22nd ASU 1988

22nd ASU 1988 problems


1. A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
2. ABCD is a convex quadrilateral. The midpoints of the diagonals and the midpoints of AB and CD form another convex quadrilateral Q. The midpoints of the diagonals and the midpoints of BC and CA form a third convex quadrilateral Q'. The areas of Q and Q' are equal. Show that either AC or BD divides ABCD into two parts of equal area.
3. Show that there are infinitely many triples of distinct positive integers a, b, c such that each divides the product of the other two and a + b = c + 1.
4. Given a sequence of 19 positive integers not exceeding 88 and another sequence of 88 positive integers not exceeding 19. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.
5. The quadrilateral ABCD is inscribed in a fixed circle. It has AB parallel to CD and the length AC is fixed, but it is otherwise allowed to vary. If h is the distance between the midpoints of AC and BD and k is the distance between the midpoints of AB and CD, show that the ratio h/k remains constant.
6. The numbers 1 and 2 are written on an empty blackboard. Whenever the numbers m and n appear on the blackboard the number m + n + mn may be written. Can we obtain (1) 13121, (2) 12131?
7. If rationals x, y satisfy x⁵ + y⁵ = 2 x² y² show that 1 - x y is the square of a rational.
8. There are 21 towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?
9. Find all positive integers n satisfying (1 + 1/n)ⁿ⁺¹ = (1 + 1/1998)¹⁹⁹⁸.
10. A, B, C are the angles of a triangle. Show that 2(sin A)/A + 2(sin B)/B + 2(sin C)/C ≤ (1/B + 1/C) sin A + (1/C + 1/A) sin B + (1/A + 1/B) sin C.
11. Form 10A has 29 students who are listed in order on its duty roster. Form 10B has 32 students who are listed in order on its duty roster. Every day two students are on duty, one from form 10A and one from form 10B. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from 10A and one from 10B) shared duty exactly once?
12. In the triangle ABC, the angle C is obtuse and D is a fixed point on the side BC, different from B and C. For any point M on the side BC, different from D, the ray AM intersects the circumcircle S of ABC at N. The circle through M, D and N meets S again at P, different from N. Find the location of the point M which minimises MP.
13. Show that there are infinitely many odd composite numbers in the sequence 1¹, 1¹ + 2², 1¹ + 2² + 3³, 1¹ + 2² + 3³ + 4⁴, ... .
14. ABC is an acute-angled triangle. The tangents to the circumcircle at A and C meet the tangent at B at M and N. The altitude from B meets AC at P. Show that BP bisects the angle MPN.
15. What is the minimal value of b/(c + d) + c/(a + b) for positive real numbers b and c and non-negative real numbers a and d such that b + c ≥ a + d?
16. n² real numbers are written in a square n x n table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are a_ij and we select row i, column h and column k, then column h becomes a_1h + a_i1, a_2h + a_i2, ... , a_nh + a_in, column k becomes a_1k - a_i1, a_2k - a_i2, ... , a_nk - a_in, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.
17. In the acute-angled triangle ABC, the altitudes BD and CE are drawn. Let F and G be the points of the line ED such that BF and CG are perpendicular to ED. Prove that EF = DG.
18. Find the minimum value of xy/z + yz/x + zx/y for positive reals x, y, z with x² + y² + z² = 1.
19. A polygonal line connects two opposite vertices of a cube with side 2. Each segment of the line has length 3 and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?
20. Let m, n, k be positive integers with m ≥ n and 1 + 2 + ... + n = mk. Prove that the numbers 1, 2, ... , n can be divided into k groups in such a way that the sum of the numbers in each group equals m.
21. A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis.
22. What is the smallest n for which there is a solution to sin x₁ + sin x₂ + ... + sin x_n = 0, sin x₁ + 2 sin x₂ + ... + n sin x_n = 100?
23. The sequence of integers a_n is given by a₀ = 0, a_n = p(a_n-1), where p(x) is a polynomial whose coefficients are all positive integers. Show that for any two positive integers m, k with greatest common divisor d, the greatest common divisor of a_m and a_k is a_d.
24. Prove that for any tetrahedron the radius of the inscribed sphere r < ab/( 2(a + b) ), where a and b are the lengths of any pair of opposite edges.