Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$. Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$. Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.

Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by $$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1$$ Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$.

The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.) (AY Belov)

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

Let $k$ be a given positive integer. The sequence $x_n$ is defined as follows: $x_1 =1$ and $x_{n+1}$ is the least positive integer which is not in $\{x_{1}, x_{2},..., x_{n}, x_{1}+k, x_{2}+2k,..., x_{n}+nk \}$. Show that there exist real number $a$ such that $x_n = \lfloor an\rfloor$ for all positive integer $n$.

For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$, we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$. For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$, $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ . We say that a positive integer $n$ is [i]special [/i]if $t(n)$ is a divisor of $n$. How many positive divisors of the number $54610$ are special?

The figure below was made by gluing together 5 non-overlapping congruent squares. The figure has area 45. Find the perimeter of the figure. [center][img]https://snag.gy/ZeKf4q.jpg[/center][/img]

The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$, $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$, $c_{n+1}=c_n^2-2c_n+2$ Prove that for all $n\ge 1$, $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$

$n$ real numbers are written in increasing order in a line. The same numbers are written in the second line below in unknown order. The third line contains the sums of the pairs of numbers above from two previous lines. It comes out, that the third line is arranged in increasing order. Prove that the second line coincides with the first one.

There is a board with three rows and $2019$ columns. In the first row are written the numbers integers from $1$ to $2019$ inclusive, ordered from smallest to largest. In the second row, $Ana$ writes those same numbers but ordered at your choice. In each box in the third row write the difference between the two numbers already written in the same column (the largest minus the smallest). $Beto$ have to paint some numbers in the third row so that the sum of the numbers painted is equal to the sum of the numbers in that row that were left unpainted. Can $Ana$ complete the second row so that $Beto$ does not achieve his goal?

In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD = BC = 5, AB = 4,$ and $DC = 10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in the right triangle $DEF$. Then $CF =$ [asy] size(200); defaultpen(fontsize(10)); pair D=origin, A=(3,4), B=(7,4), C=(10,0), E=(14,8), F=(14,0); draw(B--C--F--E--B--A--D--B^^C--D, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(5,3); label("$A$", A, N); label("$B$", B, N); label("$C$", C, S); label("$D$", D, S); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); markscalefactor=0.05; draw(rightanglemark(E,F,D), linewidth(0.7));[/asy] $\text{(A)} \ 3.25 \qquad \text{(B)} \ 3.5 \qquad \text{(C)} \ 3.75 \qquad \text{(D)} \ 4.0 \qquad \text{(E)} \ 4.25$

A pair of integers $ (m,n)$ is called [i]good[/i] if \[ m\mid n^2 \plus{} n \ \text{and} \ n\mid m^2 \plus{} m\] Given 2 positive integers $ a,b > 1$ which are relatively prime, prove that there exists a [i]good[/i] pair $ (m,n)$ with $ a\mid m$ and $ b\mid n$, but $ a\nmid n$ and $ b\nmid m$.

The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) \equal{} 6$ have? [asy]size(220); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6); real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}; real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}; draw(P1--P2--P3--P4--P5); dot("(-7, -4)",P1); dot("(-2, 6)",P2,LeftSide); dot("(1, 6)",P4); dot("(5, -6)",P5); xaxis("$x$",-7.5,7,Ticks(xticks),EndArrow(6)); yaxis("$y$",-6.5,7,Ticks(yticks),EndArrow(6));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Alice and Bob play a game where they start from a complete graph with $n$ vertices and take turns removing a single edge from the graph, with Alice taking the first turn. The first player to disconnect the graph loses. Compute the sum of all $n$ between $2$ and $100$ inclusive such that Alice has a winning strategy. (A complete graph is one where there is an edge between every pair of vertices.)

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.

