Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying:
(i) there are only finitely many $s$ in $\mathbb{R}$ such that $f(s)=0$, and
(ii) $f\left(x^{4}+y\right)=x^{3} f(x)+f(f(y))$ for all $x, y$ in $\mathbb{R}$.
Let $x$, $y$, $z$ be positive numbers such that
$$
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1.
$$
Show that
$$
\sqrt{x + y z} + \sqrt{y + z x} + \sqrt{z + x y} \geq \sqrt{x y z} + \sqrt{x} + \sqrt{y} + \sqrt{z}.
$$
Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
$$
A_{n} = \frac{a_{1} + a_{2} + \cdots + a_{n}}{n}
$$
Prove that
$$
a_{1}! a_{2}! \ldots a_{n}! \geq \left( \left\lfloor A_{n} \right\rfloor ! \right)^{n}
$$
where $\left\lfloor A_{n} \right\rfloor$ is the greatest integer less than or equal to $A_{n}$, and $a! = 1 \times 2 \times \cdots \times a$ for $a \geq 1$ (and $0! = 1$). When does equality hold?
Let $ABC$ be an equilateral triangle. Let $P$ be a point on the side $AC$ and $Q$ be a point on the side $AB$ so that both triangles $ABP$ and $ACQ$ are acute. Let $R$ be the orthocentre of triangle $ABP$ and $S$ be the orthocentre of triangle $ACQ$. Let $T$ be the point common to the segments $BP$ and $CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that triangle $TRS$ is equilateral.
Let $a$, $b$ and $c$ be positive real numbers. show that
$$
\frac{a+b}{c^2} + \frac{c+a}{b^2} + \frac{b+c}{a^2} \ge \frac{9}{a+b+c} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}
$$
Let $1 = a_1 \le a_2 \le \dots \le a_n \le \dots$ be integers such that there are infinitely many positive integers $k$ with $k = \frac{i}{a_i}$ for some $i$.
Show that for every positive integer $n$, there exists a positive integer $j$ such that $n = \frac{j}{a_j}$.
Let $ABCD$ be a rectangle with $AB \gt BC$, and $O$ be the point of intersection of its diagonals $AC$ and $BD$. Angle bisector $B\hat{A}C$ cuts $BD$ at $E$. We call $M$ the midpoint of $AB$. The perpendicular to $AB$ is drawn through $E$, which cuts $AB$ at $F$; The perpendicular to $AE$ is drawn through $E$, which cuts $AC$ at $H$. If $OH = a$ and $MF = \frac{4}{3}a$ are given, calculate the area of rectangle $ABCD$ in terms of $a$.
Let $ABC$ be a triangle with $\hat{C} = 60^\circ$. The point $P$ is the symmetrical point of $A$ with respect to the point of tangency of the circle inscribed with the side $BC$. Prove that if the bisector of the segment $CP$ cuts the line containing the bisector of the angle $\hat{B}$ at the point $Q$, then the triangle $CPQ$ is equilateral.
Let $A$ be a subset of the set $N$ of positive integers. We will say that a subset $B$ of $N$ is a basis of $A$ if the sums of the elements of each non-empty subset of $B$ are different and each element of $A$ is equal to one of these sums. Show that for each $n = 1, 2, 3, \ldots$ there exists a $k(n)$ such that each subset of $N$ with $n$ elements has a base with at most $k(n)$ elements, and determine (for each $n$) the minimum value of $k(n)$.
Given any set of 9 points in the plane of which no three are collinear, show that for each point $P$ in the set, the number of triangles that have as vertices three of the remaining eight points and $P$ in their interior is even.
