Difference between revisions of "1997 USAMO Problems"
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== Problem 2 == | == Problem 2 == | ||
Let <math>ABC</math> be a triangle, and draw isosceles triangles <math>BCD, CAE, ABF</math> externally to <math>ABC</math>, with <math>BC, CA, AB</math> as their respective bases. Prove that the lines through <math>A,B,C</math> perpendicular to the lines <math>\overleftrightarrow{EF},\overleftrightarrow{FD},\overleftrightarrow{DE}</math>, respectively, are concurrent. | Let <math>ABC</math> be a triangle, and draw isosceles triangles <math>BCD, CAE, ABF</math> externally to <math>ABC</math>, with <math>BC, CA, AB</math> as their respective bases. Prove that the lines through <math>A,B,C</math> perpendicular to the lines <math>\overleftrightarrow{EF},\overleftrightarrow{FD},\overleftrightarrow{DE}</math>, respectively, are concurrent. | ||
+ | |||
+ | == Problem 3 == | ||
+ | Prove that for any integer <math>n</math>, there exists a unique polynomial <math>Q</math> with coefficients in <math>\{0,1,...,9\}</math> such that <math>Q(-2)=Q(-5)=n</math>. |
Revision as of 07:57, 1 July 2011
Problem 1
Let be the prime numbers listed in increasing order, and let be a real number between and . For positive integer , define
where denotes the fractional part of . (The fractional part of is given by where is the greatest integer less than or equal to .) Find, with proof, all satisfying for which the sequence eventually becomes .
Problem 2
Let be a triangle, and draw isosceles triangles externally to , with as their respective bases. Prove that the lines through perpendicular to the lines , respectively, are concurrent.
Problem 3
Prove that for any integer , there exists a unique polynomial with coefficients in such that .