How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

All of the polygons in the figure below are regular. Prove that $ABCD$ is an isosceles trapezoid. [img]https://cdn.artofproblemsolving.com/attachments/e/a/3f4de32becf4a90bf0f0b002fb4d8e724e8844.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

What is the value of $2(2(2(2(2(2+1)+1)+1)+1)+1)+1$? $\textbf{(A) } 70 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 127 \qquad \textbf{(D) } 159 \qquad \textbf{(E) } 729 $

Call a subset of $\{1,2,\ldots, n\}$ [i]unfriendly[/i] if no two of its elements are consecutive. Show that the number of unfriendly subsets with $k$ elements is $\binom{n-k+1}{k}.$

On the side $AB$ of the triangle $ABC$ mark the points $M$ and $N$, such that $BM = BC$ and $AN = AC$. Then on the sides $BC$ and $AC$ mark the points$ P$ and $Q$, respectively, such that $BP = BN$ and $AQ = AM$. Prove that the points $C, Q, M, N$ and $P$ lie on the same circle.

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

You have the following pieces: one $4\times 1$ rectangle, two $3\times 1$ rectangles, three $2\times 1$ rectangles, and four $1\times 1$ squares. Ariel and Bernardo play the following game on a board of $n\times n$, where $n$ is a number that Ariel chooses. In each move, Bernardo receives a piece $R$ from Ariel. Next, Bernardo analyzes if he can place $R$ on the board so that it has no points in common with any of the previously placed pieces (not even a common vertex). If there is such a location for $R$, Bernardo must choose one of them and place $R$. The game stops if it is impossible to place $R$ in the way explained, and Bernardo wins. Ariel wins only if all $10$ pieces have been placed on the board. a) Suppose Ariel gives Bernardo the pieces in decreasing order of size. What is the smallest n that guarantees Ariel victory? b) For the $n$ found in a), if Bernardo receives the pieces in increasing order of size, is Ariel guaranteed victory? Note: Each piece must cover exactly a number of unit squares on the board equal to its own size. The sides of the pieces can coincide with parts of the edge of the board.

A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares.

Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.

Bob writes $36$ consecutive positive integers in a white paper(in ascending order), next he computes the sum of digits of each one of $36$ numbers(in the order) and writes the first $16$ results in a red paper and the last $10$ results in a blue paper. Determine if Bob can choose the $36$ integers, such that the sum of the numbers in the red paper is less than or equal to sum of the numbers in the blue paper.

In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesections are all of the same color. Find the remainder when $n$ is divided by $1000$. [i]Proposed by [b]FedeX333X[/b][/i]

Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.

Find the smallest positive real $t$ such that \[\left\{ \begin{array}{l} x_1 + x_3 = 2t x_2 \\ x_2 + x_4 = 2t x_3 \\ x_3 + x_5=2t x_4 \\ \end{array} \right. \] has a solution $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ in non-negative reals, not all zero.

In an acute triangle $ABC$, $\angle C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.