Difference between revisions of "2006 USAMO Problems"
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=== Problem 3 === | === Problem 3 === | ||
Let <math>ABCD</math> be a quadrilateral, and let <math>E</math> and <math>F</math> be points on sides <math>AD</math> and <math>BC</math> respectively, such that <math>\frac{AE}{ED}=\frac{BF}{FC}.</math> Ray <math>FE</math> meets rays <math>BA</math> and <math>CD</math> at <math>S</math> and <math>T</math> respectively. Prove that the circumcircles of triangles <math>SAE, SBF, TCF,</math> and <math>TDE</math> pass through a common point. | Let <math>ABCD</math> be a quadrilateral, and let <math>E</math> and <math>F</math> be points on sides <math>AD</math> and <math>BC</math> respectively, such that <math>\frac{AE}{ED}=\frac{BF}{FC}.</math> Ray <math>FE</math> meets rays <math>BA</math> and <math>CD</math> at <math>S</math> and <math>T</math> respectively. Prove that the circumcircles of triangles <math>SAE, SBF, TCF,</math> and <math>TDE</math> pass through a common point. | ||
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+ | [[2006 USAMO Problems/Day 2/Problem 3|Solution]] |
Revision as of 09:20, 8 July 2006
Contents
[hide]Day 1
Problem 1
Let be a prime number and let be an integer with . Prove that there exists integers and with and
if and only if is not a divisor of .
Note: For a real number, let denote the greatest integer less than or equal to , and let denote the fractional part of x.
Problem 2
For a given positive integer k find, in terms of k, the minimum value of for which there is a set of distinct positive integers that has sum greater than but every subset of size k has sum at most .
Problem 3
For integral , let be the greatest prime divisor of . By convention, we set and . Find all polynomial with integer coefficients such that the sequence
is bounded above. (In particular, this requires for
Day 2
Problem 1
Find all positive integers such that there are positive rational numbers satisfying
Problem 2
A mathematical frog jumps along the number line. The frog starts at , and jumps according to the following rule: if the frog is at integer , then it can jump either to or to where is the largest power of that is a factor of . Show that if is a positive integer and is a nonnegative integer, then the minimum number of jumps needed to reach is greater than the minimum number of jumps needed to reach
Problem 3
Let be a quadrilateral, and let and be points on sides and respectively, such that Ray meets rays and at and respectively. Prove that the circumcircles of triangles and pass through a common point.