Find all positive integers $n$ satisfying the following conditions simultaneously: (a) the number of positive divisors of $n$ is not a multiple of $8$; (b) for all integers $x$, we have \[x^n \equiv x \mod n.\] [i] Proposed by usjl[/i]

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

Positive real numbers $ x$, $ y$ satisfy the equations $ x^2 \plus{} y^2 \equal{} 1$ and $ x^4 \plus{} y^4 \equal{} \frac {17}{18}$. Find $ xy$.

Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$. [i]Proposed by Robin Park[/i]

Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$. For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$. Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$. [i]Victor Wang, inspired by 2013 ISL N7[/i]

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?

prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)

For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

A triangle $ABC$ is given. Find all the pairs of points $X,Y$ so that $X$ is on the sides of the triangle, $Y$ is inside the triangle, and four non-intersecting segments from the set $\{XY, AX, AY, BX,BY, CX, CY\}$ divide the triangle $ABC$ into four triangles with equal areas.

Determine all non empty sets $C_1, C_2, C_3, \cdots $ such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers $n$ and $m$, the number of elements in the set $C_n$ plus the number of elements in the set $C_m$ equals the sum of the elements in the set $C_{m + n}$. [i]Note:[/i] We denote $\lvert C_n \lvert$ the number of elements in the set $C_n$, and $S_k$ as the sum of the elements in the set $C_n$ so the problem's condition is that for every $n$ and $m$: \[\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}\] is satisfied.

Consider the property that each a element of a group $G$ satisfies $a ^ 2 = e$, where e is the identity element of the group. Which of the following statements is not always valid for a group $G$ with this property? (a) $G$ is commutative (b) $G$ has infinite or even order (c) $G$ is Noetherian (d) $G$ is vector space over $\mathbb{Z}_2$

Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$. Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$. If the lines $AD$ and $OE$ meet at $F$, find $|AF|/|FD|$.

Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which \[ \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) \] for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$. Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}.$ [i]Proposed by Zoltán Daróczy[/i]

Find the smallest number $n\in\mathbb{N}$, for which there exist distinct positive integers $a_i$, $i=1,2,\dots, n$ such that the expression $$\frac{(a_1+a_2+\dots+a_n)^2-2025}{a_1^2+a_2^2+\dots +a_n^2 } $$ is a positive integer. ([i]proposed by Marin Hristov[/i])

Can the numbers $1,2,...,121$ be written in the cells of an $11\times 11$ board in such a way that any two consecutive numbers are in adjacent cells (sharing a side), and all perfect squares are in the same column?

A number is written in each corner of the cube. On each step, each number is replaced with the average of three numbers in the three adjacent corners (all the numbers are replaced simultaneously). After ten such steps, every number returns to its initial value. Must all numbers have been originally equal? [i](3 points)[/i]

Find two three-digit numbers $x$ and $y$ such that the sum of all other three digit numbers is equal to $600x$.

The polynomial of seven variables $$ Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2) $$ is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients: $$ Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2. $$ Find all possible values of $P_1(1,1,\ldots,1)$. [i](A. Yuran)[/i]

Joaquim, José and João participate of the worship of triangle $ABC$. It is well known that $ABC$ is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to $ABC$, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice $A$, José vertice $B$ and João will represent vertice $C$. Thus, they must form a triangle which is similar to $ABC$, in this order. Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant $t$, Joaquim, for example, will move with constant velocity $v$ to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to $ABC$ in the desired order. The other participants act in the same way. If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal. [i]Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane[/i]

Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Prove that the angle formed by one of the legs of one of the triangles and one of the legs of the other triangle is half the measure of the angle formed by the hypotenuses.

Consider the set $ S$ of all numbers of the form $ a(n)\equal{}n^2\plus{}n\plus{}1, n \in \mathbb{N}.$ Show that the product $ a(n)a(n\plus{}1)$ is in $ S$ for all $ n \in \mathbb{N}$ and give an example of two elements $ s,t$ of $ S$ such that $ s,t \notin S$.

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$