Given five segments such that any three of them can be used to form a triangle. Show that at least one of these triangles is acute-angled. [i]Alternative formulation:[/i] Five segments have lengths such that any three of them can be sides of a triangle. Prove that there exists at least one acute-angled triangle among these triangles.

If $f(x) = x^{\left(x+1\right)}\times \left(x+2\right)^{\left(x+3\right)}$ then $f(0) + f(-1) + f(-2) + f(-3) =$ $\textbf{(A)}\ \frac{-8}{9} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{8}{9} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac{10}{9}$

Let $ABC$ be a triangle with $BC=20$ and $CA=16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m+\sqrt{n}$ for positive integers $m$ and $n$. What is $100m+n$? [i] Proposed by Tristan Shin [/i]

Consider a set of $2016$ distinct points in the plane, no four of which are collinear. Prove that there is a subset of $63$ points among them such that no three of these $63$ points are collinear.

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

A drawer has $5$ pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$? [i]Author: Ray Li[/i]

Let $\sigma(n)$ be the sum of all the positive divisors of $n$. Let $a$ be the smallest positive integer greater than or equal to $2015$ for which there exists some positive integer $n$ satisfying $\sigma(n)=a$. Finally, let $b$ be the largest such value of $n$. Compute $a+b$.

Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^o$. Find the degree measure of $\angle BAC$.

In an acute-angled triangle $ABC$ with circumcenter $O$, points $P$ and $Q$ are taken on sides $AC$ and $BC$ respectively such that $\frac{AP}{PQ} = \frac{BC}{AB}$ and $\frac{BQ}{PQ} =\frac{AC}{AB}$ . Prove that the points $O, P,Q,C$ lie on a circle.

A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called [i]straight[/i] if and only if for each integer $k$, $1\leq k\leq n-1$ the following inequality is fulfilled \[ |\sigma(k)-\sigma(k+1)|\leq 2. \] Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations. [i]Valentin Vornicu[/i]

A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is: $ \textbf{(A)}\ \frac{1}{4} \text{ the area of the original square}\\ \textbf{(B)}\ \frac{1}{2} \text{ the area of the original square}\\ \textbf{(C)}\ \frac{1}{2} \text{ the area of the circular piece}\\ \textbf{(D)}\ \frac{1}{4} \text{ the area of the circular piece}\\ \textbf{(E)}\ \text{none of these}$

(B.Frenkin, A.Zaslavsky) A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes? (Once a safe is opened, the key inside the safe can be used to open another safe.)

What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23$?

Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$. [b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.

Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than $3$ streaks? For example, if we denote Richard winning by $R$ and Shreyas winning by $S,$ $SSRSSRRRRR$ is one such outcome, with $4$ streaks.

What is the value of $1234+2341+3412+4123$? $\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt{3}$ inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey? [center] [img]https://cdn.artofproblemsolving.com/attachments/b/4/23f9bc88ea057cb2676f2b8b373330b0f5df69.png[/img][/center] $\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi$

Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.

Solve the system of equations in real numbers: \[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]

A sequence of real numbers $a_0, a_1, \dots$ satisfies the condition\[\sum\limits_{n=0}^{m}a_n\cdot(-1)^n\cdot\dbinom{m}{n}=0\]for all large enough positive integers $m$. Prove that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\ge0$.

Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.

Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.

Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ\perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}.$ Find $\angle CBA$, in degrees. [i]Proposed by Nathan Ramesh