Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

Let $n$ be a positive integer and let $x_1, x_2, \ldots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that \[\displaystyle\sum_{i=1}^n x_i^n (1 + x_i) \geq \dfrac{n}{2^{n-1}} \prod_{i=1}^n (1 + x_i).\] [i]IMO Shortlist[/i]

Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

For the positive real numbers $ a,b,c$ prove the inequality \[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1. \]

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

For any $a,b,c>0$ satisfying $a+b+c+ab+ac+bc= 3$, prove that \[2\leq a+b+c+abc\leq 3\] [i]Proposed by Mohammad Ahmadi[/i]

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 $$a^3(1+a+b)-9c(3+3a+a^2)\leq 0$$

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC =135^o$ and $$AC^2 BD^2=2AB\cdot BC \cdot CD\cdot DA.$$ Prove that the diagonals of $ABCD$ are perpendicular.

Find the maximum value of: $E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$ For $a,b$ real numbers.

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

Does there exist an infinite sequence of integers \(a_0\), \(a_1\), \(a_2\), \(\ldots\) such that \(a_0\ne0\) and, for any integer \(n\ge0\), the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\] has \(n\) distinct real roots? [i]Proposed by Amol Rama and Espen Slettnes[/i]