[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

Prove that $\sigma(n)\phi(n) < n^2$, but that there is a positive constant $c$ such that $\sigma(n)\phi(n) \ge c n^2$ holds for all positive integers $n$.

Find the area of the shaded region. [i]Lightning 1.4[/i]

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that (i) $a_n>0, \; \forall n \in \mathbb{N}$. (ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$. Prove that $|p-1| \ge 2$.

Find the number of maps $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$ such that $f(i) \le f(j)$ whenever $i < j$.

In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$. (Not all towns need to be visited.)

Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\] is at most $\binom{2023}{2}$ times divisible by $p$. [i]Proposed by Ashwin Sah[/i]

The maximum value of $k$ such that the enequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution is $\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6$

A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and [i]still[/i] have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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]