The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance two apart. Find the area of the irregularly shaped ten sided figure shown. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((-2,5)--(-3,4), linewidth(1.6)); draw((-3,4)--(-2,1), linewidth(1.6)); draw((-2,1)--(1,0), linewidth(1.6)); draw((1,0)--(2,1), linewidth(1.6)); draw((2,1)--(1,3), linewidth(1.6)); draw((1,3)--(1,4), linewidth(1.6)); draw((1,4)--(2,5), linewidth(1.6)); draw((2,5)--(0,5), linewidth(1.6)); draw((-2,5)--(-1,4), linewidth(1.6)); draw((-1,4)--(0,5), linewidth(1.6)); dot((-3,5),linewidth(6pt) + dotstyle); dot((-2,5),linewidth(6pt) + dotstyle); dot((-1,5),linewidth(6pt) + dotstyle); dot((0,5),linewidth(6pt) + dotstyle); dot((1,5),linewidth(6pt) + dotstyle); dot((2,5),linewidth(6pt) + dotstyle); dot((2,4),linewidth(6pt) + dotstyle); dot((2,3),linewidth(6pt) + dotstyle); dot((2,2),linewidth(6pt) + dotstyle); dot((2,1),linewidth(6pt) + dotstyle); dot((2,0),linewidth(6pt) + dotstyle); dot((-3,4),linewidth(6pt) + dotstyle); dot((-3,3),linewidth(6pt) + dotstyle); dot((-3,2),linewidth(6pt) + dotstyle); dot((-3,1),linewidth(6pt) + dotstyle); dot((-3,0),linewidth(6pt) + dotstyle); dot((-2,0),linewidth(6pt) + dotstyle); dot((-2,1),linewidth(6pt) + dotstyle); dot((-2,2),linewidth(6pt) + dotstyle); dot((-2,3),linewidth(6pt) + dotstyle); dot((-2,4),linewidth(6pt) + dotstyle); dot((-1,4),linewidth(6pt) + dotstyle); dot((0,4),linewidth(6pt) + dotstyle); dot((1,4),linewidth(6pt) + dotstyle); dot((1,3),linewidth(6pt) + dotstyle); dot((0,3),linewidth(6pt) + dotstyle); dot((-1,3),linewidth(6pt) + dotstyle); dot((-1,2),linewidth(6pt) + dotstyle); dot((-1,1),linewidth(6pt) + dotstyle); dot((-1,0),linewidth(6pt) + dotstyle); dot((0,0),linewidth(6pt) + dotstyle); dot((1,0),linewidth(6pt) + dotstyle); dot((1,1),linewidth(6pt) + dotstyle); dot((1,2),linewidth(6pt) + dotstyle); dot((0,2),linewidth(6pt) + dotstyle); dot((0,1),linewidth(6pt) + dotstyle); [/asy]

Let $ a,b,c,d$ be nonnegative real numbers. Prove that \[ a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.\]

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.

Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.

Let $m$, $n$ be two positive integers greater than 3. Consider the table of size $m\times n$ ($m$ rows and $n$ columns) formed with unit squares. We are putting marbles into unit squares of the table following the instructions: $-$ each time put 4 marbles into 4 unit squares (1 marble per square) such that the 4 unit squares formes one of the followings 4 pictures (click [url=http://www.mathlinks.ro/Forum/download.php?id=4425]here[/url] to view the pictures). In each of the following cases, answer with justification to the following question: Is it possible that after a finite number of steps we can set the marbles into all of the unit squares such that the numbers of marbles in each unit square is the same? a) $m=2004$, $n=2006$; b) $m=2005$, $n=2006$.

Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)|  \leq 2 $ for $ -1 \leq  x \leq 1 $.

Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$ [i]Proposed by Andrei Bâra[/i]

Find all positive integers $a\in \{1,2,3,4\}$ such that if $b=2a$, then there exist infinitely many positive integers $n$ such that $$\underbrace{aa\dots aa}_\textrm{$2n$}-\underbrace{bb\dots bb}_\textrm{$n$}$$ is a perfect square.

Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.