4th ASU 1970

4th ASU 1970 problems


1. Given a circle, diameter AB and a point C on AB, show how to construct two points X and Y on the circle such that (1) Y is the reflection of X in the line AB, (2) YC is perpendicular to XA.
2. The product of three positive numbers is 1, their sum is greater than the sum of their inverses. Prove that just one of the numbers is greater than 1.
3. What is the greatest number of sides of a convex polygon that can equal its longest diagonal?
4. n is a 17 digit number. m is derived from n by taking its decimal digits in the reverse order. Show that at least one digit of n + m is even.
5. A room is an equilateral triangle side 100 meters. It is subdivided into 100 rooms, all equilateral triangles with side 10 meters. Each interior wall between two rooms has a door. If you start inside one of the rooms and can only pass through each door once, show that you cannot visit more than 91 rooms. Suppose now the large triangle has side k and is divided into k² small triangles by lines parallel to its sides. A chain is a sequence of triangles, such that a triangle can only be included once and consecutive triangles have a common side. What is the largest possible number of triangles in a chain?
6. Given 5 segments such that any 3 can be used to form a triangle. Show that at least one of the triangles is acute-angled.
7. ABC is an acute-angled triangle. The angle bisector AD, the median BM and the altitude CH are concurrent. Prove that angle A is more than 45 degrees.
8. Five n-digit binary numbers have the property that every two numbers have the same digits in just m places, but no place has the same digit in all five numbers. Show that 2/5 ≤ m/n ≤ 3/5.
9. Show that given 200 integers you can always choose 100 with sum a multiple of 100.
10. ABC is a triangle with incenter I. M is the midpoint of BC. IM meets the altitude AH at E. Show that AE = r, the radius of the inscribed circle.
11. Given any positive integer n, show that we can find infinitely many integers m such that m has no zeros (when written as a decimal number) and the sum of the digits of m and mn is the same.
12. Two congruent rectangles of area A intersect in eight points. Show that the area of the intersection is more than A/2.
13. If the numbers from 11111 to 99999 are arranged in an arbitrary order show that the resulting 444445 digit number is not a power of 2.
14. S is the set of all positive integers with n decimal digits or less and with an even digit sum. T is the set of all positive integers with n decimal digits or less and an odd digit sum. Show that the sum of the kth powers of the members of S equals the sum for T if 1 ≤ k < n.
15. The vertices of a regular n-gon are colored (each vertex has only one color). Each color is applied to at least three vertices. The vertices of any given color form a regular polygon. Show that there are two colors which are applied to the same number of vertices.