2007 JBMO Problems

Problem 1

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.


Problem 2

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.


Problem 3

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.


Problem 4

Prove that if $p$ is a prime number, then $7p+3^{p}-4$ is not a perfect square.


See Also

2007 JBMO (ProblemsResources)
Preceded by
2006 JBMO
Followed by
2008 JBMO
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All JBMO Problems and Solutions