HMMT 二月 2008 · GEN1 赛 · 第 1 题
HMMT February 2008 — GEN1 Round — Problem 1
题目详情
英文原题
- [ 2 ] Let ABCD be a unit square (that is, the labels A, B, C, D appear in that order around the square).
Let X be a point outside of the square such that the distance from X to AC is equal to the distance
√
from X to BD , and also that AX = . Determine the value of CX .22 2
解析
英文解析
- [ 2 ] Let ABCD be a unit square (that is, the labels A, B, C, D appear in that order around the square).
Let X be a point outside of the square such that the distance from X to AC is equal to the distance
√
from X to BD , and also that AX = . Determine the value of CX .22
Answer:52
A D2
M N
B CX
Since X is equidistant from AC and BD , it must lie on either the perpendicular bisector of AB or theperpendicular bisector of AD . It turns that the two cases yield the same answer, so we will just assumethe first case. Let M be the midpoint of AB and N the midpoint of CD . Then, XM is perpendicular
√
1 3 1 10
to AB , so XM = and thus XN = , N C = . By the Pythagorean Theorem we find XC =
2 2 2 2
and the answer follows.