7th Junior Balkan Mathematical Olympiad Problems 2003

1.  Let A = 44...4 (2n digits) and B = 88...8 (n digits). Show that A + 2B + 4 is a square. 2.  A1, A2, ... , An are points in the plane, so that if we take the points in any order B1, B2, ... , Bn, then the broken line B1B2...Bn does not intersect itself. What is the largest possible value of n? 3.  ABC is a triangle. E, F are

3rd Junior Balkan Mathematical Olympiad Problems 1999

1.  a, b, c are distinct reals and there are reals x, y such that a3 + ax + y = 0, b3 + bx + y = 0 and c3 + cx + y = 0. Show that a + b + c = 0. 2.  Let an = 23n + 36n+2 + 56n+2. Find gcd(a0, a1, a2, ... , a1999). 3.

1st Junior Balkan Mathematical Olympiad Problems 1997

1.  Show that given any 9 points inside a square side 1 we can always find three which form a triangle with area < 1/8. 2.  Given reals x, y with (x2 + y2)/(x2 - y2) + (x2 - y2)/(x2 + y2) = k, find (x8 + y8)/(x8 - y8) + (x8 - y8)/(x8 + y8) in terms of k.