Difference between revisions of "Mock AIME II 2012 Problems"
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==Problem 1== | ==Problem 1== | ||
Given that <cmath>\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right)\cdots\left(\dfrac{2012^2-2007}{2012^2+2017}\right)=\dfrac{m}{n},</cmath> where <math>m</math> and <math>n</math> are positive relatively prime integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>. | Given that <cmath>\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right)\cdots\left(\dfrac{2012^2-2007}{2012^2+2017}\right)=\dfrac{m}{n},</cmath> where <math>m</math> and <math>n</math> are positive relatively prime integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>. |
Revision as of 23:57, 4 April 2012
Problem 1
Given that where
and
are positive relatively prime integers, find the remainder when
is divided by
.
Problem 2
Let be a recursion defined such that
, and
where
, and
is an integer. If
for
being a positive integer greater than
and
being a positive integer greater than 2, find the smallest possible value of
.
Problem 3
The of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the
of
is
(
). Find the
of
.