Difference between revisions of "Mock AIME 1 2013 Problems"
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== Problem 14 == | == Problem 14 == | ||
− | + | Let <cmath>P(x) = x^{2013}+4x^{2012}+9x^{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum_{j=1}^{2014}j^2x^{2014-j}.</cmath> If <math>a_1, a_2, \cdots a_{2013}</math> are its roots, then compute the remainder when <math>a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}</math> is divided by 997. | |
[[2013 Mock AIME I Problems/Problem 14|Solution]] | [[2013 Mock AIME I Problems/Problem 14|Solution]] | ||
Revision as of 12:21, 29 September 2013
Contents
Problem 1
Two circles and
, each of unit radius, have centers
and
such that
. Let
be the midpoint of
and let $C_#$ (Error compiling LaTeX. Unknown error_msg) be a circle externally tangent to both
and
.
and
have a common tangent that passes through
. If this tangent is also a common tangent to
and
, find the radius of circle
.
Problem 2
Find the number of ordered positive integer pairs such that
evenly divides
,
evenly divides
, and
.
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Let If
are its roots, then compute the remainder when
is divided by 997.
Solution