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Miscellaneous Problems 233

23. (a) Let nl, 122,... , nd be a set of positive integers exceeding 1, with
    any pair relatively prime. Let ai be the number of prime factors
    of ni counting repetitions when ni is written as a product of
    primes. Define

p(x) = f: z - c $!$L + c +;;;;^ -...

i=l i#j

where the right side is the sum of 2* - 1 terms. Prove that

p(x) is monotonically increasing on the closed interval [0,2], that

p(x) 5 1 there and p(x) can assume the value 1 if and only if

one of the ni is a power of 2.

(b) Let m and n be two positive integers exceeding 1, and let a and b,

respectively, be the number of prime factors counting repetitions

when m and n are written as the product of primes. Suppose

the least common multiple of m and n is k and that c is the

number of prime factors of k. Let

p(x) = x0/m + x*/n - xc/k.

Show that p(z) is increasing on the closed interval [0,2], p(x) 5 1

there with equality possible if and only if either m or n is a power

of 2.

24. Find polynomials p(x) and q x) over Z such that
    p(fi+d3+fi)/q(Jz+ $ 3+fi)=Jz+&


25. Consider the equation

j/igZTZ+J~=J x2+9x+3p+9 (*)

in which x and p are real and the square roots are real and nonneg-

ative. Show that, if (*) holds, then

(x2 + 2 - p)(z2 + 8x + 2p + 9) = 0.

Hence, find the set of real p such that (*) is satisfied by exactly one

real number x.

26. What condition must be satisfied by the coefficients u, v, w of the
    polynomial
    X3 -ux2+vx-w
    in order that the line segments whose lengths are the zeros of the
    polynomial can form a triangle.