MathDB

14

Part of 2006 AIME Problems

Problems(2)

Broken Tripod

Source: 2006 AIME A Problem 14

3/9/2006
A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let hh be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then hh can be written in the form mn,\frac m{\sqrt{n}}, where mm and nn are positive integers and nn is not divisible by the square of any prime. Find ⌊m+n⌋.\lfloor m+\sqrt{n}\rfloor. (The notation ⌊x⌋\lfloor x\rfloor denotes the greatest integer that is less than or equal to xx.)
floor functiongeometry3D geometrypyramidsymmetrytrigonometrytetrahedron
Short and fun.

Source: 2006 AIME II 14

3/28/2006
Let SnS_n be the sum of the reciprocals of the non-zero digits of the integers from 1 to 10n10^n inclusive. Find the smallest positive integer nn for which SnS_n is an integer.
number theoryleast common multiplerelatively primeAMC