Describe a construction of quadrilateral $ABCD$ given: (i) the lengths of all four sides; (ii) that $AB$ and $CD$ are parallel; (iii) that $BC$ and $DA$ do not intersect.

Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.

Let $ A $ be a ring that is not a division ring, and such that any non-unit of it is idempotent. Show that: [b]a)[/b] $ \left( U(A) +A\setminus\left( U(A)\cup \{ 0\} \right) \right)\cap U(A) =\emptyset . $ [b]b)[/b] Every element of $ A $ is idempotent.

The diagram below shows an equilateral triangle and a square of side length $2$ joined along an edge. What is the area of the shaded triangle? [asy] fill((2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- cycle, gray); draw((0,0) -- (2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- (0,2) -- (0,0)); draw((0,2) -- (2,2)); [/asy]

A number is called [i]nillless [/i] if it is integer and positive and contains no zeros. You can make a positive integer nillless by simply omitting the zeros. We denote this with square brackets, for example $[2050] = 25$ and $[13] = 13$. When we multiply, add, and subtract we indicate with square brackets when we omit the zeros. For example, $[[4 \cdot 5] + 7] = [[20] + 7] = [2 + 7] = [9] = 9$ and $[[5 + 5] + 9] = [[10] + 9] = [1 + 9] = [10] = 1$. The following is known about the two numbers $a$ and $b$: $\bullet$ $a$ and $b$ are nillless, $\bullet$ $1 < a < b < 100$, $\bullet$ $[[a \cdot b] - 1] = 1$. Which pairs $(a, b)$ satisfy these three requirements?

A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field? [asy] draw((0,0)--(100,0)--(100,50)--(0,50)--cycle); draw((50,0)--(50,50)); draw((0,25)--(100,25)); [/asy] $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144$

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for any real numbers $x$ and $y$ the following is true: $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$

Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$. What is the value of $p - 3q$?

Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$, we have $xy\notin A$. Determine the maximum possible size of $A$.

In a convex hexagon the value of each angle is $120^{\circ}$. The perimeter of the hexagon equals $2$. Prove that this hexagon can be covered by a triangle with perimeter at most $3$.

$a,b \in \mathbb Z$ and for every $n \in \mathbb{N}_0$, the number $2^na+b$ is a perfect square. Prove that $a=0$.

Determine all real numbers $\alpha$ such that, for every positive integer $n,$ the integer $$\lfloor\alpha\rfloor +\lfloor 2\alpha\rfloor +\cdots +\lfloor n\alpha\rfloor$$ is a multiple of $n.$ (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$ For example, $\lfloor -\pi\rfloor =-4$ and $\lfloor 2\rfloor= \lfloor 2.9\rfloor =2.$) [i]Proposed by Santiago Rodríguez, Colombia[/i]

Prove that if the real numbers $ a, b, c, d $ satisfy the equations $$ \; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},$$ then $$a^2 + ac + c^2 = b^2 + bd + d^2.$$

For positive real numbers$a,b,c$such that $ab+ac+bc=1$ prove that: $\prod\limits_{cyc} (\sqrt{bc}+\frac{1}{2a+\sqrt{bc}}) \ge 8abc$

Let $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$. Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$. Does the conclusion of the problem remain true if the sets $X_n$ are infinite? [i]Claudiu Raicu[/i]

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

Find all pairs $(x,y)$ of nonnegative integers that satisfy \[x^3y+x+y=xy+2xy^2.\]

Let $S$ be a set containing $n^{2}+n-1$ elements, for some positive integer $n$. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.

Real nonzero numbers $x, y, z$ are such that $x+y +z = 0$. Moreover, it is known that $$A =\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+ 1$$Determine $A$.

The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same. Show that we can get eight vertices with the same sum if one of the labels is changed to 13.

Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that: (a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$. (b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.

In a circle with center at $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$ a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is: $\textbf{(A) }\dfrac{3r^2}{16}\qquad \textbf{(B) }\dfrac{\pi r^2}{16}\qquad \textbf{(C) }\dfrac{\pi r^2\sqrt2}{8}\qquad \textbf{(D) }\dfrac{r^2\sqrt3}{32}\qquad \textbf{(E) }\dfrac{r^2\sqrt6}{48}$

$ABCD$ is a square and $AB = 1$. Equilateral triangles $AYB$ and $CXD$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $XY$?

Let $[AD]$ and $[CE]$ be internal angle bisectors of $\triangle ABC$ such that $D$ is on $[BC]$ and $E$ is on $[AB]$. Let $K$ and $M$ be the feet of perpendiculars from $B$ to the lines $AD$ and $CE$, respectively. If $|BK|=|BM|$, show that $\triangle ABC$ is isosceles.

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.