A car racing track consists of 6 circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \Gamma_5$ and $\Gamma_6$, interior tangents at a point $P$. The lengths of the circles are 1 km, 2 km, 4 km, 8 km, 16 km and 32 km, respectively. Two pilots, $A$ and $B$, travel along the track as follows: starting at point $P$, pilot $A$ travels $\Gamma_1$, when it reaches $P$ again it travels $\Gamma_2$, then travels $\Gamma_3$, until which completes the entire track route and reaches $P$ again. There the whole journey begins again, over and over again. It travels all the time at a constant speed, and travels through each of the circles in the same direction. Pilot $B$ begins his trip somewhat later than $A$, and travels through the circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \Gamma_5$ and $\Gamma_6$ in the same order as $A$, at the same speed as $A$, but always in the opposite direction to that of $A$.
Suppose $B$ knows the time at which $A$ started its journey. Decides whether $B$ can choose the start time of its own course so that the two cars never pass each other on the track.
Given a quadrilateral $ABCD$, isosceles triangles $ABK$, $BCL$, $CDM$ and $DAN$ are constructed, whose bases are the sides $AB$, $BC$, $CD$ and $DA$, and such that $K$, $L$, $M$ and $N$ are different points and no three of them are aligned. The perpendicular to the line $KL$ drawn by $B$ cuts the perpendicular to the line $LM$ drawn by $C$ at the point $P$; the perpendicular to the line $MN$ drawn by $D$ cuts the perpendicular to the line $NK$ drawn by $A$ at the point $Q$. Show that, if $P$ and $Q$ are different points, then $PQ$ is perpendicular to $KM$.
A police officer tries to catch a thief on a $2001 \times 2001$ board. They play alternately. Each player, on their turn, must move one square in one of the following three directions:
$\downarrow$ (below); $\rightarrow$ (right); $\vee$ (upper left diagonal).
If the policeman is in the lower right corner square, he can use his move to move directly to the upper left corner square (the thief cannot make this move). Initially the policeman is in the central square and the thief is in the upper right diagonal neighboring square to the policeman. The policeman starts the game. Show that:
a. The thief manages to move at least $10000$ times without being caught.
b. The police officer has a strategy to capture the thief.
There are 2 equal convex quadrilaterals made of paper: $ABCD$ and $A'B'C'D'$ ($AB = A'B'$, $BC = B'C'$, $CD = C'D'$, $DA = D'A'$). The quadrilateral $ABCD$ is cut along the diagonal $AC$ and the quadrilateral $A'B'C'D'$ is cut along the diagonal $B'D'$, thus obtaining four pieces of paper.
a) Indicates a procedure, that does not depend on the particular shape of the convex quadrilateral $ABCD$, that allows creating a parallelogram with the four pieces of paper.
b) If the sides of the quadrilaterals measure $3$, $3$, $4$ and $6$, it shows that the perimeter of the parallelogram is greater than $16$ and less than $28$, regardless of the order of the sides.
Let $\Gamma$ be the circumcircle and $O$ be the circumcenter of a triangle $ABC$ with $AC \neq BC$. The tangent line to $\Gamma$ drawn by $C$ intersects the line $AB$ at $M$. The line perpendicular to $OM$ drawn by $M$ cuts the lines $BC$ and $AC$ at $P$ and $Q$, respectively. Show that the segments $PM$ and $MQ$ are equal.
Let $a$, $b$ and $c$ be positive real numbers. show that
$$
\left(\frac{a}{b+c} + \frac{1}{2}\right) \left(\frac{b}{c+a} + \frac{1}{2}\right) \left(\frac{c}{a+b} + \frac{1}{2}\right) \ge 1
$$
A point $P$ is interior to the equilateral triangle $ABC$ and fulfills $\angle APC = 120^\circ$. Let $M$ be the intersection of $CP$ with $AB$ and $N$ be the intersection of $AP$ with $BC$. Find the locus of the circumcenter of the triangle $MBN$ by varying $P$.
We have a square board of $2002 \times 2002$ with squares painted black and white, like a chess board. The rows have been numbered from $1$ to $2002$ and the columns from $1$ to $2002$ (the box in row $1$ and column $1$ is black). In each box we write the product of the row number and the column number to which the box belongs. Let $A$ be the sum of the numbers written in the black boxes and $B$ be the sum of the numbers in the white boxes.
Prove that $A - B$ is a perfect square.