Define $S(N)$ to be the sum of the digits of $N$ when it is written in base $10$, and take $S^k(N) = S(S(\dots(N)\dots))$ with $k$ applications of $S$. The \textit{stability} of a number $N$ is defined to be the smallest positive integer $K$ where $S^K(N) = S^{K+1}(N) = S^{K+2}(N) = \dots$. Let $T_3$ be the set of all natural numbers with stability $3$. Compute the sum of the two least entries of $T_3$. Proposed by Albert Wang (awang11)

A square of $3 \times 3$ is subdivided into 9 small squares of $1 \times 1$. It is desired to distribute the nine digits $1, 2, . . . , 9$ in each small square of $1 \times 1$, a number in each small square. Find the number of different distributions that can be formed in such a way that the difference of the digits in cells that share a side in common is less than or equal to three. Two distributions are distinct even if they differ by rotation and/or reflection.

Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$? $ \textbf{(A)} \ 2 \qquad \textbf{(B)} \ 3 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 8 \qquad \textbf{(E)} \ 9$

Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010?

Prove that there are only two real numbers $x$ such that \[(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) = 720\]

In a triangle $ABC$, $\angle ACB=60^{\circ}$. Points $D, E$ lie on segments $BC, AC$ respectively. Points $K, L$ are such that $ADK$ and $BEL$ are equlateral, $A$ and $L$ lie on opposite sides of $BE$, $B$ and $K$ lie on the opposite siedes of $AD$. Prove that $$AE+BD=KL.$$

An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.

Let $n$ and $M$ be positive integers such that $M>n^{n-1}$. Prove that there are $n$ distinct primes $p_1,p_2,p_3 \cdots ,p_n$ such that $p_j$ divides $M + j$ for all $1 \le j \le n$.

Colour a $20000\times 20000$ square grid using 2000 different colours with 1 colour in each square. Two squares are neighbours if they share a vertex. A path is a sequence of squares so that 2 successive squares are neighbours. Mark $k$ of the squares. For each unmarked square $x$, there is exactly 1 marked square $y$ of the same colour so that $x$ and $y$ are connected by a path of squares of the same colour. For any 2 marked squares of the same colour, any path connecting them must pass through squares of all the colours. Find the maximum value of $k$.

Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$, such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$.

Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds $$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$

A cardboard circular disk of radius $5$ centimeters is placed on the table. While it is possible, Peter puts cardboard squares with side $5$ centimeters outside the disk so that: [i](1)[/i] one vertex of each square lies on the boundary of the disk; [i](2)[/i] the squares do not overlap; [i](3)[/i] each square has a common vertex with the preceding one. Find how many squares Peter can put on the table, and prove that the first and the last of them must also have a common vertex. [i](4 points)[/i]

The figure shows a line intersecting a square lattice. The area of some arising quadrilaterals are also indicated. What is the area of the region with the question mark? [img]https://cdn.artofproblemsolving.com/attachments/0/d/4d5741a63d052e3f6971f87e60ca7df7302fb0.png[/img]

Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.

We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.

Find $x + y$ if $(x+\sqrt{x^2 +1} )(y+\sqrt{y^2 +1} ) = 1.$

Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

The $n$-factorial of a positive integer $x$ is the product of all positive integers less than or equal to $z$ that are congruent to $z$ modulo $n$. For example, for the number 16, its 2-factorial is $16 \times 14 \times 12 \times 10 \times 8 \times 6 \times 4 \times 2$, its 3-factorial is $16 \times 13 \times 10 \times 7 \times 4 \times 1$ and its 18-factorial is 16. A positive integer is called [i]olympic[/i] if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers.

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

Let $c$ be the least common multiple of positive integers $a$ and $b$, and $d$ be the greatest common divisor of $a$ and $b$. How many pairs of positive integers $(a,b)$ are there such that \[ \dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1? \] $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 2 $

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

Ephramis taking his final exams. He has $7$ exams and his school holds finals over $3$ days. For a certain arrangement of finals, let $f$ be the maximum number of finals Ephram takes on any given day. Find the expected value of $f$ .

Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles? [asy] size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray); draw((11,11)--(171,11),linetype("4 4")+white+linewidth(1.5)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle,linewidth(1.125)); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle,linewidth(1.125)); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.9)*rotate(45)*"Water 1", (23,-13.5)); label(scale(.9)*rotate(45)*"Water 2", (43,-13.5)); label(scale(.9)*rotate(45)*"Water 7", (143,-13.5)); filldraw(circle((101,-13.5),.3)); filldraw(circle((97,-13.5),.3)); filldraw(circle((93,-13.5),.3)); filldraw(circle((89,-13.5),.3)); filldraw(circle((85,-13.5),.3)); label(scale(.9)*rotate(90)*"Start", (5.5,11)); label(scale(.9)*rotate(270)*"Finish", (176.5,11)); [/asy] $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 24\qquad\textbf{(D) } 48\qquad\textbf{(E) } 96$

Do there exist (a) four (b) five distinct positive integers such that the sum of any three of them is a prime number? (V Senderov)