Compute the only element of the set \[\{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \backslash \{-2, 2\}\right\}.\] [i]Proposed by Michael Tang[/i]

Let $m, n, k > 1$ be positive integers. For a set $S$ of positive integers, define $S(i,j)$ for $i<j$ to be the number of elements in $S$ strictly between $i$ and $j$. We say two sets $(X,Y)$ are a [i]fat[/i] pair if \[ X(i,j)\equiv Y(i,j) \pmod{n} \] for every $i,j \in X \cap Y$. (In particular, if $\left\lvert X \cap Y \right\rvert < 2$ then $(X,Y)$ is fat.) If there are $m$ distinct sets of $k$ positive integers such that no two form a fat pair, show that $m<n^{k-1}$. [i]Proposed by Allen Liu[/i]

Let $ABCD$ be a cyclic quadrilateral such that the ratio of its diagonals is $AC:BD=7:5.$ Let $E$ and $F$ be the intersections of lines $AB$ and $CD$ and lines $BC$ and $AD$, respectively. Let $L$ and $M$ be the midpoints of diagonals $AC$ and $BD$, respectively. Given that $EF=2020,$ the length of $LM$ can be written as $\frac{p}{q}$ where $p,q$ are relatively prime positive integers. Compute $p+q.$

Extensions of opposite sides of a convex quadrilateral $ABCD$ intersect at points $P$ and $Q$. Points are marked on the sides of $ABCD$ (one per side), which are the vertices of a parallelogram with a side parallel to $PQ$. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral $ABCD$. (E. Bakaev)

Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.

$k>2$ is an integer. $a_0,a_1,...$ is an integer sequence such that $a_0=0$, $a_{n+1}=ka_n-a_{n-1}$. Prove that for any positive integer $m$, $(2m)!|a_1a_2...a_{3m}$.

For any positive real $r$, let $d(r)$ be the distance of the nearest lattice point from the circle center the origin and radius $r$. Show that $d(r)$ tends to zero as $r$ tends to infinity.

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

Is there a positive integer $n$ such that the fractional part of \[ \left(3+\sqrt{5}\right)^n >0.99 ? \]

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

Find the largest positive integer $N$ satisfying the following properties: [list] [*]$N$ is divisible by $7$; [*]Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i\neq j$) gives an integer which is $\textit{not}$ divisible by $7$. [/list]

Let $ A\equal{}\{A_1,\dots,A_m\}$ be a family distinct subsets of $ \{1,2,\dots,n\}$ with at most $ \frac n2$ elements. Assume that $ A_i\not\subset A_j$ and $ A_i\cap A_j\neq\emptyset$ for each $ i,j$. Prove that: \[ \sum_{i\equal{}1}^m\frac1{\binom{n\minus{}1}{|A_i|\minus{}1}}\leq1\]

Consider (in the plane) three concentric circles with radii $1, 2$ and $3$ and equilateral triangle $\Delta$ such that on each of the three circles is one vertex of $\Delta$ . Calculate the length of the side of $\Delta$ . [img]https://1.bp.blogspot.com/-q40dl3TSQqE/Xy1QAcno_9I/AAAAAAAAMR8/11nsSA0syNAaGb3W7weTHsNpBeGQZXnHACLcBGAsYHQ/s0/flanders%2B2013%2Bp4.png[/img]

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$ [i]M. Zorka[/i]

Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$, and a pair of vertices $(u,v)$, define \[ L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}\]Prove that, for any choice of $V$, there exist at least $120$ distinct values of $L(u,v)$.

Let $A_0BC_0D$ be a convex quadrilateral inscribed in a circle $\omega$. For all integers $i\ge0$, let $P_i$ be the intersection of lines $A_iB$ and $C_iD$, let $Q_i$ be the intersection of lines $A_iD$ and $BC_i$, let $M_i$ be the midpoint of segment $P_iQ_i$, and let lines $M_iA_i$ and $M_iC_i$ intersect $\omega$ again at $A_{i+1}$ and $C_{i+1}$, respectively. The circumcircles of $\triangle A_3M_3C_3$ and $\triangle A_4M_4C_4$ intersect at two points $U$ and $V$. If $A_0B=3$, $BC_0=4$, $C_0D=6$, $DA_0=7$, then $UV$ can be expressed in the form $\tfrac{a\sqrt b}c$ for positive integers $a$, $b$, $c$ such that $\gcd(a,c)=1$ and $b$ is squarefree. Compute $100a+10b+c $. [i]Proposed by Eric Shen[/i]