Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

The quadrilateral $ABCD$ satisfies $\angle ACD = 2\angle CAB, \angle ACB = 2\angle CAD $ and $CB = CD.$ Show that $$\angle CAB=\angle CAD.$$

Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.

$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$. [i]Proposed by Mohammad Gharakhani[/i]

Let $a, b, c \in [-1, 1] $ such that $1 + 2abc \ge a^2 + b^2 + c^2$. Prove that $1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4$.

What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. [i]Romania[/i]

Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

A token is placed in the leftmost square in a strip of four squares. In each move, you are allowed to move the token left or right along the strip by sliding it a single square, provided that the token stays on the strip. In how many ways can the token be moved so that after exactly $15$ moves, it is in the rightmost square of the strip?

Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?

Let $p \in \mathbb{N} \setminus \{0, 1\}$ be a fixed positive integer. Prove that for every $K > 0$, there exist infinitely many $n$ and $N$ such that there are atleast $\dfrac{KN}{\log(N)}$ primes among the following $N$ numbers given by \[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\] [i]Proposed by Bimit Mandal[/i]

Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.

Suppose that $ ABC$ is an isosceles triangle with $ AB \equal{} AC$. Let $ P$ be the point on side $ AC$ so that $ AP \equal{} 2CP$. Given that $ BP \equal{} 1$, determine the maximum possible area of $ ABC$.

Let $d(k)$ be the number of positive divisors of integer $k$. A number $n$ is called [i]balanced[/i] if $d(n-1) \leq d(n) \leq d(n+1)$ or $d(n-1) \geq d(n) \geq d(n+1)$. Show that there are infinitely many balanced numbers.

Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$, $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ pass through the points $B$, $C$ and $A$ respectively and the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ pass through the points $C$, $A$ and $B$ respectively. Prove that (a) the lines $AA_2$, $B_1B_2$ and $C_1C_3$ intersect at one point. (b) the lines $AA_2$, $BB_2$ and $CC_2$ intersect at one point. (Mykhailo Plotnikov) [img]https://cdn.artofproblemsolving.com/attachments/3/d/ad2fe12ae2c82d04b48f5e683b7d54e0764baf.png[/img]

An L-shaped region is formed by attaching two $2$ by $5$ rectangles to adjacent sides of a $2$ by $2$ square as shown below. [asy] size(6cm); draw((0,0)--(7,0)--(7,2)--(2,2)--(2,7)--(0,7)--cycle); real eps = 0.45; draw(box( (0,0), (eps,eps) )); draw(box( (7,0), (7-eps,eps) )); draw(box( (7,2), (7-eps,2-eps) )); draw(box( (0,7), (eps,7-eps) )); draw(box( (2,7), (2-eps,7-eps) )); label("$7$", (0,3.5), dir(180)); label("$7$", (3.5,0), dir(270)); label("$2$", (7,1), dir(0)); label("$5$", (4.5,2), dir(90)); label("$5$", (2,4.5), dir(0)); label("$2$", (1,7), dir(90)); [/asy] The resulting shape has an area of $24$ square units. How many ways are there to tile this shape with $2$ by $1$ dominos (each of which may be placed horizontally or vertically)?

Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies \[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\] [hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]

Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.