Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than $180^{\circ}$. Let $C$ be a convex polygon with $s$ sides. The interior region of $C$ is the union of $q$ quadrilaterals, none of whose interiors overlap each other. $b$ of these quadrilaterals are boomerangs. Show that $q\ge b+\frac{s-2}{2}$.

Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

On a line $p$ which does not meet a circle $K$ with center $O$, point $P$ is taken such that $OP \perp p$. Let $X \ne P$ be an arbitrary point on $p$. The tangents from $X$ to $K$ touch it at $A$ and $B$. Denote by $C$ and $D$ the orthogonal projections of $P$ on $AX$ and $BX$ respectively. (a) Prove that the intersection point $Y$ of $AB$ and $OP$ is independent of the location of $X$. (b) Lines $CD$ and $OP$ meet at $Z$. Prove that $Z$ is the midpoint of $P$.

Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$? [i]Proposed by Nika Glunchadze, Georgia[/i]

There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$. The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4 $, $k_2$ touches the sides $AB,BC$ and $k_4$, and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$

Calculate $$\sum_{n=1}^{1989}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$

Let $n$ be a positive integer, and let $S_n = \{1, 2, \ldots, n\}$. For a permutation $\sigma$ of $S_n$ and an integer $a \in S_n$, let $d(a)$ be the least positive integer $d$ for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or $-1$ if no such integer exists). Compute the value of $n$ for which there exists a permutation $\sigma$ of $S_n$ satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017, \\ \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\] [i]Proposed by Michael Tang[/i]

A and B are friends at a summer school. When B asks A for his address, he answers: "My house is on XYZ street, and my house number is a 3-digit number with distinct digits, and if you permute its digits, you will have other 5 numbers. The interesting thing is that the sum of these 5 numbers is exactly 2017. That's all.". After a while, B can determine A's house number. And you, can you find his house number?

Let $ABC$ be an acute triangle, and $AA_1$, $BB_1$, and $CC_1$ its altitudes. Let $A_2$ be a point on segment $AA_1$ such that $\angle{BA_2C} = 90^{\circ}$. The points $B_2$ and $C_2$ are defined similarly. Let $A_3$ be the intersection point of segments $B_2C$ and $BC_2$. The points $B_3$ and $C_3$ are defined similarly. Prove that the segments $A_2A_3$, $B_2B_3$, and $C_2C_3$ are concurrent.

In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.

Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number. [i]Proposed by Gabriel Wu[/i]

Solve in $ \mathbb{Z} $ the following system of equations: $$ \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. . $$

There are $2010$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Ministry destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $11$ different cities?

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 3$. Prove that $$a^a + b^b + c^c \ge 3$$

The nation of CMIMCland consists of 8 islands, none of which are connected. Each citizen wants to visit the other islands, so the government will build bridges between the islands. However, each island has a volcano that could erupt at any time, destroying that island and any bridges connected to it. The government wants to guarantee that after any eruption, a citizen from any of the remaining $7$ islands can go on a tour, visiting each of the remaining islands exactly once and returning to their home island (only at the end of the tour). What is the minimum number of bridges needed?

Find the median of the positive divisors of $6^4-1$.

Prove that the sum of the squares of the medians of a triangle is at least $ 9/4 $ if the circumradius of the triangle, the area of the triangle and the inradius of the triangle (in this order) are in arithmetic progression. [i]Dumitru Crăciun[/i]

[b]p1.[/b] What is the largest distance between any two points on a regular hexagon with a side length of one? [b]p2.[/b] For how many integers $n \ge 1$ is $\frac{10^n - 1}{9}$ the square of an integer? [b]p3.[/b] A vector in $3D$ space that in standard position in the first octant makes an angle of $\frac{\pi}{3}$ with the $x$ axis and $\frac{\pi}{4}$ with the $y$ axis. What angle does it make with the $z$ axis? [b]p4.[/b] Compute $\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2$. [b]p5.[/b] Round $\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)$ to the nearest integer. [b]p6.[/b] Let $P$ be a point inside a ball. Consider three mutually perpendicular planes through $P$. These planes intersect the ball along three disks. If the radius of the ball is $2$ and $1/2$ is the distance between the center of the ball and $P$, compute the sum of the areas of the three disks of intersection. [b]p7.[/b] Find the sum of the absolute values of the real roots of the equation $x^4 - 4x - 1 = 0$. [b]p8.[/b] The numbers $1, 2, 3, ..., 2013$ are written on a board. A student erases three numbers $a, b, c$ and instead writes the number $$\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).$$ She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3. [b]p9.[/b] How many ordered triples of integers $(a, b, c)$, where $1 \le a, b, c \le 10$, are such that for every natural number $n$, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? Problems' source (as mentioned on official site) is Gator Mathematics Competition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The real numbers $a_1,a_2,\ldots,a_{100}$ satisfy the relationship \[ a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101. \] Prove that $|a_k|\leq 10$, for all $k=1,2,\ldots,100$.

Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

A graph $G$ with $n$ vertices is called [i]cool[/i] if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.

$16$ people sit around a circular table. After some time, they all stand up and sit down in either the chair they were previously sitting on or on a chair next to it. Determine the number of ways that this can be done. Note: two or more people cannot sit on the same chair.

Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes $1$ token or $4$ tokens from the stack. The player who removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.

Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

Some time ago there was a war across the world. In the plane $n$ lines are moving, with the regions contained by the lines being the territories of the countries at war. Each line moves parallel to itself with constant speed (each with its own speed), and no line can reverse its direction. Some of the original countries disappeared (a country disappears iff its area is converted to zero) and within the course of the time, other countries appeared. After some time, the presidents of the existing countries made a treaty to end the war, created the United Nations, and all borders ceased movement. The UN then counted the total numbers of sovereign states that were destroyed and the existing ones, obtaining a total of $k$. Prove that $k\leq \frac{n^3+5n}{6}+1$. Is is possible to have equality?