Prove the following inequality. \[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\] Own

Alan is bored one day and decides to write down all the divisors of $1260^2$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to erase all his work. Every second, he picks a random divisor which is still on the wall and instantly erases it and every number that divides it. What is the expected time it takes for Alan to erase everything on the wall?

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $ $ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$

Let $\vartriangle ABC$ be a triangle with orthocenter $H$ and altitudes $AD$, $BE$ and $CF$. Let $D'$, $E' $ and $F'$ be the intersections of the heights $AD$, $BE$ and $CF$, respectively, with the circumcircle of $\vartriangle ABC $, so that they are different points from the vertices of triangle $\vartriangle ABC$. Let $L$, $M$ and $N$ be the midpoints of $BC$, $AC$ and $AB$, respectively. Let $ P$, $Q$ and $R$ be the intersections of the circumcircle with $LH$, $MH$ and $NH$, respectively, such that $ P$ and $ A$ are on opposite sides of $BC$, $Q$ and $A$ are on opposite sides of $AC$ and $R$ and $C$ are on opposite sides of $AB$. Show that there exists a triangle whose sides have the lengths of the segments $D' P$, $E'Q$, and $F'R$.

Given triangle $ABC$. $D$ lies on $BC$ such that $AD$ bisects $BAC$. Given $AB=3$, $AC=9$, and $BC=8$. Find $AD$.

There are $N$ permutations $(a_1,a_2,\dots,a_{30})$ of $1,2,\dots,30$ such that for $m\in\{2,3,5\}$, $m$ divides $a_{n+m}-a_n$ for all integers $n$ with $1\leq n <n+m\leq 30$. Find the remainder when $N$ is divided by 1000.

What is the largest integer $n < 2018$ such that for all integers $b > 1$, $n$ has at least as many $1$'s in its base-$4$ representation as it has in its base-$b$ representation?

Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(m+f(n))=f(f(m))+f(n).\]

In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively, and let $F$ be the foot of the altitude through $A$. Show that the line $DE$, the angle bisector of $\angle ACB$ and the circumcircle of $ACF$ pass through a common point. [b]Alternate version:[/b] In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively. The line $DE$ and the angle bisector of $\angle ACB$ meet at $G$. Show that $\angle AGC$ is a right angle.

An equilateral triangle $\mathcal{T}{}$ with side 111 is divided by straight lines parallel to its sides into equilateral triangles with side 1. The vertices of these small triangles, except the centre of $\mathcal{T}{}$ are marked. Call a set of several marked points [i]linear[/i] if[list=i][*]the marked points lie on a line $\ell$ parallel to one of the sides of the triangle $\mathcal{T}$ and; [*]if two marked points on $\ell$ are in this set, every other marked point inbetween them is in the set. [/list]How many ways are there to split all the marked points into 111 linear sets?

In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.

Suppose you have ten pairs of red socks, ten pairs of blue socks, and ten pairs of green socks in your drawer. You need to go to a party soon, but the power is currently off in your room. It is completely dark, so you cannot see any colors and unfortunately the socks are identically shaped. What is the minimum number of socks you need to take from the drawer in order to guarantee that you have at least one pair of socks whose colors match?

If $A$, $B$, $C$, $D$ are four distinct points such that every circle through $A$ and $B$ intersects (or coincides with) every circle through $C$ and $D$, prove that the four points are either collinear (all on one line) or concyclic (all on one circle).

Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that: \[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\] \[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\] \[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]

Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.) [i]Proposed by Robin Park[/i]

Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers? (Alexandr Shapovalov)

There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.

Let $A$ and $B$ be points on a circle $k$. Suppose that an arc $k'$ of another circle, $\ell$, connects $A$ with $B$ and divides the area inside the circle $k$ into two equal parts. Prove that arc $k'$ is longer than the diameter of $k$.

Define the operation $ \star$ by $ a\star b \equal{} (a \plus{} b)b$. What is $ (3\star 5) \minus{} (5\star 3)$? $ \textbf{(A)}\ \minus{}16\qquad \textbf{(B)}\ \minus{}8\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

The median $ AM $ of $ ABC $ meets the incircle of $ ABC $ at $ K,L. $ The lines thru $ K $ and $ L, $ both parallel to $ BC $ meets the incircle of $ ABC $ at $ XY. $ The intersections of $ AX $ and $ AY $ with $ BC $ are $ P,Q, $ respectively. Prove that $ BP=CQ. $

Let \( \triangle ABC \) be a triangle such that \( BC > AC > AB \). A point \( X \) is marked on side \( BC \) such that \( AX = XC \). Let \( Y \) be a point on segment \( AX \) such that \( CY = AB \). Prove that \( \angle CYX = \angle ABC \).

An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

A sequence $(a_k)_{k=1}^{\infty}$ has the property that there is a natural number $n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_{n+k} = a_k$ for all $k$. Prove that there exists a natural number $N$ such that $$\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...$$

There is a board with 32 rows and 10 columns. Pablo writes 1 or -1 in each box. Matías, with Pablo's board in view, chooses one or more columns, and in each of the chosen columns, changes all of Pablo's numbers to their opposites (where there is 1 he puts -1 and where there is -1 he puts 1) . In the other columns, leave Pablo's numbers. Matías wins if he manages to make his board have each of the rows different from all the rows on Pablo's board. Otherwise, that is, if any row on Matías's board is equal to any row on Pablo's board, Pablo wins. If both play perfectly, determine which of the two is assured of victory.