For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$ prove that $$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$

Real numbers $a,b,c$ satisfy $a+b \ne 0$, $b+c \ne 0$ and $c+a \ne 0$. Show that \[\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.\]

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

If $x \in \left(0,\frac{\pi}{2}\right)$ then$$\left(\frac{\sin \left(\frac{\pi}{2}\sin x\right)}{\sin x}\right)^2+\left(\frac{\sin \left(\frac{\pi}{2}\cos x\right)}{\cos x}\right)^2\geq 3$$

A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?

The vertices of two regular octagons are numbered from $1$ to $8$, in some order, which may vary between both octagons (each octagon must have all numbers from $1$ to $8$). After this, one octagon is placed on top of the other so that every vertex from one octagon touches a vertex from the other. Then, the numbers of the vertices which are in contact are multiplied (i.e., if vertex $A$ has a number $x$ and is on top of vertex $A'$ that has a number $y$, then $x$ and $y$ are multiplied), and the $8$ products are then added. Prove that, for any order in which the vertices may have been numbered, it is always possible to place one octagon on top of the other so that the final sum is at least $162$. Note: the octagons can be rotated.

Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have \[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]

Consider $x_1,\ldots,x_n>0$. Show that there exists $a_1,a_2,\ldots,a_n\in \{-1,1\}$ such that \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\] [i]Laurențiu Panaitopol[/i]

Determine all sequences of integers $a_1, a_2,. . .,$ such that: (i) $1 \le a_i \le n$ for all $1 \le i \le n$. (ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.

Prove for postitive $a$ and natural $n$ $$a^n+\frac{1}{a^n}-2 \geq n^2(a+\frac{1}{a}-2)$$

If $a,b,c,d,e$ are real numbers, prove the inequality $a^2 +b^2 +c^2 +d^2+e^2 \ge a(b+c+d+e)$.

Let $x, y, z$ be real numbers such that $$x+y+z=2, \;\;\;\; xy+yz+zx=1$$ Find the maximum possible value of $x-y$.

Prove that for a triangle $ ABC $ with $ \angle BAC \ge 90^{\circ } , $ having circumradius $ R $ and inradius $ r, $ the following inequality holds: $$ R\sin A>2r $$ [i]Romeo Ilie[/i]

Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.

If $x,y,z$ are positive real numbers with $x^2+y^2+z^2=3$, prove that $\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3$

Determine the maximum value of the sum \[ \sum_{i < j} x_ix_j (x_i \plus{} x_j) \] over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$

Let $x_1,x_2,\cdots,x_n$ be iid rv. $S_n=\sum x_k$ (a) let $P(|x_1|\leq 1)=1$ , $E[x_1]=0$ , $E[x_1^2]=\sigma^2>0$ Prove that $\exists C>0$ , $\forall u\geq 2n\sigma^2$ $P(S_n\geq u)\leq e^{-C u \log(u/n\sigma^2)}$ (b) let $P(x_1=1)=P(x_1=-1)=\sigma^2/2$ , $P(x_1=0)=1-\sigma^2$ Prove that $\exists B_1<1,B_2>1,B_3>0$ , $\forall u\geq1, B_1 n\geq u\geq B_2 n\sigma^2$ $P(S_n\geq u)>e^{-B_3 u \log(u/n\sigma^2)}$

Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.

The solution set to the inequality $\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0$ is $\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]$

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

Show that \[ \cos x^2 + \cos y^2 - \cos xy < 3 \] for reals $x$, $y$.

Given a set $X$ and a function $f: X \rightarrow X$, for each $x \in X$ we define $f^1(x)=f(x)$ and, for each $j \ge 1$, $f^{j+1}(x)=f(f^j(x))$. We say that $a \in X$ is a fixed point of $f$ if $f(a)=a$. For each $x \in \mathbb{R}$, let $\pi (x)$ be the quantity of positive primes lesser or equal to $x$. Given an positive integer $n$, we say that $f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\}$ is [i]catracha[/i] if $f^{f(k)}(k)=k$, for every $k=1, 2, \dots n$. Prove that: (a) If $f$ is catracha, $f$ has at least $\pi (n) -\pi (\sqrt{n}) +1$ fixed points. (b) If $n \ge 36$, there exists a catracha function $f$ with exactly $ \pi (n) -\pi (\sqrt{n}) + 1$ fixed points.