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===Problem 2=== | ===Problem 2=== | ||
Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath> | Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath> | ||
− | [[ | + | [[2018 USAJMO Problems/Problem 2|Solution]] |
===Problem 3=== | ===Problem 3=== |
Revision as of 01:23, 20 April 2018
Contents
Day 1
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
For each positive integer , find the number of
-digit positive integers that satisfy both of the following conditions:
no two consecutive digits are equal, and
the last digit is a prime.
Problem 2
Let be positive real numbers such that
. Prove that
Solution
Problem 3
() Let
be a quadrilateral inscribed in circle
with
. Let
and
be the reflections of
over lines
and
, respectively, and let
be the intersection of lines
and
. Suppose that the circumcircle of
meets
at
and
, and the circumcircle of
meets
at
and
. Show that
.
Day 2
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 4
Triangle is inscribed in a circle of radius 2 with
, and
is a real number satisfying the equation
, where
. Find all possible values of
.
Problem 5
Let be a prime, and let
be integers. Show that there exists an integer
such that the numbers
produce at least
distinct remainders upon division by
.
Problem 6
Karl starts with cards labeled
lined up in a random order on his desk. He calls a pair
of these cards swapped if
and the card labeled
is to the left of the card labeled
. For instance, in the sequence of cards
, there are three swapped pairs of cards,
,
, and
.
He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had card to its left, then it now has
cards to its right. He then picks up the card labeled
and reinserts it in the same manner, and so on until he has picked up and put back each of the cards
exactly once in that order. (For example, the process starting at
would be
.)
Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2018 USAJMO (Problems • Resources) | ||
Preceded by 2017 USAJMO |
Followed by 2019 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |