2010 USAJMO 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
A permutation of the set of positive integers is a sequence
such that each element of
appears precisely one time as a term of the sequence. For example,
is a permutation of
. Let
be the number of permutations of
for which
is a perfect square for all
. Find with proof the smallest
such that
is a multiple of
.
Problem 2
Let be an integer. Find, with proof, all sequences
of positive integers with the following three properties:
;
for all
;
- given any two indices
and
(not necessarily distinct) for which
, there is an index
such that
.
Problem 3
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
.
Day 2
Problem 4
A triangle is called a parabolic triangle if its vertices lie on a parabola . Prove that for every nonnegative integer
, there is an odd number
and a parabolic triangle with vertices at three distinct points with integer coordinates with area
.
Problem 5
Two permutations and
of the numbers
are said to intersect if
for some value of
in the range
. Show that there exist
permutations of the numbers
such that any other such permutation is guaranteed to intersect at least one of these
permutations.
Problem 6
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 lengths.
完整版2010 USAJMO真题pdf版本免费下载
2010USAJMO真题参考答案及详解
AMC8/AMC10/AMC12/AIME
报名及辅导请联系


Aaron 李老师 15618605663 微信:linstitute4
Aaron 李老师 15618605663 微信:linstitute4