Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Mock AIME 1 2013 Problems"
(→‎Problem 2)
Line 5: Line 5:
  
 
== Problem 2 ==
 
== Problem 2 ==
−
Find the number of ordered positive integer pairs <math>(a,b,c) such that </math>a<math> evenly divides </math>b<math>, </math>b+1<math> evenly divides </math>c<math>, and </math>c-a=10$.
+
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]]
 
[[2013 Mock AIME I Problems/Problem 2|Solution]]
  

Revision as of 18:51, 6 May 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

Solution

Problem 15

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_1_2013_Problems&oldid=52635"