14

Part of 2004 AIME Problems

Problems(2)

Source: AIME 2004 #14

A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is

\frac{a-\sqrt{b}}c

a, b,

c

are positive integers, and

c

a+b+c

Pythagorean Theoremgeometrymathicorn

Source: AIME 2004 Alt. #14

Consider a string of

n

7777\cdots77

+

signs are inserted to produce an arithmetic expression. For example,

7+77+777+7+7=875

could be obtained from eight 7's in this way. For how many values of

n

is it possible to insert

+

signs so that the resulting expression has value 7000?

LaTeXmodular arithmetic