Find all integer solutions to \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.\]

$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ $\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 57$

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.

[asy]draw(Circle((0,0), 1)); dot((0,0)); label("$O$", (0,0), S); label("$A$", (-1,0), W); label("$B$", (1,0), E); label("$a$", (-0.5,0), S); draw((-1,-1.25)--(-1,1.25)); draw((1,-1.25)--(1,1.25)); draw((-1,0)--(1,0)); draw((-1,0)--(-1,0)+2.3*dir(30)); label("$C$", (-1,0)+2.3*dir(30), E); label("$D$", (-1,0)+1.8*dir(30), N); dot((-1,0)+.4*dir(30)); label("$E$", (-1,0)+.4*dir(30), N); [/asy] In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so that $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$ and $y$, respectively, we can deduce the relation: $\text{(A)}\ y^2=\dfrac{x^3}{2a-x} \qquad \text{(B)}\ y^2=\frac{x^3}{2a+x}\qquad \text{(C)}\ y^4=\frac{x^2}{2-x}\qquad\\ \text{(D)}\ x^2=\dfrac{y^2}{2a-x}\qquad \text{(E)}\ x^2=\frac{y^2}{2a+x}$

Find the number of positive integers less than or equal to $2020$ that are relatively prime to $588$.

In the triangle $ABC$, $h_a, h_b, h_c$ are the altitudes and $p$ is its half-perimeter. Compare $p^2$ with $h_ah_b + h_bh_c + h_ch_a$. (Gregory Filippovsky)

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a function such that $f\left(\frac{a+b}{2}\right)\in \{f(a),f(b)\},\ (\forall)a,b\in \mathbb{R}$. a) Give an example of a non-constant function that satisfy the hypothesis. b)If $f$ is continuous, prove that $f$ is constant.

Fix an integer $n \geq 3$. Determine the smallest positive integer $k$ satisfying the following condition: For any tree $T$ with vertices $v_1, v_2, \dots, v_n$ and any pairwise distinct complex numbers $z_1, z_2, \dots, z_n$, there is a polynomial $P(X, Y)$ with complex coefficients of total degree at most $k$ such that for all $i \neq j$ satisfying $1 \leq i, j \leq n$, we have $P(z_i, z_j) = 0$ if and only if there is an edge in $T$ joining $v_i$ to $v_j$. Note, for example, that the total degree of the polynomial $$ 9X^3Y^4 + XY^5 + X^6 - 2 $$ is 7 because $7 = 3 + 4$. [i]Proposed by Andrei Chiriță, Romania[/i]

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

Prove that if in a tetrahedron $ ABCD $ the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point, then $$AB \cdot CD = AC \cdot BD = AD \cdot BC$$ and that the converse also holds.

The ratio $\dfrac{(2^4)^8}{(4^8)^2}$ equals $\textbf{(A) }\dfrac14\qquad\textbf{(B) }\dfrac12\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }8$

There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

There are $13$ stones each of which weighs an integer number of grams. It is known that any $12$ of them can be put on two pans of a balance scale, six on each pan, so that they are in equilibrium (i.e., each pan will carry an equal total weight). Prove that either all stones weigh an even number of grams or all stones weigh an odd number of grams.

$AD,BE,CF$ are three heights of $\triangle ABC$, and they intersect at $H$. Let $O$ be the circumcenter of $\triangle ABC$, $ED\cap AB=M,FD\cap AC=N$. Prove: [b](a)[/b] $OB\perp DF, OC\perp DE$. [b](b)[/b] $OH\perp MN$.

The radius $OM$ of a circle rotates uniformly at a rate of $360/n$ degrees per second , where $n$ is a positive integer . The initial radius is $OM_0$. After $1$ second the radius is $OM_1$ , after two more seconds (i.e. after three seconds altogether) the radius is $OM_2$ , after $3$ more seconds (after $6$ seconds altogether) the radius is $OM_3$, ..., after $n - 1$ more seconds its position is $OM_{n-1}$. For which values of $n$ do the points $M_0, M_1 , ..., M_{n-1}$ divide the circle into $n$ equal arcs? (a) Is it true that the powers of $2$ are such values? (b) Does there exist such a value which is not a power of $2$? (V. V. Proizvolov , Moscow)

For any three positive real numbers $a$, $b$, $c$, prove the inequality \[\frac{\left(b+c\right)^{2}}{a^{2}+bc}+\frac{\left(c+a\right)^{2}}{b^{2}+ca}+\frac{\left(a+b\right)^{2}}{c^{2}+ab}\geq 6.\] Darij

For which $k$, there is no integer pair $(x,y)$ such that $x^2 - y^2 = k$? $ \textbf{(A)}\ 2005 \qquad\textbf{(B)}\ 2006 \qquad\textbf{(C)}\ 2007 \qquad\textbf{(D)}\ 2008 \qquad\textbf{(E)}\ 2009 $

The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: $ \textbf{(A)}\ 108 \qquad \textbf{(B)}\ 90 \qquad \textbf{(C)}\ 72 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 36$

Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.

For a positive integer $n$ and real numbers $a_1, a_2, \dots ,a_n$ we'll define $b_1, b_2, \dots ,b_{n+1}$ such that $b_k=a_k+\max({a_{k+1},a_{k+2}})$ for all $1\leq k \leq n$ and $b_{n+1}=b_1$. (Also $a_{n+1}=a_1$ and $a_{n+2}=a_2$) Find the least possible value of $\lambda$ such that for all $n, a_1, \dots, a_n$ the inequality $$\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}$$ holds.

Let $ [a]$ be the integer such that $ [a]\le a<[a]\plus{}1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}\plus{}[b]\plus{}\{c\}\equal{}2.9\\\{b\}\plus{}[c]\plus{}\{a\}\equal{}5.3\\\{c\}\plus{}[a]\plus{}\{b\}\equal{}4.0.\]