Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point $(i,j)$ to a point with integer coordinates that is a distance of $\sqrt{5}$ away from $(i,j)$. What is the number of ways that Harry can return to the origin after 6 knight hops? [i]Proposed by Harry Kim[/i]

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.

Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\] [i]Proposed by Md. Fuad Al Alam[/i]

Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

Consider a game on an \( n \times n \) board, where each square starts with exactly one stone. A move consists of choosing $5$ consecutive squares in the same row or column of the board and toggling the state of each of those squares (removing the stone from squares with a stone and placing a stone in squares without a stone). For which positive integers \( n \geq 5 \) is it possible to end up with exactly one stone on the board after a finite number of moves?

$\text{Prove that each integer}$ $n\ge3$ can be written as a sum of some consecutive natural numbers only and only if it isn't a power of 2

Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.

Sunaina and Malay play a game on the coordinate plane. Sunaina has two pawns on $(0,0)$ and $(x,0)$, and Malay has a pawn on $(y,w)$, where $x,y,w$ are all positive integers. They take turns alternately, starting with Sunaina. In their turn they can move one of their pawns one step vertically up or down. Sunaina wins if at any point in time all the three pawns are colinear. Find all values of $x,y$ for which Sunaina has a winning strategy irrespective of the value of $w$. Proposed by NV Tejaswi

Consider a rectangle with length $6$ and height $4$. A rectangle with length $3$ and height $1$ is placed inside the larger rectangle such that it is distance $1$ from the bottom and leftmost sides of the larger rectangle. We randomly select one point from each side of the larger rectangle, and connect these $4$ points to form a quadrilateral. What is the probability that the smaller rectangle is strictly contained within that quadrilateral?

Prove: If $f(x) > 0$ for all $x$ and $f(x) \rightarrow 0$ as $x \rightarrow \infty,$ then there exists at most a finite number of solutions of \[ f(m) + f(n) + f(p) = 1 \] in positive integers $m, n,$ and $p.$

How many different sentences with two words can be written using all letters of the word $\text{YARI\c{S}MA}$? (The Turkish word $\text{YARI\c{S}MA}$ means $\text{CONTEST}$. It will produce same result.) $ \textbf{(A)}\ 2520 \qquad\textbf{(B)}\ 5040 \qquad\textbf{(C)}\ 15120 \qquad\textbf{(D)}\ 20160 \qquad\textbf{(E)}\ \text{None of the above} $

The $n$-factorial of a positive integer $x$ is the product of all positive integers less than or equal to $z$ that are congruent to $z$ modulo $n$. For example, for the number 16, its 2-factorial is $16 \times 14 \times 12 \times 10 \times 8 \times 6 \times 4 \times 2$, its 3-factorial is $16 \times 13 \times 10 \times 7 \times 4 \times 1$ and its 18-factorial is 16. A positive integer is called [i]olympic[/i] if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers.

Let $p$ be a prime number, and let $n$ be a positive integer. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i\in \{0, 1, \ldots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that \[p^n \mid (a_1a_2 + a_3a_4 + 1).\]

If the equation: $f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$ holds true for all real x but $\pm 1$, find $f(x)$.

It is a fact that every set of 2016 consecutive integers can be partitioned in two sets with the following four properties: (i) The sets have the same number of elements. (ii) The sums of the elements of the sets are equal. (iii) The sums of the squares of the elements of the sets are equal. (iv) The sums of the cubes of the elements of the sets are equal. Let $S =\{n + 1; n + 2;$ [b]. . .[/b] $; n + k\}$ be a set of $k$ consecutive integers. (a) Determine the smallest value of $k$ such that property (i) holds for $S$. (b) Determine the smallest value of $k$ such that properties (i) and (ii) hold for $S$. (c) Show that properties (i), (ii) and (iii) hold for $S$ when $k = 8$. (d) Show that properties (i), (ii), (iii) and (iv) hold for $S$ when $k = 16$.

Given triangle $ABC$ in which $\angle CAB= 30^{\circ}$ and $\angle ACB=60^{\circ}$. On the ray $AB$ a point $D$ is chosen, and on the ray $CB$ a point $E$ is chosen such that $\angle BDE=60^{\circ}$. Lines $AC$ and $DE$ intersect at $F$. Prove that the circumcircle of $AEF$ passes through a fixed point, which is different from $A$ and does not depend on $D$.

Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.

Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ of square $ABCD$, respectively. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

Find all pairs of positive integers \((a, b)\) such that \[ \frac{a+1}{b} , \frac{b+1}{a} \] are both positive integers.

Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$ where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers

For $n\ne 0$, let f(n) be the largest $k$ such that $3^k$ divides $n$. If $M$ is a set of $n > 1$ integers, show that the number of possible values for $f(m-n)$, where $m, n$ belong to $M$ cannot exceed $n-1$.

Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$

For which integers $n \in \{1,2,\dots,15\}$ is $n^n+1$ a prime number?

Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.