A variable tetrahedron $ABCD$ has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?

Show that for any real number $x$: \[ x^2 \sin{x} + x \cos{x} + x^2 + \frac{1}{2} > 0 . \]

Define $a_k=2^{2^{k-2013}}+k$ for all integers $k$. Simplify $$(a_0+a_1)(a_1-a_0)(a_2-a_1)\cdots(a_{2013}-a_{2012}).$$

Let $r \geq 2023$ be a rational number. The real numbers $a, b$, and $c$ satisfy \[ 4a^2 + 4b^2 + 9c^2 = r. \] Does there exist a value of $r$ for which the number of rational triples $(a,b,c)$ that achieve the maximum possible value of $4ab + 6bc - 6ac$ is: a) zero b) finite, but non-zero?

In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$ Note: $(XYZ)$ is the area of triangle $XYZ$.

Find all pairs of integers $(x,y)$ such that $x(x+2)=y^2(y^2+1)$

Suppose $p$ is a prime with $p \equiv 3 \; \pmod{4}$. Show that for any set of $p-1$ consecutive integers, the set cannot be divided two subsets so that the product of the members of the one set is equal to the product of the members of the other set.

Let $AA_1$ and $BB_1$ be the altitudes of acute angled triangle $ABC$. Points $K$ and $M$ are midpoints of $AB$ and $A_1B_1$ respectively. Segments $AA_1$ and $KM$ intersect at point $L$. Prove that points $A$, $K$, $L$ and $B_1$ are noncyclic. [I]Proposed by S. Berlov[/i]

The first $360$ natural numbers are separated into $9$ blocks in such a way that the numbers in each block are consecutive. Then, the numbers in each block are added, obtaining $9$ numbers. Is it possible to fill a $3 \times 3$ grid and form a [i]magic square[/i] with these numbers? Note: In a magic square, the sum of the numbers written in any column, diagonal or row of the grid is the same.

For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then $$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.

Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?

If the sum of the real roots $x$ to each of the equations \[2^{2x}-2^{x+1}+1-\frac{1}{k^2}=0\] for $k=2,3,\dots,2023$ is $N$, what is $2^N$?

The length of the interval of solutions of the inequality $ a\le 2x\plus{}3\le b$ is $ 10$. What is $ b\minus{}a$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 30$

A function $f : [0,1] \to R$ has the following properties: (a) $f(x) \ge 0$ for $0 < x < 1$, (b) $f(1) = 1$, (c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$. Prove that $f(x) \le 2x$ for all $x \in [0,1]$.

Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard such that they do NOT attack each other?

a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely $10$ high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true. b) At a different occasion $13$ kayakers rowed in the same manner; now after arrival everybody claims to have done precisely$ 6$ high fives. Prove that at least one kayaker has miscounted.

Two squares are said to be [i]juxtaposed [/i] if their intersection is a point or a segment. Prove that it is impossible to [i]juxtapose [/i] to a square more than eight non-overlapping squares of the same size.

Denote the dot product of two sequences $\langle x_1,\dots,x_n\rangle$ and $\langle y_1,\dots,y_n\rangle$ to be $$x_1y_1+x_2y_2+\dots+x_ny_n.$$ Let $\langle a_1,\dots,a_n\rangle$ and $\langle b_1,\dots,b_n\rangle$ be two sequences of consecutive integers (i.e. for $1\leq i,i+1\leq n$, $a_i+1=a_{i+1}$ and similarly for $b_i$). Minnie permutes the two sequences so that their dot product, $m$, is minimized. Maximilian permutes the two sequences so that their dot product, $M$, is maximized. Given that $m=4410$ and $M=4865$, compute $n$, the number of terms in each sequence.

Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$

Let be given two positive integers $ m$, $ n$ with $ m < 2001$, $ n < 2002$. Let distinct real numbers be written in the cells of a $ 2001 \times 2002$ board (with $ 2001$ rows and $ 2002$ columns). A cell of the board is called [i]bad[/i] if the corresponding number is smaller than at least $ m$ numbers in the same column and at least $ n$ numbers in the same row. Let $ s$ denote the total number of [i]bad[/i] cells. Find the least possible value of $ s$.

From an urn containing one ball marked with the number 1, two balls marked with the number 2, ..., $ n $ balls marked with the number $ n $ we draw two balls without replacement. We assume that each ball is equally likely to be drawn from the urn. Calculate the probability that both balls drawn have the same number.

$n\geq3$ boxes are placed around a circle. At the first step we choose some boxes. At the second step for each chosen box we put a ball into the chosen box and into each of its two neighbouring boxes. Find the total number of possible distinct ball distributions which can be obtained in this way. (All balls are identical.)

Consider the set of $5$-tuples of positive integers at most $5$. We say the tuple ($a_1$, $a_2$, $a_3$, $a_4$, $a_5$) is [i]perfect[/i] if for any distinct indices $i$, $j$, $k$, the three numbers $a_i$, $a_j$ , $a_k$ do not form an arithmetic progression (in any order). Find the number of perfect $5$-tuples.

An integer $n\geq 2$ having exactly $s$ positive divisors $1=d_1<d_2<\cdots<d_s=n$ is said to be [i]good[/i] if there exists an integer $k$, with $2\leq k\leq s$, such that $d_k>1+d_1+\cdots+d_{k-1}$. An integer $n\geq 2$ is said to be [i]bad[/i] if it is not good. (a) Show that there are infinitely many bad integers. (b) Prove that, among any seven consecutive integers all greater than $2$, there are always at least four good integers. (c) Show that there are infinitely many sequences of seven consecutive good integers.

On a circular board there are $19$ squares numbered in order from $1$ to $19$ (to the right of $1$ is $2$, to the right of it is $3$, and so on, until $1$ is to the right of $19$). In each box there is a token. Every minute each checker moves to its right the number of the box it is in at that moment plus one; for example, the piece that is in the $7$th place leaves the first minute $7 + 1$ places to its right until the $15$th square; the second minute that same checker moves to your right $15 + 1$ places, to square $12$, etc. Determine if at some point all the tokens reach the place where they started and, if so, say how many minutes must elapse. [hide=original wording]En un tablero circular hay 19 casillas numeradas en orden del 1 al 19 (a la derecha del 1 está el 2, a la derecha de éste está el 3 y así sucesivamente, hasta el 1 que está a la derecha del 19). En cada casilla hay una ficha. Cada minuto cada ficha se mueve a su derecha el número de la casilla en que se encuentra en ese momento más una; por ejemplo, la ficha que está en el lugar 7 se va el primer minuto 7 + 1 lugares a su derecha hasta la casilla 15; el segundo minuto esa misma ficha se mueve a su derecha 15 + 1 lugares, hasta la casilla 12, etc. Determinar si en algún momento todas las fichas llegan al lugar donde empezaron y, si es así, decir cuántos minutos deben transcurrir.[/hide]