3rd ASU 1963

3rd ASU 1963 problems


1. Given 5 circles. Every 4 have a common point. Prove that there is a point common to all 5.
2. 8 players compete in a tournament. Everyone plays everyone else just once. The winner of a game gets 1, the loser 0, or each gets 1/2 if the game is drawn. The final result is that everyone gets a different score and the player placing second gets the same as the total of the four bottom players. What was the result of the game between the player placing third and the player placing seventh?
3. (a) The two diagonals of a quadrilateral each divide it into two parts of equal area. Prove it is a parallelogram. (b) The three main diagonals of a hexagon each divide it into two parts of equal area. Prove they have a common point. [If ABCDEF is a hexagon, then the main diagonals are AD, BE and CF.]
4. The natural numbers m and n are relatively prime. Prove that the greatest common divisor of m+n and m²+n² is either 1 or 2.
5. Given a circle c and two fixed points A, B on it. M is another point on c, and K is the midpoint of BM. P is the foot of the perpendicular from K to AM. (a) prove that KP passes through a fixed point (as M varies); (b) find the locus of P.
6. Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart.
7. (a) A 6 x 6 board is tiled with 2 x 1 dominos. Prove that we can always divide the board into two rectangles each of which is tiled separately (with no domino crossing the dividing line). (b) Is this true for an 8 x 8 board?
8. Given a set of n different positive reals {a₁, a₂, ... , a_n}. Take all possible non-empty subsets and form their sums. Prove we get at least n(n+1)/2 different sums.
9. Given a triangle ABC. Let the line through C parallel to the angle bisector of B meet the angle bisector of A at D, and let the line through C parallel to the angle bisector of A meet the angle bisector of B at E. Prove that if DE is parallel to AB, then CA=CB.
10. An infinite arithmetic progression contains a square. Prove it contains infinitely many squares.
11. Can we label each vertex of a 45-gon with one of the digits 0, 1, ... , 9 so that for each pair of distinct digits i, j one of the 45 sides has vertices labeled i, j?
12. Find all real p, q, a, b such that we have (2x-1)²⁰ - (ax+b)²⁰ = (x²+px+q)¹⁰ for all x.
13. We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both 1. At step n>1, place a point at the midpoint of each arc created at step n-1 and label it with the sum of the labels at the two adjacent points. What is the total sum of the labels after step n? For example, after step 4 we have: 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4.
14. Given an isosceles triangle, find the locus of the point P inside the triangle such that the distance from P to the base equals the geometric mean of the distances to the sides.