|The 12th All Soviet Union Mathematical Olympiad
- an is the nearest integer to √n. Find 1/a1 + 1/a2 + ... + 1/a1980.
- ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM.
- Show that there is no positive integer n for which 1000n - 1 divides 1978n - 1.
- If P, Q are points in space the point [PQ] is the point on the line PQ on the opposite side of Q to P and the same distance from Q. K0 is a set of points in space. Given Kn we derive Kn+1 by adjoining all the points [PQ] with P and Q in Kn.
(1) K0 contains just two points A and B, a distance 1 apart, what is the smallest n for which Kn contains a point whose distance from A is at least 1000 ?
(2) K0 consists of three points, each pair a distance 1 apart, find the area of the smallest convex polygon containing Kn.
(3) K0 consists of four points, forming a regular tetrahedron with volume 1. Let Hn be the smallest convex polyhedron containing Kn. How many faces does H1 have? What is the volume of Hn ?
- Two players play a game. There is a heap of m tokens and a heap of n < m tokens. Each player in turn takes one or more tokens from the heap which is larger. The number he takes must be a multiple of the number in the smaller heap. For example, if the heaps are 15 and 4, the first player may take 4, 8 or 12 from the larger heap. The first player to clear a heap wins. Show that if m > 2n, then the first player can always win. Find all k such that if m > kn, then the first player can always win.
- Show that there is an infinite sequence of reals x1, x2, x3, ... such that |xn| is bounded and for any m > n, we have |xm - xn| > 1/(m - n).
- Let p(x) = x2 + x + 1. Show that for every positive integer n, the numbers n, p(n), p(p(n)), p(p(p(n))), ... are relatively prime.
- Show that for some k, you can find 1978 different sizes of square with all its vertices on the graph of the function y = k sin x.
- The set S0 has the single member (5, 19). We derive the set Sn+1 from Sn by adjoining a pair to Sn. If Sn contains the pair (2a, 2b), then we may adjoin the pair (a, b). If S contains the pair (a, b) we may adjoin (a+1, b+1). If S contains (a, b) and (b, c), then we may adjoin (a, c). Can we obtain (1, 50)? (1, 100)? If We start with (a, b), with a < b, instead of (5, 19), for which n can we obtain (1, n) ?
- An n-gon area A is inscribed in a circle radius R. We take a point on each side of the polygon to form another n-gon. Show that it has perimeter at least 2A/R .
- Two players play a game by moving a piece on an n x n chessboard. The piece is initially in a corner square. Each player may move the piece to any adjacent square (which shares a side with its current square), except that the piece may never occupy the same square twice. The first player who is unable to move loses. Show that for even n the first player can always win, and for odd n the second player can always win. Who wins if the piece is initially on a square adjacent to the corner ?
- Given a set of n non-intersecting segments in the plane. No two segments lie on the same line. Can we successively add n-1 additional segments so that we end up with a single non-intersecting path? Each segment we add must have as its endpoints two existing segment endpoints.
- a and b are positive real numbers. xi are real numbers lying between a and b. Show that (x1 + x2 + ... + xn)(1/x1 + 1/x2 + ... + 1/xn) ≤ n2(a + b)2/4ab.
- n > 3 is an integer. Let S be the set of lattice points (a, b) with 0 ≤ a, b < n. Show that we can choose n points of S so that no three chosen points are collinear and no four chosen points from a parallelogram.
- Given any tetrahedron, show that we can find two planes such that the areas of the projections of the tetrahedron onto the two planes have ratio at least √2.
- a1, a2, ... , an are real numbers. Let bk = (a1 + a2 + ... + ak)/k for k = 1, 2, ... , n. Let C = (a1 - b1)2 + (a2 - b2)2 + ... + (an - bn)2, and D = (a1 - bn)2 + (a2 - bn)2 + ... + (an - bn)2. Show that C ≤ D ≤ 2C.
- Let xn = (1 + √2 + √3)n. We may write xn = an + bn√2 + cn√3 + dn√6 , where an, bn, cn, dn are integers. Find the limit as n tends to infinity of bn/an, cn/an, dn/an .