¶ 2018 USAMO Day 1 Problems and SolutionsIndividual Problems and Solutions
For problems and detailed solutions to each of the 2018 USAMO Day 1 problems, please refer below:
Problem 1: Let a , b , c a , b , c a + b + c = 4 a b c 3 a + b + c = 4 3 a b c
2 ( a b + b c + c a ) + 4 min ( a 2 , b 2 , c 2 ) ≥ a 2 + b 2 + c 2 2 ( a b + b c + c a ) + 4 min ( a 2 , b 2 , c 2 ) ≥ a 2 + b 2 + c 2
Solution:
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Problem 2: Find all functions f : ( 0 , ∞ ) → ( 0 , ∞ ) f : ( 0 , ∞ ) → ( 0 , ∞ )
f ( x + 1 y ) + f ( y + 1 z ) + f ( z + 1 x ) = 1 f ( x + y 1 ) + f ( y + z 1 ) + f ( z + x 1 ) = 1
for all x , y , z > 0 x , y , z > 0 x y z = 1 x y z = 1
Solution:
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Problem 3: For a given integer n ≥ 2 n ≥ 2 { a 1 , a 2 , … , a m } { a 1 , a 2 , … , a m } n n n n m m n n a 1 k + a 2 k + ⋯ + a m k a 1 k + a 2 k + ⋯ + a m k m m k k
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions