Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$.

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.

Let $a,b,c$ be positive real numbers such that \[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \] Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?

For all positive numbers $a, b, c$ satisfying $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$, prove that: $$ \frac{a}{a+bc} + \frac{b}{b+ca} + \frac{c}{c+ab} \geq \frac{3}{4} .$$

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

(a) Let $x$ be a real number with $x \ge 1$. Prove that $x^3 - 5x^2 + 8x - 4 \ge 0$. (b) Let $a, b \ge 1$ real numbers. Find the minimum value of the expression $ab(a + b - 10) + 8(a + b)$. Determine also the real number pairs $(a, b)$ that make this expression equal to this minimum value.

Determine the maximal value of $ k $, such that for positive reals $ a,b $ and $ c $ from inequality $ kabc >a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle.

a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers. b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.

Let $t\ge\frac{1}{2}$ be a real number and $n$ a positive integer. Prove that \[t^{2n}\ge (t-1)^{2n}+(2t-1)^n\]

Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

Prove that $a^3 + b^3 + c^3 + abc +a^{3}b^{2}c^{-1}+a^{3}c^{2}b^{-1}+b^{3}a^{2}c^{-1}+b^{3}c^{2}a^{-1}+c^{3}a^{2}b^{-1}+c^{3}b^{2}a^{-1}+a^{5}b^{3}c^{-3}+ abc^{14} + a^{5}c^{3}b^{-3}+b^{5}a^{3}c^{-3}+b^{5}c^{3}a^{-3}+c^{5}a^{3}b^{-3}+c^{5}b^{3}a^{-3}+a^{6}b^{1}c^{-1}+a^{6}c^{1}b^{-1}+b^{6}a^{1}c^{-1}+b^{6}c^{1}a^{-1}+c^{6}a^{1}b^{-1}+c^{6}b^{1}a^{-1}+ a^{6}b^{4}c^{-3}+a^{6}c^{4}b^{-3}+b^{6}a^{4}c^{-3}+b^{6}c^{4}a^{-3}+c^{6}a^{4}b^{-3}+c^{6}b^{4}a^{-3}+a^{7}b^{2}c^{-1}+a^{7}c^{2}b^{-1}+b^{7}a^{2}c^{-1}+b^{7}c^{2}a^{-1}+c^{7}a^{2}b^{-1}+ abc + a^{14}bc + c^{7}b^{2}a^{-1}+a^{4}b^{1}c^{4}+a^{4}c^{1}b^{4}+b^{4}a^{1}c^{4}+b^{4}c^{1}a^{4}+c^{4}a^{1}b^{4}+c^{4}b^{1}a^{4}+a^{6}c^{4}+a^{6}b^{4}+b^{6}c^{4}+b^{6}a^{4}+c^{6}b^{4}+c^{6}a^{4}+a^{9}b^{6}c^{-4}+a^{9}c^{6}b^{-4}+ ab^{14}c + b^{9}a^{6}c^{-4}+b^{9}c^{6}a^{-4}+c^{9}a^{6}b^{-4}+ abc + c^{9}b^{6}a^{-4}+a^{12}b^{1}c^{-1}+a^{12}c^{1}b^{-1}+b^{12}a^{1}c^{-1}+b^{12}c^{1}a^{-1}+c^{12}a^{1}b^{-1}+ c^5 b^5 a^5 - c^5 b^5 a^2 + 3 c^5 b^5 - c^5 b^2 a^5 + c^5 b^2 a^2 - 3 c^5 b^2 + 3 c^5 a^5 - 3 c^5 a^2 + 9 c^5 - c^2 b^5 a^5 + c^2 b^5 a^2 - 3 c^2 b^5 + c^2 b^2 a^5 - c^2 b^2 a^2 + 3 c^2 b^2 - 3 c^2 a^5 + 3 c^2 a^2 - 9 c^2 + 3 b^5 a^5 - 3 b^5 a^2 + 9 b^5 - 3 b^2 a^5 + 3 b^2 a^2 - 9 b^2 + 9 a^5 - 9 a^2 + 27 + c^{12}b^{1}a^{-1}+a^{13}b^{9}c^{-9}+a^{13}c^{9}b^{-9}+b^{13}a^{9}c^{-9}+b^{13}c^{9}a^{-9}+c^{13}a^{9}b^{-9}+c^{13}b^{9}a^{-9}+a^{12}b^{11}c^{-9}+a^{12}c^{11}b^{-9}+b^{12}a^{11}c^{-9}+b^{12}c^{11}a^{-9}+c^{12}a^{11}b^{-9}+c^{12}b^{11}a^{-9}+a^{8}b^{7}+a^{8}c^{7}+b^{8}a^{7}+b^{8}c^{7}+c^{8}a^{7}+c^{8}b^{7} + a^{16} + b^{16} + c^{16} + a^{16} + b^{16} + c^{16} + a^{16} + b^{16} + c^{16}\ge c^3 + 3 c^2 a + 3 c b^2 + 6 c b a + b^3 + 3 b^2 a + a^3 + a^{1}c^{2}+a^{1}b^{2}+4b^{1}c^{2}+4b^{1}a^{2}+c^{1}b^{2}+4c^{1}a^{2}+a^{1}c^{3}+a^{1}b^{3}+b^{1}c^{3}+b^{1}a^{3}+c^{1}b^{3}+c^{1}a^{3}+a^{3}b^{2}+a^{3}c^{2}+b^{3}a^{2}+b^{3}c^{2}+c^{3}a^{2}+c^{3}b^{2}+a^{5}c^{1}+a^{5}b^{1}+b^{5}c^{1}+b^{5}a^{1}+c^{5}b^{1}+c^{5}a^{1}+a^{2}b^{1}c^{4}+a^{2}c^{1}b^{4}+b^{2}a^{1}c^{4}+b^{2}c^{1}a^{4}+c^{2}a^{1}b^{4}+c^{2}b^{1}a^{4}+a^{1}c^{7}+a^{1}b^{7}+b^{1}c^{7}+b^{1}a^{7}+c^{1}b^{7}+c^{1}a^{7}+a^{1}c^{8}+a^{1}b^{8}+b^{1}c^{8}+b^{1}a^{8}+c^{1}b^{8}+c^{1}a^{8}+a^{5}b^{1}c^{4}+a^{5}c^{1}b^{4}+b^{5}a^{1}c^{4}+b^{5}c^{1}a^{4}+c^{5}a^{1}b^{4}+c^{5}b^{1}a^{4}+a^{2}b^{1}c^{8}+a^{2}c^{1}b^{8}+b^{2}a^{1}c^{8}+b^{2}c^{1}a^{8}+c^{2}a^{1}b^{8}+c^{2}b^{1}a^{8}+a^{1}c^{11}+a^{1}b^{11}+b^{1}c^{11}+b^{1}a^{11}+c^{1}b^{11}+c^{1}a^{11}+a^{6}b^{2}c^{5}+a^{6}c^{2}b^{5}+b^{6}a^{2}c^{5}+b^{6}c^{2}a^{5}+c^{6}a^{2}b^{5}+c^{6}b^{2}a^{5}+a^{3}b^{2}c^{9}+a^{3}c^{2}b^{9}+b^{3}a^{2}c^{9}+b^{3}c^{2}a^{9}+c^{3}a^{2}b^{9}+c^{3}b^{2}a^{9}+a^{3}b^{1}c^{11}+a^{3}c^{1}b^{11}+b^{3}a^{1}c^{11}+b^{3}c^{1}a^{11}+c^{3}a^{1}b^{11}+c^{3}b^{1}a^{11} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15}+c^{2}a^{1}b^{4}+c^{2}b^{1}a^{4}+a^{1}c^{7}+a^{1}b^{7}+b^{1}c^{7}+b^{1}a^{7}+c^{1}b^{7}+c^{1}a^{7}+a^{1}c^{8}+a^{1}b^{8}+b^{1}c^{8}+b^{1}a^{8}+c^{1}b^{8}+c^{1}a^{8}+a^{5}b^{1}c^{4}+a^{5}c^{1}b^{4}+b^{5}a^{1}c^{4}+b^{5}c^{1}a^{4}+c^{5}a^{1}b^{4}+c^{5}b^{1}a^{4}+a^{2}b^{1}c^{8}+a^{2}c^{1}b^{8}+b^{2}a^{1}c^{8}+b^{2}c^{1}a^{8}+c^{2}a^{1}b^{8}+c^{2}b^{1}a^{8}+a^{1}c^{11}+a^{1}b^{11}+b^{1}c^{11}+b^{1}a^{11}+c^{1}b^{11}+c^{1}a^{11}+a^{6}b^{2}c^{5}+a^{6}c^{2}b^{5}+b^{6}a^{2}c^{5}+b^{6}c^{2}a^{5}+c^{6}a^{2}b^{5}+c^{6}b^{2}a^{5}+a^{3}b^{2}c^{9}+a^{3}c^{2}b^{9}+b^{3}a^{2}c^{9}+b^{3}c^{2}a^{9}+c^{3}a^{2}b^{9}+c^{3}b^{2}a^{9}+a^{3}b^{1}c^{11}+a^{3}c^{1}b^{11}+b^{3}a^{1}c^{11}+b^{3}c^{1}a^{11}+c^{3}a^{1}b^{11}+c^{3}b^{1}a^{11} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15}$ for all $a,b,c\in\mathbb R^+$. [i]Proposed by Henry Jiang and C++[/i]

