Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: select a pair of tokens at points $A$ and $B$ and move both of them to the midpoint of $A$ and $B$. We say that an arrangement of $n$ tokens is [i]collapsible[/i] if it is possible to end up with all $n$ tokens at the same point after a finite number of moves. Prove that every arrangement of $n$ tokens is collapsible if and only if $n$ is a power of $2$.

Let $a, b > 0$ be reals such that \[ a^3=a+1\\ b^6=b+3a \] Show that $a>b$

Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?

In a dance class there are $10$ boys and $10$ girls. It is known that for each $1\le k\le 10$ and for each group of $k$ boys, the number of girls who are friends with at least one boy in the group is not less than $k$. Prove that it is possible to pair up the boys and the girls for a dance so that each pair consists of a boy and a girl who are friends.

Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $CA=8$. Let $D$ be a point on $BC$, and define points $B'$ and $C'$ on line $AD$ (or its extension) such that $BB'\perp AD$ and $CC'\perp AD$. If $B'A=B'C'$, then the ratio $BD:DC$ can be expressed in the form $m:n$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?

Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?

Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$. Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$.

$p$ passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.

Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than $\frac12$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Compute \[\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}.\]

Prove that $ N\geq 2n \minus{} 2$ integers, of absolute value not higher than $ n > 2$, and of absolute value of their sum $ S$ less than $ n \minus{} 1,$ there exist some of sum $ 0.$ Show that for $ |S| \equal{} n \minus{} 1$ this is not anymore true, and neither for $ N \equal{} 2n \minus{} 3$ (when even for $ |S| \equal{} 1$ this is not anymore true).

Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.

Three circles $S_A, S_B$, and $S_C$ in the plane with centers in $A, B$, and $C$, respectively, are mutually tangential on the outside. The touchpoint between $S_A$ and $S_B$ we call $C'$, the one $S_A$ between $S_C$ we call $B'$, and the one between $S_B$ and $S_C$ we call $A'$. The common tangent between $S_A$ and $S_C$ (passing through B') we call $\ell_B$, and the common tangent between $S_B$ and $S_C$ (passing through $A'$) we call $\ell_A$. The intersection point of $\ell_A$ and $\ell_B$ is called $X$. The point $Y$ is located so that $\angle XBY$ and $\angle YAX$ are both right angles. Show that the points $X, Y$, and $C'$ lie on a line if and only if $AC = BC$.

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.

Different points $A,B,C,D,E,F$ lie on circle $k$ in this order. The tangents to $k$ in the points $A$ and $D$ and the lines $BF$ and $CE$ have a common point $P$. Prove that the lines $AD,BC$ and $EF$ also have a common point or are parallel.

At a party, $25$ elves give each other presents. No elf gives a present to herself. Each elf gives a present to at least one other elf, but no elf gives a present to all other elves. Show that it is possible to choose a group of three elves including at least two elves who give a present to exactly one of the other two elves in the group.

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Show that, in a chessboard, it is possible to traverse to any given square from another given square using a knight. (A knight can move in a chessboard by going two steps in one direction and one step in a perpendicular direction as shown in the given figure)

Prove that, $$15x^2-7y^2=9$$ equation has any solutions in integers.

Let $n$ be a positive integer. Ana and Beto play a game on a $2 \times n$ board (with 2 rows and $n$ columns). First, Ana writes a digit from 1 to 9 in each cell of the board such that in each column the two written digits are different. Then, Beto erases a digit from each column. Reading from left to right, a number with $n$ digits is formed. Beto wins if this number is a multiple of $n$; otherwise, Ana wins. Determine which of the two players has a winning strategy in the following cases: $\bullet$ (a) $n = 1001$. $\bullet$ (b) $n = 1003$.

Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$, $$f(xy+f(x))=xf(y)+2.$$

Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]