A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular. [i]Alternative formulation.[/i] Given a convex quadrilateral $ ABCD$ with congruent diagonals $ AC \equal{} BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other. [i]Original formulation:[/i] Let $ ABCD$ be a convex quadrilateral such that $ AC \equal{} BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $ O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $ AB,BC,CD,DA$ respectively. Show that $ O_1O_3$ is perpendicular to $ O_2O_4.$

Let $a_1>a_2>a_3>\cdots$ be a sequence of real numbers which converges to 0. We put circles of radius $a_1$ into a unit square until no more can fit. (A previously laid circle must not be moved.) Then we put circles of radius $a_2$ in the remaining space until no more can fit, continuing the process for $a_3$,... What can the area covered by the circles be? a similar problem involving circles in a square: [url]https://artofproblemsolving.com/community/c7h1979044[/url]

How many $3$-element subsets of $\left\{1, 2, 3, \dots, 11\right\}$ are there, such that the sum of the three elements is a multiple of $3$?

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

For which natural numbers $n$ do there exist $n$ natural numbers $a_i\ (1\le i\le n)$ such that $\sum_{i=1}^n a_i^{-2}=1$?

Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$

How many 4-digit numbers have exactly $9$ divisors from the set $\{1,2,3,4,5,6,7,8,9,10\}$? [i]Proposed by Ethan Gu[/i]

A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

In an acute $\triangle ABC$, prove that \begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear

There is a set of weights with the following properties: 1) It contains 5 weights, pairs of different weights. 2) For any two weights, there are two other weights of the same total weight. What is the smallest number of weights that can be in this set?

If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.

Let $O$, $I$, and $\omega$ be the circumcenter, the incenter, and the incircle of nonequilateral $\triangle ABC$. Let $\omega_A$ be the unique circle tangent to $AB$ and $AC$, such that the common chord of $\omega_A$ and $\omega$ passes through the center of $\omega_A$ . Let $O_A$ be the center of $\omega_A$. Define $\omega_B, O_B, \omega_C, O_C$ similarly. If $\omega$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively, prove that the perpendiculars from $D$, $E$, $F$ to $O_BO_C , O_CO_A , O_AO_B$ are concurrent on the line $OI$. [i]Pitchayut Saengrungkongka[/i]

$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]

Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following constraints are imposed: $\bullet$ Every person must get a positive integer number of shares, and all $16$ shares must be given out. $\bullet$ No one person can have more shares than the other three people combined. Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?

11 integers are placed along the circle. It is known that any two neighbors differ by at least 20 and sum of any two neighbors is no more than 100. Find the minimal possible sum of all numbers.

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$. [i]Proposed by TuZo[/i]