Let $f$ be a polynomial with real coefficients and suppose $f$ has no nonnegative real root. Prove that there exists a polynomial $h$ with real coefficients such that the coefficients of $fh$ are nonnegative.

Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic. (Albania)

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

Prove that (a) if the natural number $n$ can be represented as $n =4k+1$ (where $k$ is an integer), then there exist $n$ odd positive integers whose sum is equal to their product, (b) if $n$ cannot be represented in this form then such a set does not exist. (M. Kontsevich)

Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that: $ (i)$ $ a\plus{}c\equal{}d;$ $ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$ $ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$. Determine $ n$.

Let $ABC$ be a triangle with $AH$ altitude. The point $K$ is chosen on the segment $AH$ as follows such that $AH =3KH$. Let $O$ be the center of the circle circumscribed around by triangle $ABC, M$ and $N$ be the midpoints of $AC$ and AB respectively. Lines $KO$ and $MN$ intersect at the point $Z$, a perpendicular to $OK$ passing through point $Z$ intersects lines $AC$ and $AB$ at points $X$ and $Y$ respectively. Prove that $\angle XKY =\angle CKB$.

Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that \[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.

Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length $16,$ find the radius of the large circle.

The figure below consists of several unit squares, $M$ of which are white and $N$ of which are green. Compute $100M+N$. [asy] size(4cm); int N = 4; path square; for (int x=-N; x<=N; ++x) { for (int y=-N+abs(x); y<=N-abs(x); ++y) { square = rotate(9)*((x+0.5,y+0.5)--(x+0.5,y-0.5)--(x-0.5,y-0.5)--(x-0.5,y+0.5)--cycle); if ((x+y) % 2 == 0) { filldraw(square, green, black); } else { filldraw(square, white, black); } } } [/asy] [i]Proposed by Evan Chen[/i]

It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.

Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$ $\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]$

Let $ \varphi(x)$ be a cubic polynomial with integer coefficients. Given that $ \varphi(x)$ has have 3 distinct real roots $u,v,w $ and $u,v,w $ are not rational number. there are integers $ a, b,c$ such that $u=av^2+bv+c$. Prove that $b^2 -2b -4ac - 7$ is a square number .

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

Let $ P$ be a point inside a triangle $ ABC$ such that \[ \angle APB \minus{} \angle ACB \equal{} \angle APC \minus{} \angle ABC. \] Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.

Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.

What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?

If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$?

[b]Problem 1[/b] Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$. Prove that there exists $x \in [0, 1]$ such that $$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$$

Peter and Basil take turns drawing roads on a plane, Peter starts. The road is either horizontal or a vertical line along which one can drive in only one direction (that direction is determined when the road is drawn). Can Basil always act in such a way that after each of his moves one could drive according to the rules between any two constructed crossroads, regardless of Peter's actions? Alexandr Perepechko

Suppose that $n^2-2m^2=m(n+3)-3$. Find all integers $m$ such that all corresponding solutions for $n$ will [i]not[/i] be real.

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$ $f(x+f(y))=2x+2f(y+1)$

The polynomial $$x^{10} + ?x^9 + ?x^8 + ... + ?x + 1$$ is written on the blackboard. Two players substitute (real) numbers instead of one of the question marks in turn. ($9$ turns total.) The first wins if the polynomial will have no real roots. Who wins?

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares. [i]Proposed by A. Golovanov[/i]