Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.

The circle $\zeta_{1}$ is inside the circle $\zeta_{2}$, and the circles touch each other at $A$. A line through $A$ intersects $\zeta_{1}$ also at $B$, and $\zeta_{2}$ also at $C$. The tangent to $\zeta_{1}$ at $B$ intersects $\zeta_{2}$ at $D$ and $E$. The tangents of $\zeta_{1}$ passing thorugh $C$ touch $\zeta_{2}$ at $F$ and $G$. Prove that $D$, $E$, $F$ and $G$ are concyclic.

Find $$\lim_{n\to\infty}\frac1{n^3}\left(\sqrt{n^2-1^2}+\sqrt{n^2-2^2}+\ldots+\sqrt{n^2-(n-1)^2}\right).$$

Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is: $ \textbf{(A)}\ x^2-bx-ac=0$ $\qquad\textbf{(B)}\ x^2-bx+ac=0$ $\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$ ${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$ ${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $

Let $A^\prime$ be the point of tangency of the excircle of a triangle $ABC$ (corrsponding to $A$) with the side $BC$. The line $a$ through $A^\prime$ is parallel to the bisector of $\angle BAC$. Lines $b$ and $c$ are analogously defined. Prove that $a, b, c$ have a common point.

Let $Z$ and $Y$ be the tangency points of the incircle of the triangle $ABC$ with the sides $AB$ and $CA$, respectively. The parallel line to $Y Z$ through the midpoint $M$ of $BC$, meets $CA$ in $N$. Let $L$ be the point in $CA$ such that $NL = AB$ (and $L$ on the same side of $N$ than $A$). The line $ML$ meets $AB$ in $K$. Prove that $KA = NC$.

Solve in positive integers the following equation: \[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] [i]Proposed by A. Golovanov[/i]

About the convex hexagon $ABCDEF$ it is known that $AB = BC =CD = DE = EF = FA$ and $AD = BE = CF$. Prove that the diagonals $AD$, $BE$, $CF$ intersect at one point.

At the exterior of the square $ ABCD $ it is constructed the isosceles triangle $ ABE $ with $ \angle ABE=120^{\circ} . M $ is the intersection of the bisector line of the angle $ \angle EAB $ with its perpendicular that passes through $ B; N $ is the intersection of the $ AB $ with its perpendicular that passe through $ M; P $ is the intersection of $ CN $ with $ MB. $ If $ G $ is the center of gravity of the triangle $ ABE, $ prove that $ PG $ and $ AE $ are parallel.

Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers $x^y$, $y^x$ is bigger ?

How many ordered pairs of integers (x; y) satisfy the equation $x^2 + y^2 + xy = 4(x + y)?$.

$f(x)$ is a polynomial of degree $2n$. Show that all polynomials $p(x)$, $q(x)$ of degree at most $n$ such that $f(x)q(x)-p(x)$ has the form \[ \sum\limits_{2n<k\leq 3n} (a^k + x^k) \] have the same $p(x)/q(x)$.

Petya and Vasya only know positive integers not exceeding $10^9-4000$. Petya considers numbers as good which are representable in the form $abc+ab+ac+bc$, where $a,b$ and $c$ are natural numbers not less than $100$. Vasya considers numbers as good which are representable in the form $xyz-x-y-z$, where $x,y$ and $z$ are natural numbers strictly bigger than $100$. For which of them are there more good numbers? [i]Proposed by I. Bogdanov[/i]

Let $ABC$ be a triangle, and let $C^{\prime}$ and $A^{\prime}$ be the feet of its altitudes issuing from the vertices $C$ and $A$, respectively. Denote by $P$ the midpoint of the segment $C^{\prime}A^{\prime}$. The circumcircles of triangles $AC^{\prime}P$ and $CA^{\prime}P$ have a common point apart from $P$; denote this common point by $Q$. Prove that: [b](a)[/b] The point $Q$ lies on the circumcircle of the triangle $ABC$. [b](b)[/b] The line $PQ$ passes through the point $B$. [b](c)[/b] We have $\frac{AQ}{CQ}=\frac{AB}{CB}$. Darij

Let $c_1<c_2<c_3$ be the three smallest positive integer values of $c$ such that the distance between the parabola $y=x^2+2020$ and the line $y=cx$ is a rational multiple of $\sqrt{2}$. Compute $c_1+c_2+c_3$.

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into n equal segments and the triangle into n congruent triangles. Each of these n triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. (a) If $n =10$, what is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (b)The same question for $n = 9$. (R Zhenodarov)

There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that \[a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 \] for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$. [i](A. Golovanov)[/i]

Show that for any real numbers $a_{3},a_{4},...,a_{85}$, not all the roots of the equation $a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0$ are real.

In regular triangular pyramid $S-ABC$, hieght $SO=3$, length of sides of bottom surface is $6$. Projection of $A$ on plane $SBC$ is $O'$. $P\in AO',\frac{AP}{PO'}=8$. Draw a plane parallel to plane $ABC$ and passes $P$. Find the area of the cross section.

We are given a triangle $ABC$ in a plane $P$. To any line $D$, not parallel to any side of the triangle, we associate the barycenter $G_D$ of the set of intersection points of $D$ with the sides of $\triangle ABC$. The object of this problem is determining the set $\mathfrak F$ of points $G_D$ when $D$ varies. (a) If $D$ goes over all lines parallel to a given line $\delta$, prove that $G_D$ describes a line $\Delta_\delta$. (b) Assume $\triangle ABC$ is equilateral. Prove that all lines $\Delta_\delta$ are tangent to the same circle as $\delta$ varies, and describe the set $\mathfrak F$. (c) If $ABC$ is an arbitrary triangle, prove that one can find a plane $P$ and an equilateral triangle $A'B'C'$ whose orthogonal projection onto $P$ is $\triangle ABC$, and describe the set $\mathfrak F$ in the general case.

Let $E$ be the point of intersection of the diagonals of convex quadrilateral $ABCD$, and let $P,Q,R,$ and $S$ be the centers of the circles circumscribing triangles $ABE,$ $BCE$, $CDE$, and $ADE$, respectively. Then $\textbf{(A) }PQRS\text{ is a parallelogram}$ $\textbf{(B) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rhombus}$ $\textbf{(C) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rectangle}$ $\textbf{(D) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a parallelogram}$ $\textbf{(E) }\text{none of the above are true}$

Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is $\tfrac{m}{n}$ for relatively prime positive integers $m$, $n$. Compute $100m + n$. [i]Proposed by Evan Chen[/i]

Four congruent and pairwise externally tangent circles are inscribed in square $ABCD$ as shown. The ray through $A$ passing through the center of the square hits the opposing circle at point $E$, shown in the diagram below. Given $AE= 5 + 2\sqrt{2}$, the area of the square can be expressed as $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is square-free. Find $a+b+c.$ [asy] size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-18.99425911800572,xmax=23.81538435842469,ymin=-15.51769962526155,ymax=6.464951807764648; pen zzttqq=rgb(0.6,0.2,0.); pair A,B,C,D,M,E; A=(0,0); B=(0,4); C=(4,4); D=(4,0); M=(2,2); E=(2.29289,2.29289); draw(A--B--C--D--cycle); draw(A--E); draw(circle((1,1),1));draw(circle((1,3),1));draw(circle((3,1),1));draw(circle((3,3),1)); dot(A);dot(B);dot(C);dot(D);dot(E);label("$A$",A,SW);label("$B$",B,NW);label("$C$",C,NE);label("$D$",D,SE);label("$E$",E,NE); [/asy] [i]Lightning 1.4[/i]

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out. Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit. [i]Dan Schwartz[/i]