Difference between revisions of "2018 USAJMO Problems"
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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 00:23, 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 |