A square of side length $1$ centimeter is cut into three convex polygons. Is it possible that the diameter of each of them does not exceed [list][b]a)[/b] $1$ centimeter; [b]b)[/b] $1.01$ centimeters; [b]c)[/b] $1.001$ centimeters?[/list]

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]

Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$. Prove that the product $a\cdot b$ is always divisible by $14$.

An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates $(0, 0), (p, 0), (p, q), (0, q)$ for some positive integers $p, q$. Show that there must exist two among them one of which is entirely contained in the other.

Let $ ABCD $ be a tetrahedron with $ \angle ABC = \angle ABD = \angle CBD = 90^\circ $ and $ AB = BC $. Let $ E, F, G $ be points on $ \overline{AD} $, $ BD $, and $ \overline{CD} $, respectively, such that each of the quadrilaterals $ AEFB $, $ BFGC $, and $ CGEA $ have an inscribed circle. Let $ r $ be the smallest real number such that $ \dfrac{[\triangle EFG]}{[\triangle ABC]} \leq r $ for all such configurations $ A, B, C, D, E, F, G $. If $ r $ can be expressed as $ \dfrac{\sqrt{a - b\sqrt{c}}}{d} $ where $ a, b, c, d $ are positive integers with $ \gcd(a, b) $ squarefree and $ c $ squarefree, find $ a + b + c + d $. Note: Here, $ [P] $ denotes the area of polygon $ P $. (This wasn't in the original test; instead they used the notation $ \text{area}(P) $, which is clear but frankly cumbersome. :P)

Suppose the numbers $a_0, a_1, a_2, ... , a_n$ satisfy the following conditions: $a_0 =\frac12$, $a_{k+1} = a_k +\frac{1}{n}a_k^2$ for $k = 0, 1, ... , n - 1$. Prove that $1 - \frac{1}{n}< a_n < 1$

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{C} \right) $ satisfying $ AB-BA=A. $ Show that: [b]a)[/b] $ \text{tr} (A) =\det (A) =0 $ [b]b)[/b] $ AB^nA=0, $ for any natural number $ n $

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.

Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days? ${ \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area $a$, the four vertices opposite to them form a quadrilateral of area $b$. Find the area of the parallelogram.

The plot of the $y=f(x)$ function, being rotated by the (right) angle around the $(0,0)$ point is not changed. a) Prove that the equation $f(x)=x$ has the unique solution. b) Give an example of such a function.

Consider a checkered board $2m \times 2n$, $m, n \in \mathbb{Z}_{>0}$. A stone is placed on one of the unit squares on the board, this square is different from the upper right square and from the lower left square. A snail goes from the bottom left square and wants to get to the top right square, walking from one square to other adjacent, one square at a time (two squares are adjacent if they share an edge). Determine all the squares the stone can be in so that the snail can complete its path by visiting each square exactly one time, except the square with the stone, which the snail does not visit.

Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

Andy and Harry are trying to make an O for the MOAA logo. Andy starts with a circular piece of leather with radius 3 feet and cuts out a circle with radius 2 feet from the middle. Harry starts with a square piece of leather with side length 3 feet and cuts out a square with side length 2 feet from the middle. In square feet, what is the positive difference in area between Andy and Harry's final product to the nearest integer? [i]Proposed by Andy Xu[/i]

Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying \[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \] for all $n \ge 2023$ then the sum \[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \] is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.

A Pythagorean triangle is any right-angled triangle for which the lengths of two legs and the length of the hypotenuse are integers. We are observing all Pythagorean triangles in which may be inscribed a quadrangle with sidelength integer number, two of which sides lie on the cathets and one of the vertices of which lies on the hypotenuse of the triangle given. Find the side lengths of the triangle with minimal surface from the observed triangles. [i]St. Doduneko[/i]

Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying (i) The side lengths of $T $ are consecutive integers. (ii) The area of $T$ is an integer.

On the horizontal line from left to right are the points $P, \, \, Q, \, \, R, \, \, S$. Construct a square $ABCD$, for which on the line $AD$ lies lies the point $P$, on the line $BC$ lies the point $Q$, on the line $AB$ lies the point $R$, on the line $CD$ lies the point $S $.

We consider tilings of a rectangular $m \times n$-board with $1\times2$-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile. (a) Prove that for every tiling of a $4 \times 2010$-board with $1\times2$-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces. (b) Prove that there exists a tiling of a $5 \times  2010$-board with $1\times 2$-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

Find all positive integer $k$s for which such $f$ exists and unique: $f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$ $f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)

Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations \begin{align*} \sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\ \sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\ \sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}. \end{align*}Then $\big[ (1-x)(1-y)(1-z) \big] ^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.