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IMO 2016 Day 2

Because IMO doesn't only last for one day.

Problem 4

A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let \(P(n) = n^2+n+1\). What is the least possible positive integer value of \(b\) such that there exists a non-negative integer \(a\) for which the set \[\{P(a+1), P(a+2), \ldots, P(a+b)\}\] is fragrant?

Problem 5

The equation \((x-1) (x-2) \ldots (x-2016) = (x-1) (x-2) \ldots (x-2016)\) is written on the board, with 2016 linear factors on each side. What is the least possible value of \(k\) for which it is possible to erase exactly \(k\) of these 4032 linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Problem 6

There are \(n \ge 2\) line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands \(n-1\) times. Every time he claps, each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will every occupy the same intersection point at the same time.

  • (a) Prove that Geoff can always fulfill his wish if \(n\) is odd.
  • (b) Prove that Geoff can never fulfill his wish if \(n\) is even.

Note by Ivan Koswara
1 year, 4 months ago

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@Geoff Pilling Congrats. You've been featured in the IMO.

Sal Gard - 1 year, 4 months ago

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Hahaha... Woo hoo! Thanks!

Geoff

Geoff Pilling - 1 year, 4 months ago

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Typo in problem 2

Shivam Jadhav - 1 year, 4 months ago

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