Difference between revisions of "Mock AIME II 2012 Problems"
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==Problem 1== | ==Problem 1== | ||
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[[Mock AIME II 2012 Problems/Problem 3| Solution]] | [[Mock AIME II 2012 Problems/Problem 3| Solution]] | ||
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+ | ==Problem 4== | ||
+ | Let <math>\triangle ABC</math> be a triangle, and let <math>I_A</math>, <math>I_B</math>, and <math>I_C</math> be the points where the angle bisectors of <math>A</math>, <math>B</math>, and <math>C</math>, respectfully, intersect the sides opposite them. Given that <math>AI_B=5</math>, <math>CI_B=4</math>, and <math>CI_A=3</math>, then the ratio <math>AI_C:BI_C</math> can be written in the form <math>m/n</math> where <math>m</math> and <math>n</math> are positive relatively prime integers. Find <math>m+n</math>. | ||
+ | |||
+ | [[Mock AIME II 2012 Problems/Problem 4| Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | A fair die with <math>12</math> sides numbered <math>1</math> through <math>12</math> inclusive is rolled <math>n</math> times. The probability that the sum of the rolls is <math>2012</math> is nonzero and is equivalent to the probability that a sum of <math>k</math> is rolled. Find the minimum value of k. | ||
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+ | [[Mock AIME II 2012 Problems/Problem 5| Solution]] | ||
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+ | |||
+ | ==Problem 6== | ||
+ | A circle with radius <math>5</math> and center in the first quadrant is placed so that it is tangent to the <math>y</math>-axis. If the line passing through the origin that is tangent to the circle has slope <math>\dfrac{1}{2}</math>, then the <math>y</math>-coordinate of the center of the circle can be written in the form <math>\dfrac{m+\sqrt{n}}{p}</math> where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math> \text{gcd}(m,p)=1 </math>. Find <math>m+n+p</math>. | ||
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+ | [[Mock AIME II 2012 Problems/Problem 6| Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | [[Mock AIME II 2012 Problems/Problem 7| Solution]] | ||
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+ | [[Mock AIME II 2012 Problems/Problem 8| Solution]] | ||
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+ | [[Mock AIME II 2012 Problems/Problem 9| Solution]] | ||
+ | |||
+ | [[Mock AIME II 2012 Problems/Problem 10| Solution]] | ||
+ | |||
+ | [[Mock AIME II 2012 Problems/Problem 11| Solution]] | ||
+ | |||
+ | [[Mock AIME II 2012 Problems/Problem 12| Solution]] | ||
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+ | [[Mock AIME II 2012 Problems/Problem 13| Solution]] | ||
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+ | [[Mock AIME II 2012 Problems/Problem 14| Solution]] | ||
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+ | [[Mock AIME II 2012 Problems/Problem 15| Solution]] |
Revision as of 02:07, 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
.
Problem 4
Let be a triangle, and let
,
, and
be the points where the angle bisectors of
,
, and
, respectfully, intersect the sides opposite them. Given that
,
, and
, then the ratio
can be written in the form
where
and
are positive relatively prime integers. Find
.
Problem 5
A fair die with sides numbered
through
inclusive is rolled
times. The probability that the sum of the rolls is
is nonzero and is equivalent to the probability that a sum of
is rolled. Find the minimum value of k.
Problem 6
A circle with radius and center in the first quadrant is placed so that it is tangent to the
-axis. If the line passing through the origin that is tangent to the circle has slope
, then the
-coordinate of the center of the circle can be written in the form
where
,
, and
are positive integers, and
. Find
.