34th IMO 1993 shortlisted problems

1.  Show that there is a finite set of points in the plane such that for any point P in the set we can find 1993 points in the set a distance 1 from P.
2.  ABC is a triangle with circumradius R and inradius r. If p is the inradius of the orthic triangle, show that p/R ≤ 1 - (1/3) (1 + r/R)2). [The orthic triangle has as vertices the feet of the altitudes of ABC.]
3.  The triangle ABC has incenter I. A circle lies inside the circumcircle and touches it. It also touches the sides CA, BC at D, E respectively. Show that I is the midpoint of DE.
4.  ABC is a triangle. D and E are points on the side BC such that ∠BAD = ∠CAE. The incircles of ABD and ACE touch BC at M and N respectively. Show that 1/MB + 1/MD = 1/NC + 1/NE.
7.  a > 0 and b, c are integers such that ac - b2 is a square-free positive integer P. [For example P could be 3·5, but not 325.] Let f(n) be the number of pairs of integers d, e such that ad2 + 2bde + ce2= n. Show that f(n) is finite and that f(n) = f(Pkn) for every positive integer k.
8.  Define the sequence a1, a2, a3, ... by a1 = 1, an = an-1 - n if an-1 > n, an-1 + n if an-1 ≤ n. Let S be the set of n such that an = 1993. Show that S is infinite. Find the smallest member of S. If the element of S are written in ascending order show that the ratio of consecutive terms tends to 3.
9.  Show that the set of positive rationals can be partitioned into three disjoint sets A, B, C such that BA = B, BB = C and BC = A, where HK denotes the set {hk: h is in H and k is in K}. Show that all positive rational cubes must lie in A. Find such a partition with the additional property that for each of the sets {1, 2}, {2, 3}, {3, 4}, ... {34, 35} at least one member is not in A.
10.  A positive integer n has property P if, for all a, n2 divides an - 1 whenever n divides an - 1. Show that every prime number has property P. Show that infinitely many other numbers have property P.
12.  Let k ≤ n be positive integers. Let S be any set of n distinct real numbers. Let T be the set of all sums of k distinct elements of S. Show that T has at least k(n - k) + 1 distinct elements.
13.  If m and n are relatively prime positive integers, denote the pair (m, n) by s and define f(s) to be the pair (k, m+n-k), where k is the largest odd integer dividing n. Show that k and m+n-k are relatively prime. Show that f(f( ... f(s)...)) = s where f is iterated t times for some t ≤ (m + n + 1)/4. Show that if m + n is prime and does not divide 2h - 1 for h = 1, 2, ... , m+n-2, then the smallest t is [(m+n-1)/4].
14.  The triangle ABC has side lengths BC = a, CA = b, AB = c, as usual. Points D, E, F on the sides BC, CA, AB respectively are such that DEF is an equilateral triangle. Show that DE √(a2 + b2 + c2 + k 4√3) ≥ k 2√2, where k is the area of ABC.
16.  A is an n-tuple of non-negative integers (a1, a2, ... , an) such that ai ≤ i-1. Given any such n-tuple, we define the successor A' = (b1, ... , bn), where b1 = 0, bi+1 is the number of earlier members of A which are at least ai. Let Ak be the sequence defined by A0 = A, Ak+1 is the successor of Ak. Show that Ak+2 = Ak for some k.
18.  Let Sn be the number of sequences of n 0s and 1s such that the sequence does not contain six consecutive identical blocks of numbers. [For example, 1000100100100100100110 is not allowed because it has six consecutive blocks 001.] Show that Sn tends to infinity.
19.  b > 1, a and n are positive integers such that bn - 1 divides a. Show that in base b, the number a has at least n non-zero digits.
20.  The n > 1 real numbers x1, x2, ... , xn satisfy 0 ≤ x1 + ... + xn ≤ n. Show that there are integers ki with sum 0 such that 1 - n ≤ xi + nki ≤ n for each i.
21.  A circle S bisects a circle S' if it cuts S' at opposite ends of a diameter. SA, SB, SC are circles with distinct centers A, B, C (respectively). Show that A, B, C are collinear iff there is no unique circle S which bisects each of SA, SB, SC. Show that if there is more than one circle S which bisects each of SA, SB, SC, then all such circles pass through two fixed points. Find these points.
23.  Show that for any finite set S of distinct positive integers, we can find a set T ⊇ S such that every member of T divides the sum of all the members of T.
24.  Show that a/(b + 2c + 3d) + b/(c + 2d + 3a) + c/(d + 2a + 3b) + d/(a + 2b + 3c) ≥ 2/3 for any positive reals.
25.  a is a real number such that |a| > 1. Solve the equations:
x12 = ax2 + 1
x22 = ax3 + 1
x9992 = ax1000 + 1
x10002 = ax1 + 1.
26.  a, b, c, d are non-negative reals with sum 1. Show that abc + bcd + cda + dab ≤ 1/27 + 176abcd/27.
Note: Problems 5, 6, 11, 15, 17 and 22 were used in the Olympiad and are not shown here.

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© John Scholes
19 Aug 2002
Last corrected/updated 9 Nov 03