A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $ 247,$ $ 275,$ and $ 756$ and end with the term $ 824.$ Let $ \mathcal{S}$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $ \mathcal{S}?$ $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

Determine the greatest possible value of the constant $c$ that satisfies the following condition: for any convex heptagon the sum of the lengthes of all it’s diagonals is greater than $cP$, where $P$ is the perimeter of the heptagon. (I. Zhuk)

From the two cities of Dobruisk and Bodruisk, the distance between which is $40$ km, two cyclists Dobi and Bodi simultaneously rode towards each other. Dobie was traveling at $23$ km/h and Bodie was traveling at $17$ km/h. Before departure, a fly landed on Dobie’s nose, which, at the moment of his departure from the city, flew towards Bodruisk at a speed of $40$ km/h. The fly met Bodie, immediately turned back and flew towards Dobruisk at a speed of $30$ km/h. (The fact is that the wind was blowing from Dobruisk to Bodruisk.) Having met Doby, the fly turned back again, etc. Determine the total path that the fly flew until the moment the cyclists met. (The speed of the fly in each direction did not change.)

Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.

Let $C$ denote the family of continuous functions on the real axis. Let $T$ be a mapping of $C$ into $C$ which has the following properties: 1. $T$ is linear, i.e. $T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2$ for $c_1$ and $c_2$ real and $\psi_1$ and $\psi_2$ in $C$. 2. $T$ is local, i.e. if $\psi_1 \equiv \psi_2$ in some interval $I$ then also $T\psi_1 \equiv T\psi_2$ holds in $I$. Show that $T$ must necessarily be of the form $T\psi(x)=f(x)\psi(x)$ where $f(x)$ is a suitable continuous function.

Let $a$ and $b$ be relatively prime positive integers, $b$ even. For each positive integer $q$, let $p=p(q)$ be chosen so that $$ \left| \frac{p}{q} - \frac{a}{b} \right|$$ is a minimum. Prove that $$ \lim_{n \to \infty} \sum_{q=1 }^{n} \frac{ q\left| \frac{p}{q} - \frac{a}{b} \right|}{n} = \frac{1}{4}.$$

Let $ABC$ be an acute triangle with $AB<AC$. Point $M{}$ from the side $(BC)$ is the foot of the bisector from the vertex $A{}$. The perpendicular bisector of the segment $[AM]$ intersects the side $(AC)$ in $E{}$, the side $(AB)$ in $D$ and the line $(BC)$ in $F{}$. Prove that $\frac{DB}{CE}=\frac{FB}{FC}=\left(\frac{AB}{AC}\right)^2$.

Let $ C_1$ and $ C_2$ be externally tangent circles with radius 2 and 3, respectively. Let $ C_3$ be a circle internally tangent to both $ C_1$ and $ C_2$ at points $ A$ and $ B$, respectively. The tangents to $ C_3$ at $ A$ and $ B$ meet at $ T$, and $ TA \equal{} 4$. Determine the radius of $ C_3$.

On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $ R$ be the region formed by the union of the square and all the triangles, and $ S$ be the smallest convex polygon that contains $ R$. What is the area of the region that is inside $ S$ but outside $ R$? $ \textbf{(A)} \; \frac{1}{4} \qquad \textbf{(B)} \; \frac{\sqrt{2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt{3} \qquad \textbf{(E)} \; 2 \sqrt{3}$

Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called [i]successful[/i] if each number is the product of its neighbors. Find the number of successful arrangements.

A [i]diameter[/i] of a finite planar set is any line segment of maximal Euclidean length having both end points in that set. A [i]lattice point[/i] in the Cartesian plane is one whose coordinates are both integral. Given an integer $n\geq 2$, prove that a set of $n$ lattice points in the plane has at most $n-1$ diameters.

