ΒΆ 2014 USAMO Day 1 Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2014 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2014 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2014 Forum](https://forums.randommath.com/c/tournaments/MAA/2014-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2014 USAMO Day 1 problems, please refer below:
Problem 1: Let a,b,c,d be real numbers such that bβdβ₯5 and all zeros x1β,x2β,x3β, and x4β of the polynomial P(x)=x4+ax3+bx2+cx+d are real. Find the smallest value the product (x12β+1)(x22β+1)(x32β+1)(x42β+1) can take.
Solution:
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Problem 2: 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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Problem 3: Prove that there exists an infinite set of points
β¦,Pβ3β,Pβ2β,Pβ1β,P0β,P1β,P2β,P3β,β¦
in the plane with the following property: For any three distinct integers a,b and c, points Paβ,Pbβ and Pcβ are collinear if and only if a+b+c=2014.
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions