49th Eötvös Competition Problems 1945

1.  Knowing that 23 October 1948 was a Saturday, which is more frequent for New Year's Day, Sunday or Monday? 2.  A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron. 3.  ABC is

42nd Eötvös Competition Problems 1938

1.  Show that a positive integer n is the sum of two squares iff 2n is the sum of two squares. 2.  Show that 1/n + 1/(n+1) + 1/(n+2) + ... + 1/n2 > 1 for integers n > 1. 3.  Show that for

41st Eötvös Competition Problems 1937

1.  a1, a2, ... , an is any finite sequence of positive integers. Show that a1! a2! ... an! < (S + 1)! where S = a1 + a2 + ... + an. 2.  P, Q, R are three points in space. The circle CP passes through Q and R, the circle CQ passes through R and P, and the circle CR passes through P and Q. The tangents to CQ and CR at P coincide. Similarly, the

40th Eötvös Competition Problems 1936

1.  Show that 1/(1·2) + 1/(3·4) + 1/(5·6) + ... + 1/( (2n-1)·2n) = 1/(n+1) + 1/(n+2) + ... + 1/(2n). 2.  ABC is a triangle. Show that, if the point P inside the triangle is such that the triangles PAB, PBC, PCA have equal area, then P must be the centroid. 3.  Find the

34th Eötvös Competition Problems 1930

1.  How many integers (1) have 5 decimal digits, (2) have last digit 6, and (3) are divisible by 3? 2.  A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line? 3.  A circle or