|
(Tag: Redirect target changed) |
| (5 intermediate revisions by 5 users not shown) |
| Line 1: |
Line 1: |
| − | == Problem 1 ==
| + | #redirect [[2013 Mock AIME I Problems]] |
| − | Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <math>A_1</math> and <math>A_2</math> such that <math>A_1A_2=6</math>. Let <math>P</math> be the midpoint of <math>A_1A_2</math> and let <math>C_#</math> be a circle externally tangent to both <math>C_1</math> and <math>C_2</math>. <math>C_1</math> and <math>C_3</math> have a common tangent that passes through <math>P</math>. If this tangent is also a common tangent to <math>C_2</math> and <math>C_1</math>, find the radius of circle <math>C_3</math>.
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 1|Solution]] | |
| − | | |
| − | == Problem 2 ==
| |
| − | Find the number of ordered positive integer pairs <math>(a,b,c)</math> such that <math>a</math> evenly divides <math>b</math>, <math>b+1</math> evenly divides <math>c</math>, and <math>c-a=10</math>.
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 2|Solution]]
| |
| − | | |
| − | == Problem 3 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 3|Solution]]
| |
| − | | |
| − | | |
| − | == Problem 4 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 4|Solution]]
| |
| − | | |
| − | | |
| − | == Problem 5 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 5|Solution]]
| |
| − | | |
| − | | |
| − | ==Problem 6==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 6|Solution]]
| |
| − | | |
| − | | |
| − | ==Problem 7==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 7|Solution]]
| |
| − | | |
| − | | |
| − | == Problem 8 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 8|Solution]]
| |
| − | | |
| − | | |
| − | ==Problem 9==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 9|Solution]]
| |
| − | | |
| − | | |
| − | ==Problem 10==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 10|Solution]]
| |
| − | | |
| − | | |
| − | == Problem 11 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 11|Solution]]
| |
| − | | |
| − | == Problem 12 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 12|Solution]]
| |
| − | | |
| − | == Problem 13 ==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 13|Solution]]
| |
| − | | |
| − | == 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]]
| |
| − | | |
| − | ==Problem 15==
| |
| − | | |
| − | [[2013 Mock AIME I Problems/Problem 15|Solution]]
| |
