38th IMO 1997 shortlist

38th IMO 1997 shortlisted problems


2. The sequences R_n are defined as follows. R₁ = (1). If R_n = (a₁, a₂, ... , a_m), then R_n+1 = (1, 2, ... , a₁, 1, 2, ... , a₂, 1, 2, ... , 1, 2, ... , a_m, n+1). For example, R₂ = (1, 2), R₃ = (1, 1, 2, 3), R₄ = (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.
3. If S is a finite set of non-zero 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.
5. 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.
6. 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.
7. 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?
9. ABC is a non-isosceles 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.
10. 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).
11. p(x) is a polynomial with real coefficients such that p(x) > 0 for x ≥ 0. Show that (1 + x)ⁿp(x) has non-negative coefficients for some positive integer n.
12. 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 p-1.
13. 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 2n-1 boys such that girl i knows boys 1, 2, ... , 2i-1 (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).
14. b > 1 and m > n. Show that if b^m - 1 and bⁿ - 1 have the same prime divisors then b + 1 is a power of 2. [For example, 7 - 1 = 2.3, 7² - 1 = 2⁴.3.]
15. If an infinite arithmetic progression of positive integers contains a square and a cube, show that it must contain a sixth power.
16. ABC is an acute-angled 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.
18. ABC is an acute-angled 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.
19. Let x₁ ≥ x₂ ≥ x₃ ... ≥ x_n+1 = 0. Show that √(x₁ + x₂ + ... + x_n) ≤ (√x₁ - √x₂) + (√2) (√x₂ - √x₃) + ... + (√n) (√x_n - √x_n+1).
20. 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.
22. Do there exist real-valued functions f and g on the reals such that f( g(x) ) = x² and g( f(x) ) = x³? Do there exist real-valued functions f and g on the reals such that f( g(x) ) = x² and g( f(x) ) = x⁴?
23. 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.
25. 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.
26. Find the minimum value of x₀ + x₁ + ... + x_n for non-negative real numbers x_i such that x₀ = 1 and x_i ≤ x_i+1 + x_i+2.
Note: problems 1, 4, 8, 17, 21, 24 were used in the Olympiad and are not shown here.