Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers $n$ such that $n$ divides the number of the year in which the $n$-th Olympiad takes place.

Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

A ball with mass $m$ projected horizontally off the end of a table with an initial kinetic energy $K$. At a time $t$ after it leaves the end of the table it has kinetic energy $3K$. What is $t$? Neglect air resistance. (A) $(3/g)\sqrt{K/m}$ (B) $(2/g)\sqrt{K/m}$ (C) $(1/g)\sqrt{8K/m}$ (D) $(K/g)\sqrt{6/m}$ (E) $(2K/g)\sqrt{1/m}$

Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$ a) for all $n\ge 2$ b) for all $n \ge 3$ (I. Gorodnin)

$x_1, x_2, ... , x_8$ is a permutation of $1, 2, ..., 8$. A move is to take $x_3$ or $x_8$ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at $1, 2, ..., 8$.

Determine all values of $n$ such that it is possible to divide a triangle in $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.

What is \[\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?\] $ \textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3} $

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

The picture shows a triangle divided into $25$ smaller triangles, numbered $1$ to $25$. Is it possible to place the same numbers in the square cells 5$\times 5$ so that any two numbers written in adjacent triangles were are also written in adjacent cells of the square? (The cells of a square are considered adjacent if they have a common side.) [img]https://cdn.artofproblemsolving.com/attachments/4/3/758fe5531ab3e576ef4712c095b393f8dff397.png[/img]

All arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?

Given an integer $c$, the sequence $a_0, a_1, a_2, ...$ is generated using the recurrence relation $a_0 = c$ and $a_i = a^i_{i-1} + 2021a_{i-1}$ for all $i \ge 1$. Given that $a_0 = c$, let $f(c)$ be the smallest positive integer $n$ such that $a_n - 1$ is a multiple of $47$. Compute $$\sum^{46}_{k=1} f(k).$$

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.

Let us call the [i]median [/i] of a system of $2n$ points of a plane a straight line passing through exactly two of them, on both sides of which there are points of this system equally. What is the smallest number of [i]medians [/i] that a system of $2n$ points, no three of which lie on the same line?

Could the set $\{1,2,3,...,3000\}$ contain a subset of $2000$ elements such that none of them is twice the size of another?

The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that: $$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$ When do we have equality in the right inequality and when in the left inequality? [i]Proposed by Walther Janous [/i]

Let $A$, $B$ be opposite vertices of a unit square with circumcircle $\Gamma$. Let $C$ be a variable point on $\Gamma$. If $C\not\in\{A, B\}$, then let $\omega$ be the incircle of triangle $ABC$, and let $I$ be the center of $\omega$. Let $C_1$ be the point at which $\omega$ meets $\overline{AB}$, and let $D$ be the reflection of $C_1$ over line $CI$. If $C \in\{A, B\}$, let $D = C$. As $C$ varies on $\Gamma$, $D$ traces out a curve $\mathfrak C$ enclosing a region of area $\mathcal A$. Compute $\lfloor 10^4 \mathcal A\rfloor$. [i]Proposed by Brandon Wang[/i]

Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

In $1748$, the great Russian mathematician Leonhard Euler published one of his most important works, Introduction to the Analysis of Infinites. In this work, in particular, Euler finds the values of two infinite sums $1 +\frac14 +\frac19+ \frac{1}{16}+...$ and $1 +\frac19+ \frac{1}{16}+...$ (the terms in the first sum are the inverses of the squares of the natural numbers, and in the second are the inverses of the squares of the odd numbers of the natural series). The value of the first sum, as Euler proved, equals $\frac{\pi^2}{6}$. Given this result, find the value of the second sum.

By inserting parentheses, it is possible to give the expression \[ 2\times3\plus{}4\times5 \]several values. How many different values can be obtained? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point. [i](Kritesh Dhakal, Nepal)[/i]

Just wondering: what exactly is Power of a Point?

Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid? [i]Proposed by David Altizio