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Difference between revisions of "Mock AIME 1 2013 Problems"
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== Problem 14 ==
 
== Problem 14 ==
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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 11:21, 29 September 2013

Problem 1

Two circles $C_1$ and $C_2$, each of unit radius, have centers $A_1$ and $A_2$ such that $A_1A_2=6$. Let $P$ be the midpoint of $A_1A_2$ and let $C_#$ (Error compiling LaTeX. Unknown error_msg) be a circle externally tangent to both $C_1$ and $C_2$. $C_1$ and $C_3$ have a common tangent that passes through $P$. If this tangent is also a common tangent to $C_2$ and $C_1$, find the radius of circle $C_3$.

Solution

Problem 2

Find the number of ordered positive integer pairs $(a,b,c)$ such that $a$ evenly divides $b$, $b+1$ evenly divides $c$, and $c-a=10$.

Solution

Problem 3

Solution


Problem 4

Solution


Problem 5

Solution


Problem 6

Solution


Problem 7

Solution


Problem 8

Solution


Problem 9

Solution


Problem 10

Solution


Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Let \[P(x) = x^{2013}+4x^{2012}+9x^{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum_{j=1}^{2014}j^2x^{2014-j}.\] If $a_1, a_2, \cdots a_{2013}$ are its roots, then compute the remainder when $a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}$ is divided by 997. Solution

Problem 15

Solution

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