2nd ASU 1968

2nd ASU 1968 problems


1. An octagon has equal angles. The lengths of the sides are all integers. Prove that the opposite sides are equal in pairs.
2. Which is greater: 31¹¹ or 17¹⁴? [No calculators allowed!]
3. A circle radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares it can pass through?
4. In a group of students, 50 speak English, 50 speak French and 50 speak Spanish. Some students speak more than one language. Prove it is possible to divide the students into 5 groups (not necessarily equal), so that in each group 10 speak English, 10 speak French and 10 speak Spanish.
5. Prove that: 2/(x² - 1) + 4/(x² - 4) + 6/(x² - 9) + ... + 20/(x² - 100) = 11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1)).
6. The difference between the longest and shortest diagonals of the regular n-gon equals its side. Find all possible n.
7. The sequence a_n is defined as follows: a₁ = 1, a_n+1 = a_n + 1/a_n for n ≥ 1. Prove that a₁₀₀ > 14.
8. Given point O inside the acute-angled triangle ABC, and point O' inside the acute-angled triangle A'B'C'. D, E, F are the feet of the perpendiculars from O to BC, CA, AB respectively, and D', E', F' are the feet of the perpendiculars from O' to B'C', C'A', A'B' respectively. OD is parallel to O'A', OE is parallel to O'B' and OF is parallel to O'C'. Also OD·O'A' = OE·O'B' = OF·O'C'. Prove that O'D' is parallel to OA, O'E' to OB and O'F' to OC, and that O'D'·OA = O'E'·OB = O'F'·OC.
9. Prove that any positive integer not exceeding n! can be written as a sum of at most n distinct factors of n!.
10. Given a triangle ABC, and D on the segment AB, E on the segment AC, such that AD = DE = AC, BD = AE, and DE is parallel to BC. Prove that BD equals the side of a regular 10-gon inscribed in a circle with radius AC.
11. Given a regular tetrahedron ABCD, prove that it is contained in the three spheres on diameters AB, BC and AD. Is this true for any tetrahedron?
12. (a) Given a 4 x 4 array with + signs in each place except for one non-corner square on the perimeter which has a - sign. You can change all the signs in any row, column or diagonal. A diagonal can be of any length down to 1. Prove that it is not possible by repeated changes to arrive at all + signs. (b) What about an 8 x 8 array?
13. The medians divide a triangle into 6 smaller triangles. 4 of the circles inscribed in the smaller triangles have equal radii. Prove that the original triangle is equilateral.
14. Prove that we can find positive integers x, y satisfying x² + x + 1 = py for an infinite number of primes p.
15. 9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the winner could have obtained?
16. {a_i} and {b_i} are permutations of {1/1,1/2, ... , 1/n}. a₁ + b₁ ≥ a₂ + b₂ ≥ ... ≥ a_n + b_n. Prove that for every m (1 ≤ m ≤ n) a_m + a_n ≥ 4/m.
17. There is a set of scales on the table and a collection of weights. Each weight is on one of the two pans. Each weight has the name of one or more pupils written on it. All the pupils are outside the room. If a pupil enters the room then he moves the weights with his name on them to the other pan. Show that you can let in a subset of pupils one at a time, so that the scales change position after the last pupil has moved his weights.
18. The streets in a city are on a rectangular grid with m east-west streets and n north-south streets. It is known that a car will leave some (unknown) junction and move along the streets at an unknown and possibly variable speed, eventually returning to its starting point without ever moving along the same block twice. Detectors can be positioned anywhere except at a junction to record the time at which the car passes and it direction of travel. What is the minimum number of detectors needed to ensure that the car's route can be reconstructed?
19. The circle inscribed in the triangle ABC touches the side AC at K. Prove that the line joining the midpoint of AC with the center of the circle bisects the segment BK.
20. The sequence a₁, a₂, ... , a_n satisfies the following conditions: a₁ = 0, \|a_i\| = \|a_i-1 + 1\| for i = 2, 3, ... , n. Prove that (a₁ + a₂ + ... + a_n)/n ≥ -1/2.
21. The sides and diagonals of ABCD have rational lengths. The diagonals meet at O. Prove that the length AO is also rational.