MathDB

8

Part of 2014 AIME Problems

Problems(2)

Four-Digit Sequence

Source: 2014 AIME I Problem 8

3/14/2014
The positive integers NN and N2N^2 both end in the same sequence of four digits abcdabcd when written in base 10, where digit aa is not zero. Find the three-digit number abcabc.
modular arithmeticSuMACquadraticsnumber theorygreatest common divisorAMCAIME
Lots of Tangency

Source: 2014 AIME II Problem #8

3/27/2014
Circle CC with radius 22 has diameter AB‾\overline{AB}. Circle DD is internally tangent to circle CC at AA. Circle EE is internally tangent to circle C,C, externally tangent to circle D,D, and tangent to AB‾\overline{AB}. The radius of circle DD is three times the radius of circle EE and can be written in the form m−n,\sqrt{m} - n, where mm and nn are positive integers. Find m+nm+n.
LaTeXtrigonometryAsymptotegeometrygeometric transformationreflectionPythagorean Theorem