PUMaC 2019 · 组合（A 组） · 第 6 题
PUMaC 2019 — Combinatorics (Division A) — Problem 6
题目详情
- The Nationwide Basketball Society (NBS) has 8001 teams, numbered 2000 through 10000. For each n , team n has n + 1 players, and in a sheer coincidence, this year each player attempted n shots and on team n , exactly one player made 0 shots, one player made 1 shot, . . . , one player made n shots. A player’s field goal percentage is defined as the percentage of shots that the player made, rounded to the nearest tenth of a percent. (For instance, 32.45% rounds to 32.5%.) A player in the NBS is randomly selected among those whose field goal percentage is 66.6%. If this player plays for team k , what is the probability that k ≥ 6000?
解析
- The Nationwide Basketball Society (NBS) has 8001 teams, numbered 2000 through 10000. For each n , team n has n + 1 players, and in a sheer coincidence, this year each player attempted n shots and on team n , exactly one player made 0 shots, one player made 1 shot, . . . , one player made n shots. A player’s field goal percentage is defined as the percentage of shots that the player made, rounded to the nearest tenth of a percent. (For instance, 32.45% rounds to 32.5%.) A player in the NBS is randomly selected among those whose field goal percentage is 66.6%. If this player plays for team k , what is the probability that k ≥ 6000? Proposed by Zackary Stier. Answer: 40007 . b Solution: We use Pick’s theorem, A = i + − 1 for A the area of an enclosed figure, i the 2 number of interior lattice points, and b the number of boundary lattice points. We draw the 2 triangle from the origin to the points P = (6655 , 10000) and Q = (6665 , 10000). The is of interest because it contains all points whose rise over run gives a fraction that we seek. We compute the number of lattice points N above the bottom edge in the trapezoid bounded by (1331 , 2000) , P, Q, (1333 , 2000), corresponding to the case of at least 2000. The bottom edge there has 5 lattice points (1333 , 2000) c for c ∈ { 1 , 2 , 3 , 4 , 5 } . We compute N = 48006, since there are 3 + 11 + 5 + 5 − 4 = 20 boundary points. We similarly compute the number of lattice points M above the bottom edge in the trapezoid bounded by (3993 , 6000) , P, Q, (3999 , 6000) (corresponding to the case of at least 6000) as M = 32008, since there are 7+11+3+3 − 4 = 20 m M 32008 16004 boundary points. = = = , which is reduced, giving our answer to be 40007. n N 48006 24003