Math: A few interesting math problems - JBMO 2007

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.	S
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 $\angle{APD}$ .	S
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.	S
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 :)).

No comments:

Subscribe to: Post Comments (Atom)