In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.

Let $a_1, a_2, \dots, a_{2019}$ be positive integers and $P$ a polynomial with integer coefficients such that, for every positive integer $n$, $$P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.$$ Prove that $P$ is a constant polynomial.

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube. [i]Dinu Serbanescu[/i]

What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).

Find the graph of the function $y=x+|1-x^3|$.

What is the sum of $3+3^2+3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ in $\mod 11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 10 $

Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and \[a^3 + b^3 = p^n\mbox{.}\] [i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]

Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers? $\textbf{(A)}\mbox{ }360\qquad\textbf{(B)}\mbox{ }388\qquad\textbf{(C)}\mbox{ }412\qquad\textbf{(D)}\mbox{ }416\qquad\textbf{(E)}\mbox{ }424$

Given $n$ points, $n > 4$. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)

Let $T(p)$ denote the number of right triangles with integer side lengths and one of its side lengths being $p$. Which of the following values of $p$ produces the greatest possible value of $T(p)$ among all five answer choices? $\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }54$

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied \[f(x+y) + f(x)f(y) = f(xy) + 1 \] $\forall x, y \in \mathbb{R}$

For how many positive integers $n \le 1000$ does the equation in real numbers $x^{\lfloor x \rfloor } = n$ have a positive solution for $x$?

In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$. (RK Gordin)

Compute the number of five-digit positive integers $\overline{vwxyz}$ for which \[ (10v+w) + (10w+x) + (10x+y) + (10y+z) = 100. \][i]Proposed by Evan Chen[/i]

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

[list] [*]There are two semi-infinite plane mirrors inclined physically at a non-zero angle with the inner surfaces being reflective.\\ Prove that all lines of incident/reflected rays are tangential to a particular circle for any given incident ray being incident on a reflective side. Assume that the incident ray lies on one of the normal planes to the mirrors.[/*] [*] There's a cone of an arbitrary base with large enough length.\\ The inner surface polished (Outer surface is absorbing in nature) and the apex is fixed to a point. The cone is being rotated around the apex at an angular speed $\omega$ around the vertical axis and assume that a large part of the inside is visible horizontally. A fixed horizontal ray is projected from outside towards the cone (which often falls inside of it), prove that all the lines of incident ray/reflected rays at all instants lie tangential to a particular sphere.\\ Try guessing the radius of the sphere with the parameters you observe.[/*]

In a triangle $ABC$ point $I$ is the incenter, $I_A$ - excenter, $W$ - midpoint of the arc $BAC$ of circumcircle $\omega$ of $ABC$. Point $H$ is the projection of $I_A$ on $IW$. The tangent line to the circumcircle $BIC$ in point $I$ intersects $\omega$ in $E, F$. Prove that the perpendicular bisector to $AI$ is tangent to the circumcircle $EFH$ [i]M. Zorka[/i]

Then number of solutions to $\cos7x=\cos5x$ in $[0,\pi]$ is________.

It holds: $N,a > 0$. Prove that $\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}$, and if $N \ge 1$ and $a = [\sqrt{N}]$. Prove that if $a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)$ is a better approximation for $\sqrt{N}$ than $a$.

Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$. [i]Proposed by Lewis Chen[/i]

At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? $ \textbf{(A)}\ 2002\ \text{and}\ 2003 \qquad \textbf{(B)}\ 2003\ \text{and}\ 2004 \qquad \textbf{(C)}\ 2004\ \text{and}\ 2005 \qquad \textbf{(D)}\ 2005\ \text{and}\ 2006 \qquad \textbf{(E)}\ 2006\ \text{and}\ 2007$

You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$. Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$? For example, if $a = [-1, 2, 2]$, then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$. [i](Proposed by Anton Trygub)[/i]

Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).