HMMT 二月 2015 · 冲刺赛 · 第 16 题
HMMT February 2015 — Guts Round — Problem 16
题目详情
- [ 8 ] Determine the number of unordered triples of distinct points in the 4 × 4 × 4 lattice grid 3 3 { 0 , 1 , 2 , 3 } that are collinear in R (i.e. there exists a line passing through the three points). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT FEBRUARY 2015, 21 FEBRUARY 2015 — GUTS ROUND Organization Team Team ID#
解析
- [ 8 ] Determine the number of unordered triples of distinct points in the 4 × 4 × 4 lattice grid 3 3 { 0 , 1 , 2 , 3 } that are collinear in R (i.e. there exists a line passing through the three points). Answer: 376 Define a main plane to be one of the xy, yz, zx planes. Define a space diagonal to be a set of collinear points not parallel to a main plane. We classify the lines as follows: (a) Lines parallel to two axes (i.e. orthogonal to a main plane). Notice that given a plane of the form v = k , where v ∈ { x, y, z } , k ∈ { 0 , 1 , 2 , 3 } , there are 8 such lines, four in one direction and four in a perpendicular direction. There are 4 × 3 = 12 such planes. However, each line lies 8 × 4 × 3 in two of these ( v, k ) planes, so there are = 48 such lines. Each of these lines has 4 points, 2 so there are 4 possible ways to choose 3 collinear points, giving 4 × 48 = 192 triplets. Guts (b) Diagonal lines containing four points parallel to some main plane. Consider a plane of the form ( v, k ), as defined above. These each have 2 diagonals that contain 4 collinear points. Each of these diagonals uniquely determines v, k so these diagonals are each counted once. There are 12 possible ( v, k ) pairs, yielding 12 × 2 × 4 = 96 triplets. (c) Diagonal lines containing three points parallel to some main plane. Again, consider a plane ( v, k ). By inspection, there are four such lines and one way to choose the triplet of points for each of these lines. This yields 4 × 12 = 48 triplets. (d) Main diagonals. There are four main diagonals, each with 4 collinear points, yielding 4 × 4 = 16 triplets. 3 (e) Space diagonals containing three points. Choose one of the points in the set { 1 , 2 } to be the midpoint of the line. Since these 8 possibilities are symmetric, say we take the point (1 , 1 , 1). There are four space diagonals passing through this point, but one is a main diagonal. So each of the 8 points has 3 such diagonals with 3 points each, yielding 8 × 3 = 24 ways. Adding all these yields 192 + 96 + 48 + 16 + 24 = 376.