Difference between revisions of "2016 USAJMO Problems"
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==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
− | Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math>{1, 2, | + | Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math>\{1, 2,\dots,N\}</math>, one can still find <math>2016</math> distinct numbers among the remaining elements with sum <math>N</math>. |
[[2016 USAJMO Problems/Problem 4|Solution]] | [[2016 USAJMO Problems/Problem 4|Solution]] | ||
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[[2016 USAJMO Problems/Problem 6|Solution]] | [[2016 USAJMO Problems/Problem 6|Solution]] | ||
+ | {{USAJMO box|year=2016|before=[[2015 USAJMO Problems]]|after=[[2017 USAJMO Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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Latest revision as of 15:43, 5 August 2023
Contents
[hide]Day 1
Problem 1
The isosceles triangle , with
, is inscribed in the circle
. Let
be a variable point on the arc
that does not contain
, and let
and
denote the incenters of triangles
and
, respectively.
Prove that as varies, the circumcircle of triangle
passes through a fixed point.
Problem 2
Prove that there exists a positive integer such that
has six consecutive zeros in its decimal representation.
Problem 3
Let be a sequence of mutually distinct nonempty subsets of a set
. Any two sets
and
are disjoint and their union is not the whole set
, that is,
and
, for all
. Find the smallest possible number of elements in
.
Day 2
Problem 4
Find, with proof, the least integer such that if any
elements are removed from the set
, one can still find
distinct numbers among the remaining elements with sum
.
Problem 5
Let be an acute triangle, with
as its circumcenter. Point
is the foot of the perpendicular from
to line
, and points
and
are the feet of the perpendiculars from
to the lines
and
, respectively.
Given that prove that the points
and
are collinear.
Problem 6
Find all functions such that for all real numbers
and
,
2016 USAJMO (Problems • Resources) | ||
Preceded by 2015 USAJMO Problems |
Followed by 2017 USAJMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.