Pigeonhole principle12/28/2023 ![]() ![]() (Hint : again look at the sequence of fibonacci numbers modulo $10^n$ and try to prove that this sequence is a periodic sequence. look at this sequence modulo n and by PHP find the solution in the form of x-y where x and y are in this sequence.)ģ- for any positive integer n, prove there exist a Fibonacci number divisible by $10^n$. (Hint: integer coordinates can be odd or even, and you are given 5 points! now look at the middle of lines.)Ģ- for any positive integer n, prove that there exist a multiple of n which its presentation in base 10 has only 0 and 1. Prove that there exist another lattice point on at least one of these lines.(By "lattice point" I mean points of the plane with integer coordinates) so we draw 10 lines, between these points. Hope you'll find them useful:ġ-Given five lattice points on the plane, we connect any two of them by drawing a line between them. I'll mention three of them here and some hints about solutions. There are great applications of pigeonhole principle (PHP) in some olympiad problems and some theorems, both in finite and infinite structures. (Example-)Problem: Given a $n\times n$ square, prove that if $5$ points are placed randomly inside the square, then two of them are at most $\frac$$ I'll illustrate this with an example I've always liked.
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