Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.

The sequence $x_{n}$ is defined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers? [i]Proposed by Jacob Weiner[/i]

Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to $1$. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).

A round robin tournament is held with $2016$ participants. Each player plays each other player once and no games result in ties. We say a pair of players $A$ and $B$ is a [i]dominant pair[/i] if all other players either defeat $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number dominant pairs. [i]Proposed by Nathan Ramesh

For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically. Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$. (a) Find all $n$ such that $f(n)=n$. (b) Find all $n$ such that $f(n) = n+1$.

Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?

Alice and Bob are racing. Alice runs at a rate of $2\text{ m/s}$. Bob starts $10\text{ m}$ ahead of Alice and runs at a rate of $1.5\text{ m/s}$. How many seconds after the race starts will Alice pass Bob?

Point $E$ is the midpoint of the base $AD$ of the trapezoid $ABCD$. Segments $BD$ and $CE$ intersect at point $F$. It is known that $AF$ is perpendicular to $BD$. Prove that $BC = FC$.

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

On a straight road, points $1, 2, \ldots, n$ are marked. The distance between any two adjacent points is 1. A "placement" refers to the arrangement of $n$ cars, numbered with the same numbers, at the marked points (there can be multiple cars at one point). The "distance" between two placements is defined as the minimum total length of sections that need to be paved so that cars from the first placement can drive on the asphalt, forming the second one (cars can change places on the road). Prove that for any $\alpha<1$, there exists an integer number $n$ for which there are $100^n$ placements, the pairwise distances between which are greater than $\alpha n$. [i]Proposed by Ilya Bogdanov[/i]

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$. (a) Prove that the lines $AB$ and $FG$ are parallel. (b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.

Prove that the fraction $ \dfrac{21n \plus{} 4}{14n \plus{} 3}$ is irreducible for every natural number $ n$.

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

In a triangle $ABC$, the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$. Prove that, $$ \frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}. $$

Let $\Gamma$ be a circle with the center $O$ and let $A$, $A ^ \prime $ be two diametrically opposite points in $\Gamma$. Let $P$ be the midpoint of $OA ^ \prime$ and $\ell$ a line that passes through $P$, different from the line $AA ^ \prime$ and different from the line perpendicular on $AA ^ \prime$. Let $B$ and $C$ be the intersection points of $\ell$ with $\Gamma$, let $H$ be the foot of the altitude from $A$ on $BC$, let $M$ be the midpoint of $BC$, and let $D$ be the intersection of the line $A ^ \prime M$ with $AH$. Show that the angle $\angle ADO = 90 ^ \circ $.

Given a finite set $K_0$ of points (in the plane or space). The sequence of sets $K_1, K_2, ... , K_n, ...$ is constructed according to the rule: [i]we take all the points of $K_i$, add all the symmetric points with respect to all its points, and, thus obtain $K_{i+1}$.[/i] a) Let $K_0$ consist of two points $A$ and $B$ with the distance $1$ unit between them. For what $n$ the set $K_n$ contains the point that is $1000$ units far from $A$? b) Let $K_0$ consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing $K_n. K_0$ below is the set of the unit volume tetrahedron vertices. c) How many faces contain the minimal convex polyhedron containing $K_1$? d) What is the volume of the above mentioned polyhedron? e) What is the volume of the minimal convex polyhedron containing $K_n$?

For $2n$ numbers in a row, Bob could perform the following operation: $$S_i=(a_1,a_2,\ldots,a_{2n})\mapsto S_{i+1}=(a_1,a_3,\ldots,a_{2n-1},a_2,a_4,\ldots,a_{2n}).$$ Let $T$ be the order of this operation. In other words, $T$ is the smallest positive integer such that $S_i=S_{i+T}$. Prove that $T<2n$.

A regular hexagon is split into $6$ congruent equilateral triangles by drawing in the $3$ main diagonals. Each triangle is colored $1$ of $4$ distinct colors. Rotations and reflections of the figure are considered nondistinct. Find the number of possible distinct colorings.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$