Let $n \geq 2$ be a positive integer and let $A, B$ be two finite sets of integers such that $|A| \leq n$. Let $C$ be a subset of the set $\{(a, b) | a \in A, b \in B\}$. Achilles writes on a board all possible distinct differences $a-b$ for $(a, b) \in C$ and suppose that their count is $d$. He writes on another board all triplets $(k, l, m)$, where $(k, l), (k, m) \in C$ and suppose that their count is $p$. Show that $np \geq d^2.$

Of the following (1) $ a(x\minus{}y)\equal{}ax\minus{}ay$ (2) $ a^{x\minus{}y}\equal{}a^x\minus{}a^y$ (3) $ \log (x\minus{}y)\equal{}\log x\minus{}\log y$ (4) $ \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y}$ (5) $ a(xy)\equal{}ax\times ay$ $\textbf{(A)}\ \text{Only 1 and 4 are true} \qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true} \qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true} \qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true} \qquad\\ \textbf{(E)}\ \text{Only 1 is true}$

There are 100 persons in a hall, everyone knowing at least 66 of the others. Prove that there is a case in which among any four some two don’t know each other.

There exist two distinct positive integers, both of which are divisors of $10^{10}$, with sum equal to $157$. What are they?

Triangle $ABC$ is isosceles with $AB=AC$. The bisectors of angles $ABC$ and $ACB$ meet at $I$. If the measure of angle $CIA$ is $130^\circ$, compute the measure of angle $CAB$. [i]Proposed by Connor Gordon[/i]

Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at his or her own roll, and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma and Fred each state the expected sum of the dice rolls based on the information they have. All six people say different numbers; in particular, Alice, Bob, Charlie, and Diana say $19$, $22$, $21$, and $23$, respectively. Compute the product of the dice rolls.

Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways? Notes: 1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube. 2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.

Anis, Banu, and Cholis are going to play a game. They are given an $n\times n$ board consisting of $n^2$ unit squares, where $n$ is an integer and $n > 5$. In the beginning of the game, the number $n$ is written on each unit square. Then Anis, Banu, and Cholis take turns playing the game, repeatedly in that order, according to the following procedure: On every turn, an arrangement of $n$ squares on the same row or column is chosen, and every number from the chosen squares is subtracted by $1$. The turn cannot be done if it results in a negative number, that is, no arrangement of $n$ unit squares on the same column or row in which all of its unit squares contain a positive number can be found. The last person to get a turn wins. Determine which player will win the game.

If the numbers $2a + 2$ and $2b + 2$ add up to $2004$, find the sum of the numbers $\frac{a}{2} - 2$ and $\frac{b}{2} - 2$

In the isosceles triangle $ABC$ ($AB=AC$), let $l$ be a line parallel to $BC$ through $A$. Let $D$ be an arbitrary point on $l$. Let $E,F$ be the feet of perpendiculars through $A$ to $BD,CD$ respectively. Suppose that $P,Q$ are the images of $E,F$ on $l$. Prove that $AP+AQ\le AB$ [i]Proposed by Morteza Saghafian[/i]

Let $ABC$ be a triangle, $H$ the orthocenter and $M$ the midpoint of $AC$. Let $\ell$ be the parallel through $M$ to the bisector of $\angle AHC$. Prove that $\ell$ divides the triangle in two parts of equal perimeters. [i]Pedro Marrone, Panamá[/i]

Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds: $$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$

For an integer $n \ge 0$, let $f(n)$ be the smallest possible value of $ |x + y|$, where $x$ and $y$ are integers such that $3x - 2y = n$. Evaluate $f(0) + f(1) + f(2) +...+ f(2013)$.

Find all positive integers $k<202$ for which there exist a positive integers $n$ such that $$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

A point $M (x, y)$ of the Cartesian plane of $xOy$ coordinates is called [i]lattice [/i] if it has integer coordinates. Each lattice point is colored red or blue. Prove that in the plan there is at least one rectangle with lattice vertices of the same color.

Consider a right cylinder with height $5\sqrt3$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the dierence of the lengths of the major and minor axes of this ellipse. [i]Note:[/i] An ellipse is a regular oval shape resulting when a cone is cut by an oblique plane which does not intersect the base. The major axis is the longer diameter and the minor axis the shorter.

It is given triangle $ABC$ and parallelogram $ASCR$ with diagonal $AC$. Let line constructed through point $B$ parallel with $CS$ intersects line $AS$ and $CR$ in $M$ and $P$, respectively. Let line constructed through point $B$ parallel with $AS$ intersects line $AR$ and $CS$ in $N$ and $Q$, respectively. Prove that lines $RS$, $MN$ and $PQ$ are concurrent

A positive integer $n{}$ is called [i]discontinuous[/i] if for all its natural divisors $1 = d_0 < d_1 <\cdots<d_k$, written out in ascending order, there exists $1 \leqslant i \leqslant k$ such that $d_i > d_{i-1}+\cdots+d_1+d_0+1$. Prove that there are infinitely many positive integers $n{}$ such that $n,n+1,\ldots,n+2019$ are all discontinuous.

Given a triangle $ABC$, $D$ is the reflection from the perpendicular foot from $A$ to $BC$ through the midpoint of $BC$. $E$ is the reflection from the perpendicular foot from $B$ to $CA$ through the midpoint of $CA$. $F$ is the reflection from the perpendicular foot from $C$ to $AB$ through the midpoint of $AB$. Prove: $DE \perp AC$ if and only if $DF \perp AB$

Given two parallelograms $ABCD$ and $AECF$ with common diagonal $AC$, where $E$ and $F$ lie inside parallelogram $ABCD$. Show: The circumcircles of the triangles $AEB$, $BFC$, $CED$ and $DFA$ have one point in common.

Alice has $n > 1$ one variable quadratic polynomials written on paper she keeps secret from Bob. On each move, Bob announces a real number and Alice tells him the value of one of her polynomials at this number. Prove that there exists a constant $C > 0$ such that after $Cn^5$ questions, Bob can determine one of Alice’s polynomials. [i]Proposed by Rohan Goyal and Anant Mudgal[/i]

Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.

Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$