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>. | ||
+ | |||
+ | == Problem 4 == | ||
+ | To ''clip'' a convex <math>n</math>-gon means to choose a pair of consecutive sides <math>AB, BC</math> and to replace them by three segments <math>AM, MN,</math> and <math>NC,</math> where <math>M</math> is the midpoint of <math>AB</math> and <math>N</math> is the midpoint of <math>BC</math>. In other words, one cuts off the triangle <math>MBN</math> to obtain a convex <math>(n+1)</math>-gon. A regular hexagon <math>P_6</math> of area <math>1</math> is clipped to obtain a heptagon <math>P_7</math>. Then <math>P_7</math> is clipped (in one of the seven possible ways) to obtain an octagon <math>P_8</math>, and so on. Prove that no matter how the clippings are done, the area of <math>P_n</math> is greater than <math>\frac{1}{3}</math>, for all <math>n\ge6</math>. | ||
+ | |||
+ | == Problem 5 == | ||
+ | Prove that, for all positive real numbers <math>a, b, c,</math> | ||
+ | |||
+ | <math>(a^3+b^3+abc)^{-1}+(b^3+c^3+abc)^{-1}+(a^3+c^3+abc)^{-1}\le(abc)^{-1}</math>. | ||
+ | |||
+ | == Problem 6 == | ||
+ | Suppose the sequence of nonnegative integers <math>a_1,a_2,...,a_{1997}</math> satisfies | ||
+ | |||
+ | <math>a_i+a_j\lea_{i+j}\lea_i+a_j+1</math> | ||
+ | |||
+ | for all <math>i, j\ge1</math> with <math>i+j\le1997</math>. Show that there exists a real number <math>x</math> such that <math>a_n=\lfloor{nx}\rfloor</math> (the greatest integer <math>\lenx</math>) for all <math>1\len\le1997</math>. |
Revision as of 08:34, 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).