The 38th International Mathematical Olympiad Shortlist Problems
1997年第38届国际数学奥林匹克备选题 
 was used in the Olympiad.
 The sequences R_{n} are defined as follows. R_{1} = (1). If R_{n} = (a_{1}, a_{2}, ... , a_{m}), then R_{n+1} = (1, 2, ... , a_{1}, 1, 2, ... , a_{2}, 1, 2, ... , 1, 2, ... , a_{m}, n+1). For example, R_{2} = (1, 2), R_{3} = (1, 1, 2, 3), R_{4} = (1, 1, 1, 2, 1, 2, 3, 4). Show that for n > 1, the kth term from the left in R_{n} is 1 iff the kth term from the right is not 1.
 was used in the Olympiad.
 If S is a finite set of nonzero vectors in the plane, then a maximal subset is a subset whose vector sum has the largest possible magnitude. Show that if S has n vectors, then there are at most 2n maximal subsets of S. Give a set of 4 vectors with 8 maximal subsets and a set of 5 vectors with 10 maximal subsets.
 Let ABCD be a regular tetrahedron. Let M be a point in the plane ABC and N a point different from M in the plane ADC. Show that the segments MN, BN and MD can be used to form a triangle.
 Let a, b, c be positive integers such that a and b are relatively prime and c is relatively prime to a or b. Show that there are infinitely many solutions to m^{a} + n^{b} = k^{c}, where m, n, k are distinct positive integers.
 was used in the Olympiad.
 ABCDEF is a convex hexagon with AB = BC, CD = DE, EF = FA. Show that BC/BE + DE/DA + FA/FC ≥ 3/2. When does equality occur?
 ABC is a nonisosceles triangle with incenter I. The smaller circle through I tangent to CA and CB meets the smaller circle through I tangent to BC and BA at A' (and I). B' and C' are defined similarly. Show that the circumcenters of AIA', BIB' and CIC' are collinear.
 Find all positive integers n such that if p(x) is a polynomial with integer coefficients such that 0 ≤ p(k) ≤ n for k = 0, 1, 2, ... , n+1 then p(0) = p(1) = ... = p(n+1).
 p(x) is a polynomial with real coefficients such that p(x) > 0 for x ≥ 0. Show that (1 + x)^{n}p(x) has nonnegative coefficients for some positive integer n.
 p is prime. q(x) is a polynomial with integer coefficients such that q(k) = 0 or 1 mod p for every positive integer k, and q(0) = 0, q(1) = 1. Show that the degree of q(x) is at least p1.
 In town A there are n girls and n boys and every girl knows every boy. Let a(n,r) be the number of ways in which r girls can dance with r boys, so that each girl knows her partner. In town B there are n girls and 2n1 boys such that girl i knows boys 1, 2, ... , 2i1 (and no others). Let b(n,r) be the number of ways in which r girls from town B can dance with r boys from town B so that each girl knows her partner. Show that a(n,r) = b(n,r).
 b > 1 and m > n. Show that if b^{m}  1 and b^{n}  1 have the same prime divisors then b + 1 is a power of 2. [For example, 7  1 = 2.3, 7^{2}  1 = 2^{4}.3.]
 If an infinite arithmetic progression of positive integers contains a square and a cube, show that it must contain a sixth power.
 ABC is an acuteangled triangle with incenter I and circumcenter O. AD and BE are altitudes, and AP and BQ are angle bisectors. Show that D, I, E are collinear iff P, O, Q are collinear.
 was used in the Olympiad.
 ABC is an acuteangled triangle. The altitudes are AD, BE and CF. The line through D parallel to EF meets AC at Q and AB at R. The line EF meets BC at P. Show that the midpoint of BC lies on the circumcircle of PQR.
 Let x_{1} ≥ x_{2} ≥ x_{3} ... ≥ x_{n+1} = 0. Show that √(x_{1} + x_{2} + ... + x_{n}) ≤ (√x_{1}  √x_{2}) + (√2) (√x_{2}  √x_{3}) + ... + (√n) (√x_{n}  √x_{n+1}).
 ABC is a triangle. D is a point on the side BC (not at either vertex). The line AD meets the circumcircle again at X. P is the foot of the perpendicular from X to AB, and Q is the foot of the perpendicular from X to AC. Show that the line PQ is a tangent to the circle on diameter XD iff AB = AC.
 was used in the Olympiad.
 Do there exist realvalued functions f and g on the reals such that f( g(x) ) = x^{2} and g( f(x) ) = x^{3}? Do there exist realvalued functions f and g on the reals such that f( g(x) ) = x^{2} and g( f(x) ) = x^{4}?
 ABCD is a convex quadrilateral and X is the point where its diagonals meet. XA sin A + XC sin C = XB sin B + XD sin D. Show that ABCD must be cyclic.
 was used in the Olympiad.
 ABC is a triangle. The bisectors of A, B, C meet the circumcircle again at K, L, M respectively. X is a point on the side AB (not one of the vertices). P is the intersection of the line through X parallel to AK and the line through B perpendicular to BL. Q is the intersection of the line through X parallel to BL and the line through A perpendicular to AK. Show that KP, LQ and MX are concurrent.
 Find the minimum value of x_{0} + x_{1} + ... + x_{n} for nonnegative real numbers x_{i} such that x_{0} = 1 and x_{i} ≤ x_{i+1} + x_{i+2}.

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