Rectangle $ABCD$ has an area of 30. Four circles of radius $r_1 = 2$, $r_2 = 3$, $r_3 = 5$, and $r_4 = 4$ are centered on the four vertices $A$, $B$, $C$, and $D$ respectively. Two pairs of external tangents are drawn for the circles at A and $C$ and for the circles at $B$ and $D$. These four tangents intersect to form a quadrilateral $W XY Z$ where $\overline{W X}$ and $\overline{Y Z}$ lie on the tangents through the circles on $A$ and $C$. If $\overline{W X} + \overline{Y Z} = 20$, find the area of quadrilateral $W XY Z$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/cb3b3457f588a15ffb4c875b1646ef2aec8d11.png[/img]

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Let $ABC$ be a triangle with incentre $I$. The incircle of the triangle $ABC$ touches the sides $AC$ and $AB$ at points $E$ and $F$ respectively. Let $\ell_B$ and $\ell_C$ be the tangents to the circumcircle of $BIC$ at $B$ and $C$ respectively. Show that there is a circle tangent to $EF, \ell_B$ and $\ell_C$ with centre on the line $BC$. [i]Proposed by Navneel Singhal, India[/i]

Evaluate $$lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} ([\frac{2n}{k}] -2[\frac{n}{k}])$$ and express your answer in the form $\log a-b,$ with $a$ and $b$ positive integers. Here $[x]$ is defined to be the integer such that $[x] \leq x <[x]+1$ and $\log x$ is the logarithm of $x$ to base $e.$

Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$. [i](Karl Czakler)[/i]

Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Let $H_a, H_b, H_c$ be the orthocenter of triangles $TBC, TCA, TAB$, respectively. a) Prove that: $T$ is the centroid of the $\triangle H_aH_bH_c$. b) Denote $D, E, F$ respectively by the intersections of $H_cH_b$ and the segment $BC$, $H_cH_a$ and the segment $CA$, $H_aH_b$ and the segment $AB$. Prove that: the triangle $DEF$ is equilateral. c) Prove that: the lines passing through $D, E, F$ and are respectively perpendicular to $BC, CA, AB$ are concurrent at a point. Let that point be $S$. d) Prove that: $TS$ is parallel to the [i]Euler[/i] line of the triangle $ABC$.

Let $x,y,z$ be positive numbers such that \[ {1\over x}+{1\over y}+{1\over z}=1. \] Show that \[ \sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z} \]

Let $ABCDEF$ be a regular hexagon with side length $2$. Point $M$ is the midpoint of diagonal $AE$. The pentagon $ABCDE$ is folded along segments $BD$, $BM$, and $DM$ in such a way that points $A$, $C$, and $E$ coincide. As a result of this operation, a tetrahedron is obtained. Determine its volume.

Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

Solve the system \[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\] in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.

Find the value of $x$ such that $2^{x+3} - 2^{x-3} = 2016$.

Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

$A$ and $B$ are two points in the plane $\alpha$, and line $r$ passes through points $A, B$. There are $n$ distinct points $P_1, P_2, \ldots, P_n$ in one of the half-plane divided by line $r$. Prove that there are at least $\sqrt n$ distinct values among the distances $AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.$

Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$, and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$. Show that there is a point $ y\in\mathcal{H}$ such that \[ \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\}\] is an orthonormal system of vectors in $ \mathcal{H}$.

Show that there exists some [b]positive[/b] integer $k$ with $$\gcd(2012,2020)=\gcd(2012+k,2020)$$$$=\gcd(2012,2020+k)=\gcd(2012+k,2020+k).$$

Let us call a figure on a sheet of squares an arbitrary finite set of connected squares, i.e. a set of squares in which it is possible to go from any square to any other by walking only on the squares of this figure. Prove that for every natural n there exists such a figure on the sheet of squares that it can be cut into "corners" (Fig. 1) exactly in $F_n$ ways, where $F_n$ s the $n$-th Fibonacci number (in the series of Fibonacci numbers $F_1 = F_2 = 1$ and for each $i > 1$ holds $F_{i+2} = F_{i+1} + F_i$). For example, a rectangle of $2 \times 3$ squares can be cut at the corners in exactly two ways (Fig. $2$). [img]https://cdn.artofproblemsolving.com/attachments/6/5/c82340623ff5f92a410bd73755ba8cbdc501ff.png[/img]

A table of the type $~$ $ (n_1, n_2, ... , n_m) ,\ n_1 \ge n_2 \ge ... \ge n_m $ $~$ is defined in the following way: $~$ $n_1$ $~$ squares are ordered horizontally one next to another, then $~$ $n_2$ $~$ squares are ordered horizontally beneath the already ordered $~$ $n_1$ $~$ squares. The procedure continues until a net composed of $~$ $n_1$ $~$ squares in the first row, $~$ $n_2$ $~$ in the second, $~$ $n_i$ $~$ in the $~$ $i$-th row is obtained, such that there are totally $~$ $n=n_1+n_2+...+n_m$ $~$ squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from $~$ $1$ $~$ till $~$ $n$ $~$ in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type $~$ $(5,4,2,1)$ $~$ and one possible way of filling it is attached to the post. Find the number of ways the table of type $~$ $(4,3,2)$ $~$ can be filled.

Two positive integers $m$ and $n$ are written on the board. We replace one of two numbers in each step on the board by either their sum, or product, or ratio (if it is an integer). Depending on the numbers $m$ and $n$, specify all the pairs that can appear on the board in pairs. (Radovan Švarc)

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

How many integers between $2$ and $100$ inclusive [i]cannot[/i] be written as $m \cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$? Note that there are $25$ primes less than $100.$

Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.

Find all the natural numbers $a, b, c$ satisfying the following equation: $$a^{12} + 3^b = 1788^c$$.

Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

There are $3n$ participants in the Mathematical Olympiad competition. They are assigned seats in three rows, with $n$ seats in each, and are admitted into the hall one at a time, after which they immediately take their seats. Calculate the probability that until the last competitor takes his seat, at any moment for each two rows the difference in the number of players sitting in them is no greater than 1.

Isaac writes each fraction $\frac{1^2}{300}$ , $\frac{2^2}{300}$ , $...$, $\frac{300^2}{300}$ in reduced form. Compute the sum of all denominators over all the reduced fractions that Isaac writes down.