Difference between revisions of "Mock AIME II 2012 Problems"
(Created page with "==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}...") |
|||
Line 3: | Line 3: | ||
[[Mock AIME II 2012 Problems/Problem 1 |Solution]] | [[Mock AIME II 2012 Problems/Problem 1 |Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | Let <math>\{a_n\}</math> be a recursion defined such that <math>a_1=1, a_2=20</math>, and <math>a_n=\sqrt{\left| a_{n-1}^2-a_{n-2}^2 \right|}</math> where <math>n\ge 3</math>, and <math>n</math> is an integer. If <math>a_m=k</math> for <math>k</math> being a positive integer greater than <math>1</math> and <math>m</math> being a positive integer greater than 2, find the smallest possible value of <math>m+k</math>. | ||
+ | |||
+ | [[Mock AIME II 2012 Problems/Problem 2| Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | The <math>\textit{digital root}</math> 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 <math>\textit{digital root}</math> of <math>237</math> is <math>3</math> (<math>2+3+7=12, 1+2=3</math>). Find the <math>\textit{digital root}</math> of <math>2012^{2012^{2012}}</math>. | ||
+ | |||
+ | [[Mock AIME II 2012 Problems/Problem 3| Solution]] |
Revision as of 00:56, 5 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
.