1.Prove that the following holds for all x in R :
|cos(x)|+|cos(2x)|+ … +|cos(2n+1)x| >= n*|sin(x)|+ 1/2*|sin(2nx)|
2. Prove that if x,y,z>0 are real numbers then the following
inequality holds :
(xy+xz+yz)[ 1/(x+y)^2 + 1/(x+z)^2 + 1/(y+z)^2 ] >= 9/4 .
3. Given an infinite sequence a_1 < a_2 < a_3 < … of natural numbers, show that it is possible to select from the sequence (i,j in N) an infinite subsequence having the property that no member of this subsequence divides any other member.
4. Consider a triangle ABC with side AB=AC, and angle BAC = 20 degrees. D is a point on side AC. AD=BC. Find angle DBC.
5. The coefficients of a 6th degree polynomial P(x) are integers and its roots are 6 different prime numbers. There exist two integers A and B such that P(A) = 65536 and P(B) = 45441. P(75) < 0. Find the 6 roots.
6. Which MxN boards can be tiled by 1×4 tiles.
7. Can you find 6 lattice points (integral coordinates) which are the vertices of a regular hexagon?
8. Is there a continuous function f:R->R such that f(f(x))=-x?
9. Show that there can be at most countably many disjoint T’s drawn in the plane.
10. Suppose that you must draw a straight line between points a mile apart. Your tools are a geometer’s straight-edge and compass of normal size. How do you proceed?