Characterise all numbers that cannot be written as a sum of $1$ or more consecutive odd numbers.

Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.

Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.

$n$ students, numbered from $1$ to $n$ are sitting next to each other in a class. In the beginning the $1$st student has $n$ pieces of paper in one pile. The goal of the students is to distribute the $n$ pieces in a way that everyone gets exactly one. The teacher claps once in a minute and for each clap the students can choose one of the following moves (or do nothing): $\bullet$ They divide one of their piles of paper into two smaller piles. $\bullet$ They give one of their piles of paper to the student with the next number. At least how many times does the teacher need to clap in order to make it possible for the students to distribute all the pieces of paper amongst themselves?

With an unmarked ruler only, reconstruct the trapezoid $ABCD$ ($AD \parallel BC$) given the vertices $A$ and $B$, the intersection point $O$ of the diagonals of the trapezoid and the midpoint $M$ of the base $AD$. (Yukhim Rabinovych)

The sum of the three nonnegative real numbers $ x_1, x_2, x_3$ is not greater than $\frac12$. Prove that $(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12$

A random triangle is produced as follows. A pair of standard dice is rolled independently three times to get three random numbers between 2 and 12, inclusive, by adding the numbers that come up on each pair rolled. Call these three random numbers $a$, $b$, and $t$. The random triangle has two sides of lengths $a$ and $b$ with the angle between them measuring $15(t - 1)$ degrees. What is the probability that the triangle is a right triangle?

a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]

For integers $k\ (0\leq k\leq 5)$, positive numbers $m,\ n$ and real numbers $a,\ b$, let $f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx$, $p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)$. Find the values of $m,\ n,\ a,\ b$ for which $E$ is minimized.

Let $ABC$ an isosceles triangle with $CA=CB$ and $\Gamma$ it´s circumcircle. The perpendicular to $CB$ through $B$ intersect $\Gamma$ in points $B$ and $E$. The parallel to $BC$ through $A$ intersect $\Gamma$ in points $A$ and $D$. Let $F$ the intersection of $ED$ and $BC, I$ the intersection of $BD$ and $EC, \Omega$ the cricumcircle of the triangle $ADI$ and $\Phi$ the circumcircle of $BEF$.If $O$ and $P$ are the centers of $\Gamma$ and $\Phi$, respectively, prove that $OP$ is tangent to $\Omega$

An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?

If an arc of $60^\circ$ on circle I has the same length as an arc of $45^\circ$ on circle II, the ratio of the area of circle I to that of circle II is: $\textbf{(A)}\ 16:9 \qquad \textbf{(B)}\ 9:16 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:4 \qquad \textbf{(E)}\ \text{None of these} $

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list] Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). [i]Proposed by Linus Tang[/i]

Evaluate \[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\] [i]Proposed by Nathan Xiong[/i]

A sequence $(a_n)$ of positive integers satisfies$(a_m,a_n) = a_{(m,n)}$ for all $m,n$. Prove that there is a unique sequence $(b_n)$ of positive integers such that $a_n = \prod_{d|n} b_d$

Positive integer $q$ is the $k{}$-successor of positive integer $n{}$ if there exists a positive integer $p{}$ such that $n+p^2=q^2$. Let $A{}$ be the set of all positive integers $n{}$ that have at least a $k{}$-successor, but every $k{}$-successor does not have $k{}$-successors of its own. Prove that $$A=\{7,12\}\cup\{8m+3\mid m\in\mathbb{N}\}\cup\{16m+4\mid m\in\mathbb{N}\}.$$

Given an integer $n$, an integer $1 \le a \le n$ is called $n$-[i]well[/i] if \[ \left\lfloor\frac{n}{\left\lfloor n/a \right\rfloor}\right\rfloor = a. \] Let $f(n)$ be the number of $n$-well numbers, for each integer $n \ge 1$. Compute $f(1) + f(2) + \ldots + f(9999)$. [i]Proposed by Ashwin Sah[/i]

An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the tunnels connecting them. Find the maximum number of good groups.

Define a $ k$-[i]clique[/i] to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.

Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set $$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$ Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $ $ \# $ [i]denotes the cardinal.[/i] [i]Eugen Păltânea[/i]

There are $3N+1$ students with different heights line up for asking questions. Prove that the teacher can drive $2N$ students away such that the remain students satisfies: No one has neighbors whose heights are consecutive.

Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$. Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)

Let ABCD be a trapezoid with $AB \parallel CD, AB = 5, BC = 9, CD = 10,$ and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \tfrac ab$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

Show that there are infinitely many positive integers $a$ that cannot be written as $a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},$ where the $a_i$ are positive integers. State and prove a generalization.

Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$th positive integer which is not a perfect square. [i]Proposed by David Stoner[/i]