Two equal circles in the same plane cannot have the following number of common tangents: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{none of these}$

[b]p1.[/b] Twelve tables are set up in a row for a Millenium party. You want to put $2000$ cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one (for example, if the $5$th table has $20$ cupcakes, then the $4$th table has either $19$ or $21$ cupcakes, and the $6$th table has either $19$ or $21$ cupcakes). a) Find a way to do this. b) Suppose a Y2K bug eats one of the cupcakes, so you have only $1999$ cupcakes. Show that it is impossible to arrange the cupcakes on the tables according to the above conditions. [b]p2.[/b] Let $P$ and $Q$ lie on the hypotenuse $AB$ of the right triangle $CAB$ so that $|AP|=|PQ|=|QB|=|AB|/3$. Suppose that $|CP|^2+|CQ|^2=5$. Prove that $|AB|$ has the same value for all such triangles, and find that value. Note: $|XY|$ denotes the length of the segment $XY$. [b]p3.[/b] Let $P$ be a polynomial with integer coefficients and let $a, b, c$ be integers. Suppose $P(a)=b$, $P(b)=c$, and $P(c)=a$. Prove that $a=b=c$. [b]p4.[/b] A lattice point is a point $(x,y)$ in the plane for which both $x$ and $y$ are integers. Each lattice point is painted with one of $1999$ available colors. Prove that there is a rectangle (of nonzero height and width) whose corners are lattice points of the same color. [b]p5.[/b] A $1999$-by-$1999$ chocolate bar has vertical and horizontal grooves which divide it into $1999^2$ one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; and then either putting both rectangles down or eating one piece and putting the other piece down. The players move alternately. The one who cannot make a move at his turn (because there are only one-by-one squares left) loses. Caesar starts. Which player has a winning strategy? Describe a winning strategy for that player. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

Point $P$ lies inside square $ABCD$ such that the areas of $\triangle{PAB}, \triangle{PBC}, \triangle{PCD},$ and $\triangle{PDA}$ are $1, 2, 3,$ and $4,$ in some order. Compute $PA \cdot PB \cdot PC \cdot PD.$

We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$. For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$. Show that an odd perfect number has at least $3$ distinct prime divisors. [i]Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.[/i]

Given a natural number $n,$ find all polynomials $P(x)$ of degree less than $n$ satisfying the following condition \[ \sum^n_{i=0} P(i) \cdot (-1)^i \cdot \binom{n}{i} = 0. \]

Triangle $ABC$ is right-angled at $C$. $AQ$ is drawn parallel to $BC$ with $Q$ and $B$ on opposite sides of $AC$ so that when $BQ$ is drawn, intersecting $AC$ at $P$, we have $PQ = 2AB$. Prove that $\angle ABC = 3\angle PBC$.

For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.

In triangle $ABC$, the point $D$ lies on segment $AB$ such that $CD$ is the angle bisector of angle $\angle C$. The perpendicular bisector of segment $CD$ intersects the line $AB$ in $E$. Suppose that $|BE| = 4$ and $|AB| = 5$. (a) Prove that $\angle BAC = \angle BCE$. (b) Prove that $2|AD| = |ED|$. [asy] unitsize(1 cm); pair A, B, C, D, E; A = (0,0); B = (2,0); C = (1.8,1.8); D = extension(C, incenter(A,B,C), A, B); E = extension((C + D)/2, (C + D)/2 + rotate(90)*(C - D), A, B); draw((E + (0.5,0))--A--C--B); draw(C--D); draw(interp((C + D)/2,E,-0.3)--interp((C + D)/2,E,1.2)); dot("$A$", A, SW); dot("$B$", B, S); dot("$C$", C, N); dot("$D$", D, S); dot("$E$", E, S); [/asy]

There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds \[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]

Find all triples $(a, b, c)$ of integers such that $a+ b + c = 2010 \cdot 2011 $ and the solutions to the equation $$2011x^3 +ax^2 +bx+c = 0$$ are all nonzero integers.

Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying: (a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer. (b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$