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Xn belong to the interval [−1, 1] and the sum of their cubes is zero. Prove that their sum does not exceed n/3. 3. (IMO 1972/2) Let x1 , . . , x5 be positive reals such that (x2i+1 − xi+3 xi+5 )(x2i+2 − xi+3 xi+5 ) ≤ 0 for i = 1, . . , 5, where xn+5 = xn for all n. Prove that x1 = · · · = x5 . 4. (USAMO 1979/3) Let x, y, z ≥ 0 with x + y + z = 1. Prove that 1 x3 + y 3 + z 3 + 6xyz ≥ . 4 34 5. (Taiwan, 1995) Let P (x) = 1 + a1 x + · · · + an−1 xn−1 + xn be a polynomial with complex coefficients.

The bad news about this criterion is that determining whether a matrix is positive definite is not a priori an easy task: one cannot check M x · x ≥ 0 for every vector, so it seems one must compute all of the eigenvalues of M , which can be quite a headache. The good news is that there is a very nice criterion for positive definiteness of a symmetric matrix, due to Sylvester, that saves a lot of work. Theorem 28 (Sylvester’s criterion). An n × n symmetric matrix of real numbers is positive definite if and only if the determinant of the upper left k × k submatrix is positive for k = 1, .

Rabinowitz, Index to Mathematical Problems 1980-1984, Mathpro Press, Westford (Massachusetts), 1992.

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A is less than B by Kedlaya K.S.


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