Five segments have lengths such that any three of them can be sides of a - possibly degenerate - triangle. Also, the lengths of these segments are nonzero and pairwisely different. Prove that there exists at least one acute-angled triangle among these triangles.

Find the least positive multiple of 999 that does not have a 9 as a digit.

In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle? $ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $

Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$ has root in interval $\left(0,\frac{\pi}{2}\right)$

An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network [i]feasible[/i] if it satisfies the following conditions: [list] [*] All connections operate in both directions [*] If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.[/list] Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

A [i]mixing[/i] of the sequence $a_1,a_2,\dots ,a_{3n}$ is called the following sequence: $a_3,a_6,\dots ,a_{3n},a_2,a_5,\dots ,a_{3n-1},a_1,a_4,\dots ,a_{3n-2}$. Is it possible after finite amount of [i]mixings[/i] to reach the sequence $192,191,\dots ,1$ from $1,2,\dots ,192$?

How many different compounds have the formula $\text{C}_3\text{H}_8\text{O}$? $ \textbf{(A)}\ \text{one} \qquad\textbf{(B)}\ \text{two}\qquad\textbf{(C)}\ \text{three} \qquad\textbf{(D)}\ \text{four} \qquad$

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

Let $ ABC $ be a triangle whose vertices have integer coordinates and inside of which there is exactly one point $ O $ with integer coordinates. Let $ D $ be the intersection of the lines $ BC $ and $ AO. $ Find the largest possible value of $ \frac {\overline{AO}} {\overline{OD}} $.

A fire that starts in the steppe spreads in all directions at a speed of $1$ km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of $1$ km. The grader moves at a speed of $14$ km per hour and cuts a strip with a plow that cuts off the fire. Indicate the path along which the grader should move so that the total area of the burnt steppe does not exceed: a) $4 \pi $ km$^2$; b) $3 \pi $ km$^2$. (We can assume that the grader’s path consists of straight segments and circular arcs.)

Two players by turns draw diagonals in a regular $(2n+1)$-gon ($n>1$). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy? [i]K. Sukhov[/i]

Find the sum of all positive integers $b<1000$ such that the base-$b$ integer $36_b$ is a perfect square and the base-$b$ integer $27_b$ is a perfect cube.

A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.

Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.

Prove that $\overline{a0... 09}$ (in which $a > 0$ is a digit and there is at least one zero) is not a perfect square. (VA Senderov)

Michael draws $\triangle ABC$ in the sand such that $\angle ACB=90^\circ$ and $\angle CBA=15^\circ$. He then picks a point at random from within the triangle and labels it point $M$. Next, he draws a segment from $A$ to $BC$ that passes through $M$, hitting $BC$ at a point he labels $D$. Just then, a wave washes over his work, so Michael redraws the exact same diagram farther from the water, labeling all the points the same way as before. If hypotenuse $AB$ is $4$ feet in length, let $p$ be the probability that the number of feet in the length of $AD$ is less than $2\sqrt3-2$. Compute $\lfloor1000p\rfloor$.

If $x = \frac{a}{b}$, $a \ne b$ and $b \ne 0$, then $\frac{a + b}{a - b} = $ $ \textbf{(A)}\ \frac{x}{x + 1}\qquad\textbf{(B)}\ \frac{x + 1}{x - 1}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ x - \frac{1}{x}\qquad\textbf{(E)}\ x + \frac{1}{x} $

Find all integer values of $x$ for which the value of the expression \[x^2+6x+33\] is a perfect square.

Let $ABC$ be a triangle with $AB\ne AC$. Its incircle has center $I$ and touches the side $BC$ at point $D$. Line $AI$ intersects the circumcircle $\omega$ of triangle $ABC$ at $M$ and $DM$ intersects again $\omega$ at $P$. Prove that $\angle API= 90^o$.

Let $p>2$ be a prime. How many residues $\pmod p$ are both squares and squares plus one?

Let $f(x)=x^{2021}+15x^{2020}+8x+9$ have roots $a_i$ where $i=1,2,\cdots , 2021$. Let $p(x)$ be a polynomial of the sam degree such that $p \left(a_i + \frac{1}{a_i}+1 \right)=0$ for every $1\leq i \leq 2021$. If $\frac{3p(0)}{4p(1)}=\frac{m}{n}$ where $m,n \in \mathbb{Z}$, $n>0$ and $\gcd(m,n)=1$. Then find $m+n$.

The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.

An object has the shape of a square and has side length $a$. Light beams are shone on the object from a big machine. If $m$ is the mass of the object, $P$ is the power $\emph{per unit area}$ of the photons, $c$ is the speed of light, and $g$ is the acceleration of gravity, prove that the minimum value of $P$ such that the bar levitates due to the light beams is \[ P = \dfrac {4cmg}{5a^2}. \] [i]Problem proposed by Trung Phan[/i]