A two-digit integer is [i]reversible [/i] if, when written backwards in base $10$, it has the same number of positive divisors. Find the number of reversible integers.

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$? [asy] size(200); defaultpen(linewidth(.6pt)+fontsize(12pt)); dotfactor=4; draw((0,0)--(0,2)); draw((0,0)--(1,0)); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((0,0)--(1,2)); draw((0,2)--(2,1)); draw((0,2)--(2,2)); draw((2,1)--(2,2)); label("$A$",(0,2),NW); label("$B$",(1,2),N); label("$C$",(4/5,1.55),W); dot((0,2)); dot((1,2)); dot((4/5,1.6)); dot((2,1)); dot((0,0)); [/asy] $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt2}{4} $

At the altitude $AH_1$ of an acute non-isosceles triangle $ABC$ chose a point $X$ , from which draw the perpendiculars $XN$ and $XM$ on the sides $AB$ and $AC$ respectively. It turned out that $H_1A$ is the angle bisector $MH_1N$. Prove that $X$ is the point of intersection of the altitudes of the triangle $ABC$.

Let $ f,F:\mathbb{R}\longrightarrow\mathbb{R} $ be two functions such that $ f $ is nondecreasing, $ F $ admits finite lateral derivates in every point of its domain, $$ \lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} , $$ for all real numbers $ y, $ and $ F(0)=0. $ Prove that $ F(x)=\int_0^x f(t)dt, $ for all real numbers $ x. $

Let $ABC$ be an acute triangle with side lengths $AB = 7$, $ BC = 12$, $AC = 10$, and let $\omega$ be its incircle. If $\omega$ is touching $AB$, $AC$ at $F, E$, respectively, and if $EF$ intersects $BC$ at $X$, suppose that the ratio in which the angle bisector of $\angle BAC$ divides the segment connecting midpoint of $EX$ and $C$ is $\frac{a}{b}$ , where $a, b$ are relatively prime integers. Find $a + b$.

In the playboard shown beside, players $A$ and $B$ alternately fill the empty cells by integers, player $A$ starting. In each step the empty cell and the integer can be chosen arbitrarily. Show that player $A$ can always achieve that all the equalities hold after the last step. [img]https://cdn.artofproblemsolving.com/attachments/c/0/524195b1a8ab8457b72005a162f8124c2b1bd2.png[/img]

Given is a triangle $ABC$ and let $AA_1$, $BB_1$ be angle bisectors. It turned out that $\angle AA_1B=24^{\circ}$ and $\angle BB_1A=18^{\circ}$. Find the ratio $\angle BAC:\angle ACB:\angle ABC$.

How many base 10 four-digit numbers, $N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions? (i) $4,000 \leq N < 6,000;$ (ii) $N$ is a multiple of 5; (iii) $3 \leq b < c \leq 6$. $ \mathbf{(A)}\; 10\qquad \mathbf{(B)}\; 18\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 36\qquad \mathbf{(E)}\; 48$

In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set $S$ of cities is said [i]beautiful[/i] if it contains at least three different cities and from each city $A$ in $S$ at least two buses depart, each one goes directly to a different city in $S$ and none of them is $A$ (if there is a direct bus from $A$ to a city $B$ in $S$, there is not necessarily a direct bus from $B$ to $A$). Show that if there exists a beautiful set of cities $S$, then there exists a beautiful $T$ subset of $S$, such that for any two cities in $T$, you can get from one to another by taking buses that only pass through cities in $T$. Note: A bus goes directly from one city to another if it does not pass through any other city.

Find all $4$-digit numbers $\overline{abba}$ that are equal to the product of some consecutive prime numbers.

Let $ABC$ be a triangle and $P$ be in its interior. Let $Q$ be the isogonal conjugate of $P$. Show that $BCPQ$ is cyclic if and only if $AP=AQ$.

Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.

Let $n$ be a given positive integer. Find the smallest positive integer $k$ for which the following statement is true: for any given simple connected graph $G$ and minimal cuts $V_1, V_2,\ldots, V_n$, at most $k$ vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer $1\le i\le n$ such that $V_i$ separates the two vertices. A partition of the vertices of $G$ into two disjoint non-empty sets is called a [i]minimal cut[/i] if the number of edges crossing the partition is minimal. [i]Proposed by András Imolay, Budapest[/i]

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

We say that a positive integer $k$ is tricubic if there are three positive integers $a, b, c,$ not necessarily different, such that $k=a^3+b^3+c^3.$ a) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: exactly one of the three numbers $n, n+2$ and $n+28$ is tricubic. b) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: exactly two of the three numbers $n, n+2$ and $n+28$ are tricubic. c) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: the three numbers $n, n+2$ and $n+28$ are tricubic.

Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.