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Saturday, January 5, 2008

A few interesting math problems - JBMO 2007

Here are problems from Junior Balkan MO 2007:

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

These are not very hard, but it took me 2 hours to solve 3. one, cuz I`m not very good at analysis.
Here are solutions to all of them(click on here :)).

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