Reference no: EM131084612
Putnam TNG - Polynomials and Binomials
1: Define polynomials fn(x) for n ≥ 0 by f0(x) = 1, fn(0) = 0 for n ≥ 1, and d/dxfn+1(x) = (n + 1)fn(x + 1) for n ≥ 0. Find, with proof, the explicit factorization of f100(1) into powers of distinct primes.
2: Prove that the number of odd binomial coefficients in any finite binomial expression is a power of 2.
3: Consider all lines which meet the graph y = 2x4 + 7x3 + 3x - 5 in four distinct points, say (xi, yi), i = 1, 2, 3, 4. Show that x1 + x2 + x3 + x4/4, is independent of the line, and find its value.
4: Let a, b, c be real numbers such that a + b + c = 0. Prove that
a5 + b5 + c5/5 = (a3 + b3 + c3/3)·(a2 + b2 + c2/2).
5: Let p be a prime number. Show that
6: Let x(n) = x(x - 1)· · ·(x - n + 1) for n a positive integer, and let x(0) = 1. Prove that for all real numbers x and y
7: Let p(x) be a polynomial that is nonnegative for all real x. Prove that for some k, there are polynomials f1(x), ..., fk(x) such that
p(x) = j=1∑k(fj (x))2.
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Sense of modernization and integration
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Explain the differences between opinion and analysis
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Should you have separate goals for different part of company
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Discuss how article can enhance your professional decisions
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Find the explicit factorization with proof
: Define polynomials fn(x) for n ≥ 0 by f0(x) = 1, fn(0) = 0 for n ≥ 1, and d/dxfn+1(x) = (n + 1)fn(x + 1) for n ≥ 0. Find, with proof, the explicit factorization of f100(1) into powers of distinct primes
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What amount income is recognized by the beneficiaries
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Given the random variables x and y
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Loss adjustment expenses-net profitability in dollars
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Integrated circuit produces an acceptable circuit
: Each test of an integrated circuit produces an acceptable circuit with probability p, independent of the outcome of the test of any other circuit. In testing n circuits, let K denote the number of circuits rejected and let X denote the number of a..
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