28 2 Algebra
85.Show that for an odd integern≥5,
(
n
0)
5 n−^1 −(
n
1)
5 n−^2 +(
n
2)
5 n−^3 −···+(
n
n− 1)
is not a prime number.
86.Factor 5^1985 −1 into a product of three integers, each of which is greater than 5^100.
87.Prove that the number
5125 − 1
525 − 1
is not prime.
88.Letaandbbe coprime integers greater than 1. Prove that for non≥0isa^2 n+b^2 n
divisible bya+b.
89.Prove that any integer can be written as the sum of five perfect cubes.
90.Solve in real numbers the equation√ (^3) x− 1 +√ (^3) x+√ (^3) x+ 1 = 0.
91.Find all triples(x,y,z)of positive integers such that
x^3 +y^3 +z^3 − 3 xyz=p,
wherepis a prime number greater than 3.
92.Leta, b, cbe distinct positive integers such thatab+bc+ca≥ 3 k^2 −1, wherek
is a positive integer. Prove that
a^3 +b^3 +c^3 ≥ 3 (abc+ 3 k).
93.Find all triples(m, n, p)of positive integers such thatm+n+p=2002 and the
system of equationsyx+yx=m,yz+yz=n,zx+xz=phas at least one solution
in nonzero real numbers.
2.1.2 x^2 ≥0..................................................
We now turn to inequalities. The simplest inequality in algebra says that the square of
any real number is nonnegative, and it is equal to zero if and only if the number is zero.
We illustrate how this inequality can be used with an example by the second author of
the book.