HMMT 二月 2013 · 冲刺赛 · 第 6 题

HMMT February 2013 — Guts Round — Problem 6

[ 5 ] Let R be the region in the Cartesian plane of points ( x, y ) satisfying x ≥ 0, y ≥ 0, and x + y + b x c + b y c ≤ 5. Determine the area of R .

解析

[ 5 ] Let R be the region in the Cartesian plane of points ( x, y ) satisfying x ≥ 0, y ≥ 0, and x + y + ⌊ x ⌋ + ⌊ y ⌋ ≤ 5. Determine the area of R . 9 Answer: We claim that a point in the first quadrant satisfies the desired property if the point 2 is below the line x + y = 3 and does not satisfy the desired property if it is above the line. To see this, for a point inside the region, x + y < 3 and ⌊ x ⌋ + ⌊ y ⌋ ≤ x + y < 3 However, ⌊ x ⌋ + ⌊ y ⌋ must equal to an integer. Thus, ⌊ x ⌋ + ⌊ y ⌋ ≤ 2. Adding these two equations, x + y + ⌊ x ⌋ + ⌊ y ⌋ < 5, which satisfies the desired property. Conversely, for a point outside the region, ⌊ x ⌋ + ⌊ y ⌋ + { x } + { y } = x + y > 3 However, { x } + { y } < 2. Thus, ⌊ x ⌋ + ⌊ y ⌋ > 1, so ⌊ x ⌋ + ⌊ y ⌋ ≥ 2, implying that x + y + ⌊ x ⌋ + ⌊ y ⌋ > 5. Guts Round To finish, R is the region bounded by the x-axis, the y-axis, and the line x + y = 3 is a right triangle 9 whose legs have length 3. Consequently, R has area . 2