HMMT 十一月 2013 · 冲刺赛 · 第 26 题
HMMT November 2013 — Guts Round — Problem 26
题目详情
英文原题
- [ 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
AB = , what is a ?3
2 2 24
解析
英文解析
- [ 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
AB = , what is a ?3
√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
√tt +
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