Prove that: [b](a)[/b] $ 5<\sqrt {5}\plus{}\sqrt [3]{5}\plus{}\sqrt [4]{5}$ [b](b)[/b] $ 8>\sqrt {8}\plus{}\sqrt [3]{8}\plus{}\sqrt [4]{8}$ [b](c)[/b] $ n>\sqrt {n}\plus{}\sqrt [3]{n}\plus{}\sqrt [4]{n}$ for all integers $ n\geq 9 .$ [b][Weightage 16/100][/b]

Of the following statements, the only one that is incorrect is: $ \textbf{(A)}$ An inequality will remain true after each side is increased, decreased, multiplied or divided (zero excluded) by the same positive quantity. $ \textbf{(B)}$ The arithmetic mean of two unequal positive quantities is greater than their geometric mean. $ \textbf{(C)}$ If the sum of two positive quantities is given, ther product is largest when they are equal. $ \textbf{(D)}$ If $ a$ and $ b$ are positive and unequal, $ \frac {1}{2}(a^2 \plus{} b^2)$ is greater than $ [\frac {1}{2}(a \plus{} b)]^2$. $ \textbf{(E)}$ If the product of two positive quantities is given, their sum is greatest when they are equal.

Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

Let $n$ be an integer, $n \ge 2$. For each number $k = 1, 2, ....., n,$ denote by $a _ k$ the number of multiples of $k$ in the set $\{1, 2,. .., n \}$ and let $x _ k = \frac {1} {1} + \frac {1} {2} + \frac {1} {3} _... + \frac {1} {a _ k}$ . Show that: $$\frac {x _ 1 + x _ 2 + ... + x _ n} {n} \le \frac {1} {1 ^ 2} + \frac {1} {2 ^ 2} + ... + \frac {1} {n ^ 2} $$.

For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$

Given positive numbers $a,b$. Prove that the following sentences are equivalent: ($1$) $ \sqrt{a} + 1 > \sqrt{b} $; ($2$) for every $ x > 1, ax + \frac{x}{x - 1} > b$.

Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality: \[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]

Let $x_1,x_2,x_3$ be three positive real roots of the equation $x^3+ax^2+bx+c=0$ $(a,b,c\in R)$ and $x_1+x_2+x_3\leq 1. $ Prove that $$a^3(1+a+b)-9c(3+3a+a^2)\leq 0$$

The coefficients of the equation $ ax^2\plus{}bx\plus{}c\equal{}0$, where $ a\ne 0$, satisfy the inequality $ (a\plus{}b\plus{}c)(4a\minus{}2b\plus{}c)<0$. Prove that this equation has $ 2$ real distinct solutions.

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then $ \textbf{(A)}\ 1<p\le 11 \qquad\textbf{(B)}\ 11<p \le 21 \qquad\textbf{(C)}\ 21< p \le 31 \\ \qquad\textbf{(D)}\ 31< p \le 41 \qquad\textbf{(E)}\ 41< p \le 51 $

Let $x, y, z$ be positive real numbers. Prove that $$\sqrt{(z + x)(z + y)} - z \ge \sqrt{xy}.$$

Let $x, y, z$ be three real distinct numbers such that $$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$. Prove that $ a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$

Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ? Justify your answer.

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+...+1=q^2+q+1$.

A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles. [i]Calvin Deng.[/i]

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

Let $H_n=\sum_{r=1}^{n} \frac{1}{r}$. Show that $$n-(n-1)n^{-1\slash (n-1)}>H_n>n(n+1)^{1\slash n}-n$$ for $n>2$.

Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that $$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$ $$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$

Let $n$ be a natural number. For all real numbers $x,y,z$, $(x^2+y^2+z^2)^2\geq n(x^4+y^4+z^4)$, then the minumum value of $n$ is________.