The Art and Craft of Problem Solving

156 CHAPTER 5 ALGEBRA

Problems and Exercises

5.2.24 (AIME 1987) Find 3.?l if x, y are integers

such thatl+3x^2 l = 30x^2 +517.

5.2.25 Find all positive integer solutions (x,y) to

(a) x^2 - l = 20.

(b) xy +5x+3y = 200.

5.2.26 (Mathpath 2006 Qualifying Quiz) Suppose

that a, b, c, d are real numbers such that

Show that

a^2 +b^2 = I

c^2 +d^2 = I

ac+bd =0.

a^2 +c^2 = I

b^2 +d^2 = I

ab+cd =0.

This problem can get very messy, but doesn't have to.

Strive to find an elegant and complete solution.

5.2.27 (AIME 1988) Find the smallest positive inte­

ger whose cube ends in 888 (of course, do this without

a calculator or computer).

5.2.28 Find the minimum value of xy + yz + xz, given

that x,y, z are real and x^2 + l + z^2 = 1. No calculus,

please!

5.2.29 (AIME 1991) Find x^2 + l if x, yEN and

xy +x+ y = 71, .?y +xi = 880.

5.2.30 Find all integer solutions (n,m) to

n^4 + 2n^3 + 2n^2 + 2n + I = m^2.

5.3 Sums and Products

Notation

5.2.31 (AIME 1989) Assume that XI ,X 2 , ... ,X 7 are

real numbers such that

Xl +4X 2 +9X 3 + 16x 4 + 25x 5 + 36x6 + 49x 7 = I

(^4) Xl + 9X 2 + 16x 3 + 25x 4 + 36x 5 + 49x6 + 64X 7 = 12
9 Xl + 16x 2 + 25x 3 + 36x 4 + 49x 5 +64X6 + 81x 7 = 123.
Find the value of
5.2.32 Show that each number in the sequence
49,4489,444889,44448889, ...
is a perfect square.
5.2.33 (Crux Mathematicorum, June/July 1978) Show
that n^4 - 20n^2 + 4 is composite when n is any integer.
5.2.34 If x^2 + l + z^2 = 49 and X + y + z = x^3 + y^3 +
z^3 = 7, find xyz.
5.2.35 Find all real values of X that satisfy (16x^2 -
W + (9x^2 - 16)^3 = (25x^2 - 25)^3.
5.2.36 Find all ordered pairs of positive integers (x,y)
that satisfy x^3 -y^3 = 721.
5.2.37 (Crux Mathematicorum, April 1979) Deter­
mine the triples of integers (x,y, z) satisfying the equa­
tion
x^3 +y^3 +z^3 = (x+ y+z)^3.
5.2.38 (AIME 1987) Compute
(10^4 + 324)(22^4 + 324) ··· (58^4 + 324)
(4^4 + 324)(16^4 +324) ... (52^4 + 324)
.
The upper-case Greek letters L (sigma) and II (pi) are used respectively for sums
n n
and products. We abbreviate the sum XI +X 2 + ... +xn by ,6 Xi· Likewise, [lXi
indicates the product XIX 2 " ·Xn• The variable i is called the index, and can of course
be denoted by any symbol and assume any upper and lower limits, including infinity.
If the indices are not consecutive integers, one can specify them in other ways. Here
are a few examples.