ΒΆ 2017 USAMO Day 1 Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 2017 USAMO Day 1 problems, please refer below:
Problem 1: Prove that there are infinitely many distinct pairs (a,b) of relatively prime integers a>1 and b>1 such that ab+ba is divisible by a+b.
Solution:
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Problem 2: Let m1β,β¦,mnβ be a collection of n positive integers, not necessarily distinct. For any sequence of integers A=(a1β,β¦,anβ) and any permutation w=w1β,β¦,wnβ of m1β,β¦,mnβ, define an A-inversion of w to be a pair of entries wiβ,wjβ with i<j for which one of the following conditions holds:
- aiββ₯wiβ>wjβ,
- wjβ>aiββ₯wiβ, or
- wiβ>wjβ>aiβ.
Show that, for any two sequences of integers A=(a1β,β¦,anβ) and B=(b1β,β¦,bnβ), and for any positive integer k, the number of permutations of m1β,β¦,mnβ having exactly kA-inversions is equal to the number of permutations of m1β,β¦,mnβ having exactly kB-inversions.
Solution:
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Problem 3: Let ABC be a scalene triangle with circumcircle Ξ© and incenter I. Ray AI meets BC at D and meets Ξ© again at M; the circle with diameter DM cuts Ξ© again at K. Lines MK and BC meet at S, and N is the midpoint of IS. The circumcircles of β³KID and β³MAN intersect at points L1β and L2β. Prove that Ξ© passes through the midpoint of either IL1ββ or IL2ββ.
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions