3rd ASU 1969

3rd ASU 1969 problems


1. In the quadrilateral ABCD, BC is parallel to AD. The point E lies on the segment AD and the perimeters of ABE, BCE and CDE are equal. Prove that BC = AD/2.
2. A wolf is in the center of a square field and there is a dog at each corner. The wolf can run anywhere in the field, but the dogs can only run along the sides. The dogs' speed is 3/2 times the wolf's speed. The wolf can kill a single dog, but two dogs together can kill the wolf. Prove that the dogs can prevent the wolf escaping.
3. A finite sequence of 0s and 1s has the following properties: (1) for any i < j, the sequences of length 5 beginning at position i and position j are different; (2) if you add an additional digit at either the start or end of the sequence, then (1) no longer holds. Prove that the first 4 digits of the sequence are the same as the last 4 digits.
4. Given positive numbers a, b, c, d prove that at least one of the inequalities does not hold: a + b < c + d; (a + b)(c + d) < ab + cd; (a + b)cd < ab(c + d).
5. What is the smallest positive integer a such that we can find integers b and c so that ax² + bx + c has two distinct positive roots less than 1?
6. n is an integer. Prove that the sum of all fractions 1/rs, where r and s are relatively prime integers satisfying 0 < r < s ≤ n, r + s > n, is 1/2.
7. Given n points in space such that the triangle formed from any three of the points has an angle greater than 120 degrees. Prove that the points can be labeled 1, 2, 3, ... , n so that the angle defined by i, i+1, i+2 is greater than 120 degrees for i = 1, 2, ... , n-2.
8. Find 4 different three-digit numbers (in base 10) starting with the same digit, such that their sum is divisible by 3 of the numbers.
9. Every city in a certain state is directly connected by air with at most three other cities in the state, but one can get from any city to any other city with at most one change of plane. What is the maximum possible number of cities?
10. Given a pentagon with equal sides. (a) Prove that there is a point X on the longest diagonal such that every side subtends an angle at most 90 degrees at X. (b) Prove that the five circles with diameter one of the pentagon's sides do not cover the pentagon.
11. Given the equation x³ + ax² + bx + c = 0, the first player gives one of a, b, c an integral value. Then the second player gives one of the remaining coefficients an integral value, and finally the first player gives the remaining coefficient an integral value. The first player's objective is to ensure that the equation has three integral roots (not necessarily distinct). The second player's objective is to prevent this. Who wins?
12. 20 teams compete in a competition. What is the smallest number of games that must be played to ensure that given any three teams at least two play each other?
13. A regular n-gon is inscribed in a circle radius R. The distance from the center of the circle to the center of a side is h_n. Prove that (n+1)h_n+1 - nh_n > R.
14. Prove that for any positive numbers a₁, a₂, ... , a_n we have: a₁/(a₂+a₃) + a₂/(a₃+a₄) + ... + a_n-1/(a_n+a₁) + a_n/(a₁+a₂) > n/4.