Difference between revisions of "1997 USAMO Problems"
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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>. | 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>. | ||
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+ | [http://www.artofproblemsolving.com/Wiki/index.php/Problem_6 Solution] |
Revision as of 08:39, 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).