MAΘ Problem Set 3
January 20, 2004
The Mississippi School for Mathematics and Science
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1. Given that r is irrational, prove that r is also irrational. (A rational
number is one that can be expressed in the form a/b where a and b are
integers).
2. For what integers n do fractions of the form m/n have decimal representations that terminate?
3. Two common identities are (a + b)2 = a2 + 2ab + b2 and (a + b)3 =
a3 +3a2 b+3ab2 +b3 . Generalizing this to all powers, the binomial theorem
states that
n
X
(nj )an−j bj
(a + b)n =
j=0
How many digits does (1011 + 1)6 − (1011 − 1)6 have?
4. There exists a fraction equivalent to the following with an integral denominator : √3+√15−√7 . What is the denominator of this fraction when it is
written in lowest terms?
5. The AM-GM inequality states that the arithmetic mean of a set of positive
numbers is greater than or equal to the geometric mean of the set. (Recall
that, for a set of n terms, the geometric mean of a set is the nth root of
the product of the terms). Prove that (a + b + c)3 − 25abc is positive for
any positive a, b, and c.
6. A graceful gazelle is standing 108 meters west of a hungry cheetah. The
fleeing gazelle begins running south at a hasty 24 meters per second. The
cheetah pursues at a blistering pace of 30 meters per second. Assuming the
cheetah understands the math involved in optimizing the path of pursuit,
what is the minimum amount of time that it will have to sustain its pace
in order to obtain its meal?