Reference no: EM131084493
Putnam TNG - Polynomials
1: Find polynomials f(x), g(x), and h(x), if they exist, such that for all x,
2: Let k be a fixed positive integer. The n-th derivative of 1/xk-1 has the form Pn(x)/(xk-1)n+1 where Pn(x) is a polynomial. Find Pn(1).
3: Determine necessary and sufficient conditions for the equations:
x + y + z = A x2 + y2 + z2 = B x3 + y3 + z3 = C
to have a solution (possibly complex) for (x, y, z) with at least one of x, y or z equal to zero.
4: Let f(x) be a polynomial with real coefficients. If f(x) ≥ 0 for all real x, prove there are polynomials p1(x), p2(x), . . . , pk(x) for which
f(x) = i=1∑kpi2(x).
5: Let f be a non-constant polynomial with positive integer coefficients. Prove that if n is a positive integer, then f(n) divides f(f(n) + 1) if and only if n = 1.
6: Let P(x) be a polynomial of degree n with real coefficients and only real zeroes. Prove that for every real x, (n - 1) [P'(x)]2 ≥ nP(x)P''(x), and determine for which P(x) equality is obtained.
7: Let p(x) = x5 + x and q(x) = x5 + x2. Find all pairs (w, z) of complex numbers with w ≠ z for which p(w) = p(z) and q(w) = q(z).
8: Let n be a positive integer. Find the number of pairs P, Q of polynomials with real coefficients such that (P(X))2 + (Q(X))2 = X2n + 1 and deg P > deg Q.
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Find the expected value vector
: Find the expected value vector E[Y], the correlation matrix RY, and the covariance matrix CY.
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Describe the importance of a project charter
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Is that conclusion reasonable
: A newspaper article recently reported that couples who live together before marriage are less likely to stay married than are couples who don't live together before marriage. The article suggested that people who want to have long- lasting marriag..
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Perhaps observational techniques
: Please provide a discussion of one of these methods identified in the assignment you will be completing this week. Does your chosen study use a survey to collect participant data, or perhaps observational techniques are used?
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Determine necessary and sufficient conditions for equations
: Let k be a fixed positive integer. The n-th derivative of 1/xk-1 has the form Pn(x)/(xk-1)n+1 where Pn(x) is a polynomial. Find Pn(1). Determine necessary and sufficient conditions for the equations: x + y + z = A, x2 + y2 + z2 = B, x3 + y3 + z3 = C
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American founding principles
: Consider combined hazards. Describe an appropriate response to the hazard (s) that you selected as most dangerous. Be sure to link back to one or more American Founding Principles.
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Why should you be skeptical
: Having read about correlation and causation, why should you be skeptical? What are some possible reasons that there could be a correlation between sitting up front and getting good grades? Could you design an experiment to test whether sitting up ..
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Question regarding the principles of design
: Describe how the principles of design are similar to the way we accomplish order in our daily lives. For example, you could relate rhythm to the things you pass on your way to work.
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Contrast risk of the us portfolio and the global portfolio
: Determine what percentage of funds should be invested in each US and foreign stockthat to provide the lowest risk domestic portfolio - Contrast the risk of the US portfolio and theglobal portfolio
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