MathDB

5

Part of 2006 AIME Problems

Problems(2)

Square Roots

Source: 2006 AIME A Problem 5

3/9/2006
The number 1046+46810+14415+2006 \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} can be written as a2+b3+c5,a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, where a,b,a, b, and cc are positive integers. Find a⋅b⋅ca\cdot b\cdot c.
USAMTSAMCRadicals
Dice rolling

Source: 2006 AIME II 5

3/28/2006
When rolling a certain unfair six-sided die with faces numbered 1,2,3,4,51, 2, 3, 4, 5, and 66, the probability of obtaining face FF is greater than 16\frac{1}{6}, the probability of obtaining the face opposite is less than 16\frac{1}{6}, the probability of obtaining any one of the other four faces is 16\frac{1}{6}, and the sum of the numbers on opposite faces is 77. When two such dice are rolled, the probability of obtaining a sum of 77 is 47288\frac{47}{288}. Given that the probability of obtaining face FF is mn\frac{m}{n}, where mm and nn are relatively prime positive integers, find m+nm+n.
probabilityquadraticsnumber theoryrelatively primealgebraquadratic formulaAMC