Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

Let $ABC$ be a triangle and $K$ be its circumcircle. Let $P$ be the point of intersection of $BC$ with tangent in $A$ to $K$. Let $D$ and $E$ be the symmetrical points of $B$ and $A$, respectively, from $P$. Let $K_1$ be the circumcircle of triangle $DAC$ and let $K_2$ the circumscribed circle of triangle $APB$. We denote with $F$ the second intersection point of the circles $K_1$ and $K_2$ Then denote with $G$ the second intersection point of the circle $K_1$ with $BF$. Show that the lines $BC$ and $EG$ are parallel.

In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour. [i]Vasile Pop[/i]

Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]

$A,B,C,D,E$ are points on a circle $O$ with radius equal to $r$. Chords $AB$ and $DE$ are parallel to each other and have length equal to $x$. Diagonals $AC,AD,BE, CE$ are drawn. If segment $XY$ on $O$ meets $AC$ at $X$ and $EC$ at $Y$ , prove that lines $BX$ and $DY$ meet at $Z$ on the circle.

Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.

Suppose that \[ \frac {2x}{3} \minus{} \frac {x}{6} \]is an integer. Which of the following statements must be true about $ x$? $ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$ $ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$ $ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$ $ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$

Let $ABCD$ be a cyclic quadrangle, let the diagonals $AC$ and $BD$ cross at $O$, and let $I$ and $J$ be the incentres of the triangles $ABC$ and $ABD$, respectively. The line $IJ$ crosses the segments $OA$ and $OB$ at $M$ and $N$, respectively. Prove that the triangle $OMN$ is isosceles.

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? $\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

A square and a line intersect at a $45^o$ angle. The line bisects the square into two unequal pieces such that the area of one piece is twice that of the other. If the square has side length $6$, compute the length of the cut due to the line. [img]https://cdn.artofproblemsolving.com/attachments/6/4/2eb33fb9766497d25d342001cdbae9a7ffd4b4.png[/img]

The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to $1, 1$ and $2$, respectively. Determine the length of this hexagon's side. (Mikhail Evdokimov)

If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is $\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$

Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with \[\frac{x + y}{2} \quad \text{and} \quad 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}.\] After $2017$ minutes the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.

A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a [i]final configuration[/i]. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$. [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=119189]IMO ShortList 2001, combinatorics problem 7, alternative[/url]

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

An [i]$ n\minus{}m$ society[/i] is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every [i]$ n_0\minus{}m_0$ society[/i] contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.

p1. Given a set of $n$ the first natural number. If one of the numbers is removed, then the average number remaining is $21\frac14$ . Specify the number which is deleted. p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring $3 \times 3$. Ipin gets first turn by playing $X$. Upin plays $O$. They must fill in the $X$ or $O$ mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the $4$th step. For example, one of the positions the end is like the picture on the side. [img]https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png[/img] p3. Numbers $ 1$ to $10$ are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are $16$. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png[/img] p4. Define $$S(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$ Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$ p5. Consider the cube $ABCD.EFGH$ with side length $2$ units. Point $A, B, C$, and $D$ lie in the lower side plane. Point $I$ is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid $I.ABCD$. If the pyramid $I.ABCD$ is cut by a diagonal plane connecting the points $A, B, G$, and $H$, determine the volume of the truncated pyramid low part.

There are 1995 segments such that a triangle can be formed from any three of them. Prove that using these $1995 $ segments, it is possible to assemble $664$ acute-angled triangles so that each segment is part of no more than one triangle.

Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines? $\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$

Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$