Once upon a time there are $n$ pairs of princes and princesses who are in love with each other. One day a witch comes along and turns all the princes into frogs; the frogs can be distinguished by sight but the princesses cannot tell which frog corresponds to which prince. The witch tells the princesses that if any of them kisses the frog that corresponds to the prince very that she loves then that frog will immediately transform back into a prince. If each princess can stand kissing at most $k$ frogs, what is the maximum number of princes they can be sure to save? (The princesses may take turns kissing in any order, communicate with each other and vary their strategy for future kisses depending on information gained from past kisses.)

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

Find the greatest prime number $p$ for which there exists a prime number $q$ such that $p$ divides $4^q + 1$ and $q$ divides $4^p + 1$.

Ababa plays with a word made up of the letters of his name and has set certain rules: If you find an $A$ followed immediately by a $B$, you can substitute $BAA$ for them. If you find two consecutive $B$'s, you can delete them. If you find three consecutive $A$'s, you can delete them. Ababa begins with the word $ABABABAABAAB$. With the above rules, how many letters do you have the shortest word you can come up with? Why can't you come up with one more word shorter?

Ana writes a three-digit code, and Beto has to guess it. To do so, he can ask about a sequence of three digits, and Ana will respond "warm" if the sequence Beto proposes has at least one correct digit in the correct position, and she will respond "cold" if none of the digits are correct. For example, if the correct code is $014$, then if Beto asks $099$ or $014$, he receives the answer "warm", and if he asks $140$ or $322$, he receives the answer "cold". Determine the minimum number of questions Beto needs to ask in order to know the correct code with certainty.

In acute triangle $ABC$ ($AB \neq AC$), $I$ is its incenter and $J$ is the $A$-excenter. $X, Y$ are on minor arcs $\widehat{AB}$ and $\widehat{AC}$ respectively such that $\angle{AXI}=\angle{AYJ}=90^{\circ}$. $K$ is on line $BC$ such that $KI=KJ$. Proof that line $AK$ bisects $\overline{XY}$.

Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

Find a matrix $A_{(2 \times 2)}$ for which \[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{R}$ such that $f(1)=\tfrac{5}{2}$ and that \[f(x)f(y)=f(x+y)+f(x-y)\] for all integers $x$ and $y$.

Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$. Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot n^{\frac 1 4}$ holds for all $n > 1.$ Pierre.

Let $ f$ be a linear function for which $ f(6)\minus{}f(2)\equal{}12$. What is $ f(12)\minus{}f(2)$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 36$

Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins. Determine who has a winning strategy depending on $n, k$. [i](Proposed by Fedir Yudin)[/i]

Calculate the minimum value of $ \text{tr} (A^tA) , $ where $ A $ in the cases where is a matrix of pairwise distinct nonnegative integers and: [b]a)[/b] $ \det A\equiv 1\pmod 2 $ [b]b)[/b] $ \det A=0 $ [i]Vlad Mihaly[/i]

In an $8 \times 8$ grid of squares, each square was colored black or white so that no $2\times 2$ square has all its squares in the same color. A sequence of distinct squares $x_1,\dots, x_m$ is called a [b]snake of length $m$[/b] if for each $1\leq i <m$ the squares $x_i, x_{i+1}$ are adjacent and are of different colors. What is the maximum $m$ for which there must exist a snake of length $m$?

Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.

There are $N{}$ mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) $N = 2{}$ and b) $N = 10$? [i]Alexey Zaslavsky[/i]

Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.) [list] [*] $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$ [*] $|A| = |B| \geq 1$ [*] Sum of elements in $A$ and sum of elements in $B$ are equal. [/list]

For how many primes $p$ less than $15$, there exists integer triples $(m,n,k)$ such that \[ \begin{array}{rcl} m+n+k &\equiv& 0 \pmod p \\ mn+mk+nk &\equiv& 1 \pmod p \\ mnk &\equiv& 2 \pmod p. \end{array} \] $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

For each positive integer $n$, let $f_n$ be a real-valued symmetric function of $n$ real variables. Suppose that for all $n$ and all real numbers $x_1,\ldots,x_n, x_{n+1},y$ it is true that $\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,$ $\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),$ $\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).$ Prove that $f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.$

Let $\mathcal{S}$ be the set of all nonconstant polynomials $P$ with integer coefficients satisfying $P(\sqrt{3}+\sqrt{2})=P(\sqrt{3}-\sqrt{2}).$ If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0).$

Find all numbers $x$ such that: $1+2\cdot2^x+3\cdot3^x<6^x$

Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board. Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers.