Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that \[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality $$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. [asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

Prove that \[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \] for all positive real numbers $a$ and $b.$

Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality? [i](Walther Janous)[/i]

Prove that $ a < b$ implies that $ a^3 \minus{} 3a \leq b^3 \minus{} 3b \plus{} 4.$ When does equality occur?

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Prove that, abc≤1 and a,b,c>0 \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c} \]

Find the largest real constant $k$ for which the inequality \[(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) + k(a - 1)(b - 1)(c - 1)(d - 1) \ge 0\] holds for all real numbers $a$, $b$, $c$, and $d$.

Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that \[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]

If $a, b$ are positive reals such that $a+b<2$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}$$ and determine all $a, b$ yielding equality. [i]Proposed by Gottfried Perz[/i]

Let $a_1, a_2, \ldots, a_{2016}$ be real numbers such that $9a_i\ge 11a^2_{i+1}$ $(i=,2,\cdots,2015)$. Find the maximum value of $(a_1-a^2_2)(a_2-a^2_3)\cdots (a_{2015}-a^2_{2016})(a_{2016}-a^2_{1}).$

Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

Let $x,y,z$ be nonnegative positive integers. Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$