ΒΆ 2014 USAJMO Day 1 Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 2014 USAJMO Day 1 problems, please refer below:
Problem 1: Let a,b,c be real numbers greater than or equal to 1 . Prove that
min(b2β5b+1010a2β5a+1β,c2β5c+1010b2β5b+1β,a2β5a+1010c2β5c+1β)β€abc
Solution:
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Problem 2: Let β³ABC be a non-equilateral, acute triangle with β A=60β, and let O and H denote the circumcenter and orthocenter of β³ABC, respectively.
(1) Prove that line OH intersects both segments AB and AC.
(2) Line OH intersects segments AB and AC at P and Q, respectively. Denote by s and t the respective areas of triangle APQ and quadrilateral BPQC. Determine the range of possible values for s/t.
Solution:
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Problem 3: Let Z be the set of integers. Find all functions f:ZβZ such that
xf(2f(y)βx)+y2f(2xβf(y))=xf(x)2β+f(yf(y))
for all x,yβZ with xξ =0.
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions