The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that $\textbf{(A) }\textstyle -2<x<1\qquad\textbf{(B) }-\frac{4}{3}<x<\frac{1}{2}\qquad\textbf{(C) }-\frac{1}{2}<x<\frac{4}{3}\qquad$ $\textbf{(D) }x<\textstyle\frac{1}{2}\text{ or }x>-\frac{4}{3}\qquad\textbf{(E) }x<-\frac{4}{3}\text{ or }x>\frac{1}{2}$

Given 2n+3 points in the plane, no three on a line and no four on a circle, prove that it is always possible to find a circle C that goes through three of the given points and splits the other 2n in half, that is, has n on the inside and n on the outside.

On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

There are three stacks of tokens on the table: the first contains $a,$ the second contains $b,$ and the third contains $c$ where $a \ge b \ge c.$ Players $A$ and $B$ take turns playing a game of swapping tokens. $A$ starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of $a, b, c$ who has the winning strategy ($A$ or $B$)?

On the table there are $20$ coins of weights $1,2,3,\ldots,15,37,38,39,40$ and $41$ grams. They all look alike but their colours are all distinct. Now Miss Adams knows the weight and colour of each coin, but Mr. Bean knows only the weights of the coins. There is also a balance on the table, and each comparison of weights of two groups of coins is called an operation. Miss Adams wants to tell Mr. Bean which coin is the $1$ gram coin by performing some operations. What is the minimum number of operations she needs to perform?

A [i]mirrored polynomial[/i] is a polynomial $f$ of degree $100$ with real coefficients such that the $x^{50}$ coefficient of $f$ is $1$, and $f(x) = x^{100} f(1/x)$ holds for all real nonzero $x$. Find the smallest real constant $C$ such that any mirrored polynomial $f$ satisfying $f(1) \ge C$ has a complex root $z$ obeying $|z| = 1$.

An infinite number of squares on an infinitely square grid paper are painted red. Show that you can draw a number of squares on the paper, with sides along the grid lines, such that: (1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to more than one square) (2) each red square is located in one of the squares and the number of red squares in such square is at least $1/5$ and at most $4/5$ of the number of squares in the square. [hide=original wording] Ett andligt antal rutor pa ett oandligt rutat papper ar malade roda. Visa att man pa papperet kan rita in ett antal kvadrater, med sidor utefter rutnatets linjer, sadana att : (1) ingen ruta i natet tillhor mer an en kvadrat (en kant kan daremot tillhora mer an en kvadrat), (2) varje rod ruta ligger i nagon av kvadraterna och antalet roda rutor i en sadan kvadratar minst 1/5 och hogst 4/5 av antalet rutor i kvadraten. [url=http://www.mattetavling.se/wp-content/uploads/2011/01/Final10.pdf]source[/url][/hide] PS. I always post the original wording when I doubt about my (using Google) translation.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Determine the largest $3$ digit prime number that is a factor of ${2000 \choose 1000}$.

An integer $n \geq 3$ is [i]fabulous[/i] when there exists an integer $a$ with $2 \leq a \leq n - 1$ for which $a^n - a$ is divisible by $n$. Find all the [i]fabulous[/i] integers.

Show that the system of equations \begin{align*} \lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\ 3x+y &=2, \end{align*} has infinitely many solutions and all these solutions satisfy bounds \begin{align*} 0<\ &x <4, \\ -9\le\ &y\le 1. \end{align*}

Let $n$ be a positive integer. Find the value of the following sum \[\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}},\] where $e_k\in\{0,1\}$ for $1\leq k \leq n$, and the sum $\sum_{(n)}$ is taken over all $2^n$ possible choices of $e_1,\ldots ,e_n$.

Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$

Let $P(x) \in \mathbb {Z}[X]$ be a polynomial of degree $2016$ with no rational roots. Prove that there exists a polynomial $T(x) \in \mathbb {Z}[X]$ of degree $1395$ such that for all distinct (not necessarily real) roots of $P(x)$ like $(\alpha ,\beta):$ $$T(\alpha)-T(\beta) \not \in \mathbb {Q}$$ Note: $\mathbb {Q}$ is the set of rational numbers.

A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.

The square polynomial $f(x)=ax^2+bx+c$ is of such a sort, that the equation $f(x)=x$ does not have real roots. Prove that the equation $f(f(x))=0$ does not have real roots also.

Given a parallelogram $ABCD$. The line perpendicular to $AC$ passing through $C$ and the line perpendicular to $BD$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $PC$ intersects the line $BC$ at point $X$, ($X \ne C$) and the line $DC$ at point $Y$ , ($Y \ne C$). Prove that the line $AX$ passes through the point $Y$ .

Given a positive integer $ n$, denote by $ \sigma (n)$ the sum of all positive divisors of $ n$. We say that $ n$ is $ abundant$ if $ \sigma (n)>2n.$ (For example, $ 12$ is abundant since $ \sigma (12)\equal{}28>2 \cdot 12$.) Let $ a,b$ be positive integers and suppose that $ a$ is abundant. Prove that $ ab$ is abundant.

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$. (Nagel Igor)

Let $m, n,$ and $k$ be natural numbers, where $n$ is odd. Prove that $\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}$ is not a natural number.

Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle. by A.Khachaturyan