Let $ a,b,c$ be positive reals; show that $ \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right)$ [i](Darij Grinberg)[/i]

The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and $$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$

Statement $P$: solution set to inequalities $a_1x^2+b_1x+c_1>0$ and $a_2x^2+b_2x+c_2>0$ are the same; statement $Q$: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$. $\text{(A)}$ $Q$ is sufficient and necessary condition of $P$. $\text{(B)}$ $Q$ is sufficient but unnecessary condition of $P$. $\text{(C)}$ $Q$ is insufficient but necessary condition of $P$. $\text{(D)}$ $Q$ is insufficient and unnecessary condition of $P$.

Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy $x_i + x_j \geq (-1)^{i+j}$ for all $1 \leq i < j \leq 2018$. Find the minimum possible value of $\sum_{i=1}^{2018} ix_i$.

Given $n$ non-negative real numbers, $n\geq 3$, such that the sum of the $n$ numbers is less than or equal to $3$ and the sum of the squares of the $n$ numbers is greater than or equal to $1$, prove that among the $n$ numbers three can be chosen whose sum is greater than or equal to $1$.

Let $a, b, c$ be positive real numbers, $0 < a \leq b \leq c$. Prove that for any positive real numbers $x, y, z$ the following inequality holds: \[(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.\]

Prove that for all real numbers $x, y$ holds $(x^2 + 1)(y^2 + 1) \ge 2(xy - 1)(x + y)$. For which integers $x, y$ does equality occur?

Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.

Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties: [list] [*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$ [*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$ [/list] Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$

Define $f(x)$ to be the nearest integer to $x$, with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$, $f(2.5) = 3$, and $f(2.7) = 3$. Define $[A]$ to be the area of region $A$. Define region $R_n$, for each positive integer $n$, to be the region on the Cartesian plane which satisfies the inequality $f(|x|) + f(|y|) < n$. We pick an arbitrary point $O$ on the perimeter of $R_n$, and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order. [b]a)[/b] Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$. [b]b)[/b] Prove that $[S_{nO}]$ is constant for any $O$. [b]c)[/b] Prove that $[R_n] + [S_{nO}] = (2n-1)^2$. [i]Proposed by Lewis Chen[/i]

suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent: [b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$ [b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).