2011 JBMO Problems

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Problem 1

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:

$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

Problem 2

Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$

Problem 3

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

Problem 4

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that $$\tfrac{AB}{AE}=\tfrac{CD}{DF}=n$$

If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$