Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any $8$ real numbers $a,b,c,d,x,y,z,t$? Edit: Fixed a mistake! Thanks @below.

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

Find all solutions in integers of the equation $$x^2 + 2^2 = y^3 + 3^3.$$

Prove that in a topological space X , if all discrete subspaces have compact closure , then X is compact.

A circle is divided into $n$ sectors. Pawns stand on some of the sectors; the total number of pawns equals $n + 1$. This configuration is changed as follows. Any two of the pawns standing on the same sector move simultaneously to the neighbouring sectors in different directions. Prove that after several such transformations a configuration in which no less than half of the sectors are occupied by pawns, will inevitably appear. (D. Fomin, St Petersburg)

The sum and the product of two purely periodic decimal fractions $a$ and $b$ are purely periodic decimal fractions of period length $T$. Show that the lengths of the periods of the fractions $a$ and $b$ are not greater than $T$. [i]Note.[/i] A [i]purely periodic decimal fraction[/i] is a periodic decimal fraction without a non-periodic starting part.

Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Regular hexagon $ABCDEF$ has area $1$. Starting with edge $AB$ and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side $AB$, the new point, $G$, is drawn so $AG = \tfrac12 AB$. This forms hexagon $GHIJKL$, as shown. What is the area of this new hexagon? [asy] size(120); pair A = (-1/2, sqrt(3)/2); pair B = (1/2, sqrt(3)/2); pair C = (1,0); pair D = (1/2, -sqrt(3)/2); pair EE = (-1/2, -sqrt(3)/2); pair F = (-1,0); pair G = (A+B)/2; pair H = (B+C)/2; pair I = (C+D)/2; pair J = (D+EE)/2; pair K = (EE+F)/2; pair L = (F+A)/2; draw(A--B--C--D--EE--F--cycle); draw(G--H--I--J--K--L--cycle); dot(A^^B^^C^^D^^EE^^F^^G^^H^^I^^J^^K^^L); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,SE); label("$E$",EE,SW); label("$F$",F,W); label("$G$",G,N); label("$H$",H,NE); label("$I$",I,SE); label("$J$",J,S); label("$K$",K,SW); label("$L$",L,NW); [/asy] $\textbf{(A) }\dfrac35\qquad\textbf{(B) }\dfrac57\qquad\textbf{(C) }\dfrac34\qquad\textbf{(D) }\dfrac79\qquad\textbf{(E) }\dfrac45$

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

A square of side length $5$ is inscribed in a square of side length $7$. If we construct a grid of $1\times1$ squares for both squares, as shown to the right, then we find that the two grids have $8$ lattice points in common. If we do the same construction by inscribing a square of side length $1489$ in a square of side length $2009$, and construct a grid of $1\times1$ squares in each large square, then how many lattice points will the two grids of $1\times1$ squares have in common? [asy] import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=11.88,ymin=-4.69,ymax=8.77; pair H_2=(0,3), I_2=(3,7), J_2=(7,4), K_2=(4,0), L_2=(3.01,1.99), M_2=(2.01,4), N_2=(4.01,5.01), O_2=(5.01,3); draw((0,0)--(0,7)); draw((0,7)--(7,7)); draw((7,7)--(7,0)); draw((7,0)--(0,0)); draw((0,6)--(7,6)); draw((0,5)--(7,5)); draw(J_2--(0,4)); draw(H_2--(7,3)); draw((0,2)--(7,2)); draw((0,1)--(7,1)); draw((1,0)--(1,7)); draw((2,7)--(2,0)); draw((3,0)--I_2); draw(K_2--(4,7)); draw((5,0)--(5,7)); draw((6,7)--(6,0)); draw(H_2--I_2); draw(I_2--J_2); draw(J_2--K_2); draw(K_2--H_2); draw(H_2--I_2); draw(I_2--J_2); draw((2.41,6.21)--(6.4,3.2)); draw((5.8,2.4)--(1.81,5.41)); draw((1.2,4.61)--(5.2,1.6)); draw((4.6,0.8)--(0.6,3.8)); draw((3.8,6.4)--(0.8,2.4)); draw((1.61,1.79)--(4.6,5.8)); draw((5.4,5.2)--(2.41,1.19)); draw((3.21,0.59)--(6.2,4.6)); draw((0,7)--(7,7),linewidth(1.2)); draw((7,7)--(7,0),linewidth(1.2)); draw((0,0)--(7,0),linewidth(1.2)); draw((0,7)--(0,0),linewidth(1.2)); dot(H_2,linewidth(4pt)+ds); dot(I_2,linewidth(4pt)+ds); dot(J_2,linewidth(4pt)+ds); dot(K_2,linewidth(4pt)+ds); dot(L_2,linewidth(4pt)+ds); dot(M_2,linewidth(4pt)+ds); dot(N_2,linewidth(4pt)+ds); dot(O_2,linewidth(4pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]

Let $ABC$ be a triangle with $m(\widehat{A})=90^\circ$, $|AB|=4$, and $|AC|=3$. Let $D$ be the foot of perpendicular from $A$ to $[BC]$. If $P$ a point on $[BD]$ such that $5|AP|=13|PD|$, what is $|CP|$? $ \textbf{(A)}\ \frac{9 + 4\sqrt 3}{5} \qquad\textbf{(B)}\ \frac{56}{15} \qquad\textbf{(C)}\ \frac{14}{5} \qquad\textbf{(D)}\ \frac{37}{13} \qquad\textbf{(E)}\ \frac{5\sqrt 5 + 3}{5} $