Difference between revisions of "2024 USAJMO Problems"
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Let <math>a(n)</math> be the sequence defined by <math>a(1)=2</math> and <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>. | Let <math>a(n)</math> be the sequence defined by <math>a(1)=2</math> and <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>. | ||
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Revision as of 20:35, 19 March 2024
Contents
[hide]Day 1
Problem 1
Let be a cyclic quadrilateral with and . Points and are selected on line segment so that . Points and are selected on line segment so that . Prove that is a quadrilateral.
Problem 2
Let and be positive integers. Let be the set of integer points with and . A configuration of rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Problem 3
Let be the sequence defined by and for each integer . Suppose that is prime and is a positive integer. Prove that some term of the sequence is divisible by .