Let $ x,y,z$ be positive real numbers such that $ xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z$. Prove the inequality $ \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1$ When does the equality hold?

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that: $$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$

Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that \[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\] and find all cases of equality.

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent: (a) $\displaystyle 1+1=0$; (b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.

Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where$$f(x-y)+xf(x-1)+f(y)=x^2$$for all reals $x,y.$ Proposed by [b]cj13609517288[/b]

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of $$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.$$

A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic. [img]http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg[/img] For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer. 1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10. 2) Find all rulers that do the multiplication in the interval $[1,10]$. 3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.

Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$

Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?

Prove that for all $a,b>0$ with $a+b=2$, we have $$\left(a^n+1\right)\left(b^n+1\right)\ge4$$ for all $n\in\mathbb N_{\ge2}$.

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

Let $n$ and $k$ be positive integers. Let $A$ be a set of $2n$ distinct points on the Euclidean plane such that no three points in $A$ are collinear. Some pairs of points in $A$ are linked with a segment so that there are $n^2 + k$ distinct segments on the plane. Prove that there exists at least $\frac{4}{3}k^{3/2}$ distinct triangles on the plane with vertices in $A$ and sides as the aforementioned segments. [i] Proposed by Ho-Chien Chen[/i]