Find all triples of integers $(x, y, z)$ not zero and relative primes in pairs such that $\frac{(y+z-x)^2}{4x}$, $\frac{(z+x-y)^2}{4y}$ and $\frac{(x+y-z)^2}{4z}$ are all integers.

Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games - that is, into unordered pairs of unordered pairs of people?

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

A standard deck of $52$ cards is shuffled and randomly arranged in a queue, with each card having a suit $(\diamondsuit,\clubsuit,\heartsuit,\spadesuit)$ and a rank $(\text{Ace},2,3,4,5,6,7,8,9,10,\text{Jack},\text{Queen},\text{ King})$. For example, a card with the $\diamondsuit$ suit and the $7$ rank would be denoted as $\diamondsuit7$, and a card with the $\spadesuit$ and the $\text{Ace}$ rank would be denoted as $\spadesuit\text{Ace}$. In the queue, there exists a card with a rank of $\text{Ace}$ that appears for the first time in the queue. Let the card immediately following the above card be denoted as card $C$. Is the probability that $C$ is a $\spadesuit\text{A}$ higher than, equal to, or lower than the probability that $C$ is a $\clubsuit2$?

Given a square-free positive integer $n$. Show that there do not exist coprime positive integers $x,y$ such that $x^n+y^n$ is a multiple of $(x+y)^3$.

Let $b_1,b_2,... $ be a sequence of positive real numbers such that for each $ n\ge 1$, $$b_{n+1}^2 \ge \frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+...+\frac{b_n^2}{n^3}$$ Show that there is a positive integer $M$ such that $$\sum_{n=1}^M \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}$$

Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$. A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a [i]decision function[/i] if (a) the value of the function changes if we change all of its arguments; and (b) the values does not change if we replace any of the arguments by the function value. A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a [i]dictatoric function[/i], if there is an index $i$ such that the value of the function equals its $i$th argument. The [i]democratic function[/i] is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments. Prove that any decision function is a composition of dictatoric and democratic functions.

Let $n$ and $k$ be two positive integers. Prove that there exist infinitely many perfect squares of the form $n \cdot 2^k - 7$.

Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements.

We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$ Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\ (Ivan Novak)

Let $ ABC$ be an acute triangle with $ \angle BAC\equal{}60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB\equal{}\angle BPC\equal{}\angle CPA\equal{}120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$. a) Prove that $ \angle YXZ\equal{}60^{\circ}$ b) Prove that $ X,P,M$ are collinear.

Find the unique integer $a > 1$ that satisfies \[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]

Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.

How many integer $x$ are there such that $\frac{x^2-6}{x-6}$ is a positive integer?

Given a $8$x$8$ square board a) Prove that: for any ways to color the board, we are always be able to find a rectangle consists of $8$ squares such that these squares are not colored. b) Prove that: we can color $7$ squares on the board such that for any rectangles formed by $\geq 9$ squares, there are at least $1$ colored square.

Prove that the medians from the vertices $A$ and $B$ of a triangle $ABC$ are orthogonal if and only if $BC^2+AC^2=5AB^2$.

The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.

For real numbers $(x, y, z)$ satisfying the following equations, find all possible values of $x+y+z$ $x^2y+y^2z+z^2x=-1$ $xy^2+yz^2+zx^2=5$ $xyz=-2$

Let $d(n)$ be the number of divisors of $n,$ where $n$ is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence $\left\{a_{1}, a_{2}, \cdots\right\}$ if $\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\}$ is an nonconstant geometric series then $\left\{a_{1}, a_{2}, \cdots\right\}$ does not bear same colour.

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

Real numbers $a, b, c$ satisfy $(2b - a)^2 + (2b - c)^2 = 2(2b^2 - ac)$. Prove that $a + c = 2b$.

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality \[AE\cdot ED + BE^2=CD\cdot AE.\] Show that $\angle EBA=\angle DCB$.

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]