Difference between revisions of "1997 USAMO Problems"
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== Problem 3 == | == 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>. | 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>. | ||
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+ | [http://www.artofproblemsolving.com/Wiki/index.php/Problem_3 Solution] | ||
== Problem 4 == | == Problem 4 == |
Revision as of 08:36, 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 .
Problem 4
To clip a convex -gon means to choose a pair of consecutive sides and to replace them by three segments and where is the midpoint of and is the midpoint of . In other words, one cuts off the triangle to obtain a convex -gon. A regular hexagon of area is clipped to obtain a heptagon . Then is clipped (in one of the seven possible ways) to obtain an octagon , and so on. Prove that no matter how the clippings are done, the area of is greater than , for all .
Problem 5
Prove that, for all positive real numbers
.
Problem 6
Suppose the sequence of nonnegative integers satisfies
$a_i+a_j\lea_{i+j}\lea_i+a_j+1$ (Error compiling LaTeX. Unknown error_msg)
for all with . Show that there exists a real number such that (the greatest integer $\lenx$ (Error compiling LaTeX. Unknown error_msg)) for all $1\len\le1997$ (Error compiling LaTeX. Unknown error_msg).