Math Problems : SMT 2011
8:54 PM
Muhammad Yusuf
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1. Let a, b ∈ ℂ such that a + b = a2 + b2 = ⅔√3i. Compute |Re(a)|.
2. Consider the curves x2 + y2 = 1 and 2x2 + 2xy + y2 - 2x - 2y = 0. These curves intersect at two points, one of which is (1, 0). Find the other one.
3. If r, s, t, and u denote the roots of the polynomial f(x) = x4 + 3x3 + 3x + 2, find
1/r2 + 1/s2 + 1/t2 + 1/u2
4. Find the 2011th-smallest x, with x > 1, that satisfies the following relation:
sin(ln x) + 2 cos(3 ln x) sin(2 ln x) = 0
5. Find the remainder when (x + 2)2011 - (x + 1)2011 is divided by x2 + x + 1.
6. There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012.
Let x be one of these numbers. Find the maximum of x + x-1.
7. Let P(x) be a polynomial of degree 2011 such that P(1) = 0, P(2) = 1, P(4) = 2, ... , and P(22011) = 2011. Compute the coeficient of the x1 term in P(x).
8. Find the maximum of
for reals a, b, c, and d not all zero.
9. It is a well-known fact that the sum of the first n k-th powers can be represented as a polynomial in n. Let Pk(n) be such a polynomial for integers k and n. For example,
so one has
Evaluate P7(-3) + P6(-4).
10. How many polynomials P of degree 4 satisfy P(x2) = P(x)P(-x)?
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