43rd IMO 2002 shortlisted problems


Number theory

N1.  Express 20022002 as the smallest possible number of (positive or negative) cubes.
N3.  If N is the product of n distinct primes, each greater than 3, show that 2N + 1 has at least 4n divisors.
N4.  Does the equation 1/a + 1/b + 1/c + 1/(abc) = m/(a + b + c) have infinitely many solutions in positive integers a, b, c for any positive integer m?
N5.  m, n are integers > 1. a1, a2, ... , an are integers, and none is a multiple of mn-1. Show that there are integers ei, not all zero, with |ei| < m, such that e1a1 + e2a2 + ... + enan is a multiple of mn.


G1.  B is a point on the circle S. A is a point (distinct from B) on the tangent to S at B. C is a point not on S such that the line segment AC meets the circle S at two distinct points. S' is a circle which touches AC at C and S at D, where B and D are on opposite sides of the line AC. Show that the circumcenter of BCD lies on the circumcircle of ABC.
G2.  The sides of the triangle ABC subtend the same angles at the point F inside the triangle. The lines BF, CF meet the sides AC, AB at D, E respectively. Show that AB + AC ≥ 4 DE.
G4.  S and S' are circles intersecting at P and Q. A, B are distinct variable points on the circle S not at P or Q. The lines AP, BP meet the circle S' again at A', B' respectively. The lines AB, A'B' meet at C. Show that the circumcenter of AA'C lies on a fixed circle (as A, B vary).
G5.  S is a set of 5 coplanar points, no 3 collinear. M(S) is the largest area of a triangle with vertices in S. Similarly, m(S) is the smallest area. What is the smallest possible value of M(S)/m(S) as S varies?
G7.  ABC is an acute-angled triangle. The incircle touches BC at K. The altitude AD has midpoint M. The line KM meets the incircle again at N. Show that the circumcircle of BCN touches the incircle of ABC at N.
G8.  The circles S and S' meet at A and B. A line through A meets S, S' again at C, D respectively. M is a point on CD. The line through M parallel to BC meets BD at K, and the line through M parallel to BD meets BC at N. The perpendicular to BC at N meets S at the point E on the opposite side of BC to A. The perpendicular to BD at K meets S' at F on the opposite side of BD to A. Show that ∠EMF = 90o.


A1.  Find all real-valued functions f(x) on the reals such that f(f(x) + y) = 2x + f(f(y) - x) for all x, y.
A2.  The infinite real sequence x1, x2, x3, ... satisfies |xi - xj| ≥ 1/(i + j) for all unequal i, j. Show that if all xi lie in the interval [0, c], then c ≥ 1.
A3.  p(x) = ax3 + bx2 + cx + d is a polynomial with integer coefficients and a non-zero. We have x p(x) = y p(y) for infinitely many pairs (x, y) of unequal integers. Show that p(x) has an integer root.
A5.  Given a positive integer n which is not a cube, define a = n1/3, b = 1/(a - [a]), c = 1/(b - [b]). Show that there are infinitely many such n for which we can find integers r, s, t, not all zero, such that ra + sb + tc = 0.
A6.  A is a non-empty set of positive integers. There are positive integers b1, b2, ... , bn, c1, c2, ... , cn such that each set biA + ci = {bia + ci : a belongs to A} is a subset of A, and the n sets biA + ci are pairwise disjoint. Prove that 1/b1 + 1/b2 + ... + 1/bn ≤ 1.


C2.  n is an odd integer. The squares of an n x n chessboard are colored alternately black and white, with the four corner squares black. A tromino is an L shape formed from three squares. For which n can the black squares all be covered by non-overlapping trominoes. What is the minimum number required?
C3.  A sequence of n positive integers is full if for eack k > 1, k only occurs if k-1 occurs before the last occurrence of k. How many full sequences are there for each n?
C4.  T is the set of all triples (x, y, z) with x, y, z non-negative integers < 10. A chooses a member (X, Y, Z) of T. B seeks to identify it. He is allowed to name a triple (a, b, c) in T. A must then reply with |X + Y - a - b| + |Y + Z - b - c| + |Z + X - c - a|. How many triples does B need to name to be sure of determining A's triple?
C5.  r > 1 is a fixed positive integers. F is an infinite family of sets, each of size r, no two of which are disjoint. Show that some set of size r - 1 has non-empty intersection with every element of F.
C6.  n is even. Show that there is a permutation x1, x2, ... , xn of 1, 2, ... n such that for each i, xi+1 belongs to {2xi, 2xi - 1, 2xi - n, 2xi - n - 1}, where, as usual we take xn+1 to mean x1.
C7.  A graph has 120 points. A weak quartet is a set of four points with just one edge. What is the maximum possible number of weak quartets?
Note: problems N2, N6, G3, G6, A4, C1 were used in the Olympiad and are not shown here.

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© John Scholes
1 Aug 2003
Last corrected 1 Aug 2003