HMMT 十一月 2013 · 冲刺赛 · 第 26 题
HMMT November 2013 — Guts Round — Problem 26
题目详情
- [ 13 ] Trapezoid ABCD is inscribed in the parabola y = x such that A = ( a, a ), B = ( b, b ), 2 2 C = ( − b, b ), and D = ( − a, a ) for some positive reals a, b with a > b . If AD + BC = AB + CD , and 3 AB = , what is a ? 4 2 2 2
解析
- [ 13 ] Trapezoid ABCD is inscribed in the parabola y = x such that A = ( a, a ), B = ( b, b ), 2 2 C = ( − b, b ), and D = ( − a, a ) for some positive reals a, b with a > b . If AD + BC = AB + CD , and 3 AB = , what is a ? 4 √ 27 3 2 2 2 2 Answer: Let t = , so 2 a + 2 b = 2 ( a − b ) + ( a − b ) = 2 t gives a = b = t and 40 4 t √ t + 3 3 2
1+ t 2 2 2 2 2 27 4 5 t = ( a − b ) [1 + ( a + b ) ] = ( a − b ) [1 + t ]. Thus a = = = . 2 2 40 2 2 2