three points are marked on the plane: the midpoints of its altitudes $BH$ and $BP$ and the midpoint of the side $AD$.

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

Let circles $\omega$ and $\Gamma$, centered at $O_1$ and $O_2$ and having radii $42$ and $54$ respectively, intersect at points $X,Y$, such that $\angle O_1XO_2 = 105^{\circ}$. Points $A$, $B$ lie on $\omega$ and $\Gamma$ respectively such that $\angle O_1XA = \angle AXB = \angle BXO_2$ and $Y$ lies on both minor arcs $XA$ and $XB$. Define $P$ to be the intersection of $AO_2$ and $BO_1$. Suppose $XP$ intersects $AB$ at $C$. What is the value of $\frac{AC}{BC}$? [i]Proposed by Puhua Cheng[/i]

In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$

Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.