1997 USAMO Problems
Contents
Day 1
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 .
Day 2
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
for all with . Show that there exists a real number such that (the greatest integer ) for all .
See Also
1997 USAMO (Problems • Resources) | ||
Preceded by 1996 USAMO |
Followed by 1998 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.