2010 USAMO Problems真题及答案
完整版真题免费下载
+答案解析请参考文末
Day 1
Problem 1
Let be a convex pentagon inscribed in a semicircle of diameter
. Denote by
,
,
,
the feet of the perpendiculars from
onto lines
,
,
,
, respectively. Prove that the acute angle formed by lines
and
is half the size of
, where
is the midpoint of segment
.
Problem 2
There are students standing in a circle, one behind the other. The students have heights
. If a student with height
is standing directly behind a student with height
or less, the two students are permitted to switch places. Prove that it is not possible to make more than
such switches before reaching a position in which no further switches are possible.
Problem 3
The 2010 positive numbers satisfy the inequality
for all distinct indices
. Determine, with proof, the largest possible value of the product
.
Day 2
Problem 4
Let be a triangle with
. Points
and
lie on sides
and
, respectively, such that
and
. Segments
and
meet at
. Determine whether or not it is possible for segments
,
,
,
,
,
to all have integer side lengths.
Problem 5
Let where
is an odd prime, and let
Prove that if
for relatively prime integers
and
, then
is divisible by
.
Problem 6
A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer at most one of the pairs
and
is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number
of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
完整版2010USAMO真题pdf版本免费下载
2010USAMO真题参考答案及详解
AMC8/AMC10/AMC12/AIME
报名及辅导请联系
免费领取备考资料,更有讲座不定时和你见面,和一群志同道合的朋友一起努力!一起学习!
还可领取国际竞赛最新年份真题及解析!