Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and radius $10$. Let sides $AB$, $BC$, $CD$, and $DA$ have midpoints $M, N, P$, and $Q$, respectively. If $MP = NQ$ and $OM + OP = 16$, then what is the area of triangle $\vartriangle OAB$?

Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$

Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$.

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that \[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \] for all real number $x$.

Let $ABCD$ be a quadrilateral with $AD=BC$ and $\widehat{A}+\widehat{B}=120^{0}$. Let us draw equilateral $ACP,DCQ,DBR$ away from $AB$ . Prove that the points $P,Q,R$ are collinear.

A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$. Prove that there exists a unique point on a polyhedron which belongs to all such balls.

We call a number [i]pal[/i] if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example, $122$ and $34$ are pal but $304$ and $12$ are not pal. Prove that there exists a pal number with $n$ digits, $n > 1$.

Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]

Let $m$ and $n$ be positive integers. A people planted two kind of different trees on a plot tabular grid size $ m \times n $ (each square plant one tree.) A plant called [i]inpressive[/i] if two conditions following conditions are met simultaneously: i) The number of trees in each of kind is equal; ii) In each row the number of tree of each kind is diffrenent not less than a half of number of cells on that row and In each colum the number of tree of each kind is diffrenent not less than a half of number of cells on that colum. a) Find an inpressive plant when $m=n=2016$; b) Prove that if there at least a inpressive plant then $4|m$ and $4|n$.

Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$. Prove the inequality \[\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\]

Let $W,A,B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent. [list] [*] For all $x, y, z\ge 0$ satisfying $x+y\le z+W, x+z\le y+W, y+z\le x+W$ we have $Axyz+B\ge x^2+y^2+z^2$. [*] $B\ge W^2$ and $AW^3+B\ge 3W^2$. [/list] [i]Proposed by Ákos Somogyi, London[/i]

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

Prove that $$\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1$$ This means $1/(2+ (1/(3+ (1/(4+(...+1/1991)))))) +1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)$ (G. Galperin, Moscow-Tel Aviv)

From digits $0$, $1$, $3$, $4$, $7$ and $9$ were written $5$ digit numbers which all digits are different. How many numbers from them are divisible with $5$

Let $ p\ge 2 $ be a fixed natural number, and let the sequence of functions $ \left( f_n\right)_{n\ge 2}:[0,1]\longrightarrow\mathbb{R} $ defined as $ f_n (x)=f_{n-1}\left( f_1 (x)\right) , $ where $ f_1 (x)=\sqrt[p]{1-x^p} . $ Find $ a\in (0,1) $ such that: [b]a)[/b] exists $ b\ge a $ so that $ f_1:[a,b]\longrightarrow [a,b] $ is bijective. [b]b)[/b] $ \forall x\in [0,1]\quad\exists y\in [0,1]\quad m\in\mathbb{N}\implies \left| f_m(x)-f_m(y)\right| >a|x-y| $

One company from Tesanj has last year produced profit for $112 \%$ of expected one . Determine how many percents expected profit is from produced one

In any rectangular game board with black and white squares, call a row $X$ a mix of rows $Y$ and $Z$ whenever each cell in row $X$ has the same colour as either the cell of the same column in row $Y$ or the cell of the same column in row $Z$. Let a natural number $m \ge 3$ be given. In some rectangular board, black and white squares lie in such a way that all the following conditions hold. 1) Among every three rows of the board, one is a mix of two others. 2) For every two rows of the board, their corresponding cells in at least one column have different colours. 3) For every two rows of the board, their corresponding cells in at least one column have equal colours. 4) It is impossible to add a new row with each cell either black or white to the board in a way leaving both conditions 1) and 2) still in force Find all possibilities of what can be the number of rows of the board.

Several pupils wrote a solution of a math problem on the blackboard on the break. When the teacher came in, a pupil was just clearing the blackboard, so the teacher could only observe that there was a rectangle with the sides of integer lenghts and a diagonal of lenght $ 2002$. Then the teacher pointed out that there was a computation error in pupils' solution. Why did he conclude that?

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

A composite natural number $n$ is called [i]happy [/i] if at most one of the numbers $2^{2^n}+ 1$ and $6^{2^n}+ 1$ is prime. Show that there are infinitely many happy numbers.

In $\triangle ABC$, angle $B$ is obtuse, $AB = 42$ and $BC = 69$. Let $M$ and $N$ be the midpoints of $AB$ and $BC$, respectively. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet $BC$ and $CA$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the midpoints of $AD$ and $AN$ respectively. Let $CY$ and $BX$ meet $AB$ and $CA$ at $P$ and $Q$. If $EM$ and $PQ$ meet on $BC$, find $CA$. [i]Proposed by Sid Doppalapudi[/i]

In $\triangle ABC$, let $O$, $H$, and $N$ be its circumcenter, orthocenter, and nine-point center respectively. Let $AN$ meet the circumcircle of $\triangle ABC$ at $S$. Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $D$. Show that $\angle DSH = \angle DOA$.

Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.