MathDB

6

Part of 2002 Putnam

Problems(2)

Putnam 2002 A6

Source:

12/8/2008
Fix an integer b≥2 b \geq 2. Let f(1) \equal{} 1, f(2) \equal{} 2, and for each n≥3 n \geq 3, define f(n) \equal{} n f(d), where d d is the number of base-b b digits of n n. For which values of b b does \sum_{n\equal{}1}^\infty \frac{1}{f(n)} converge?
Putnamgeometryintegrationlogarithmsfunctioncollege contestsPutnam calculus
Putnam 2002 B6

Source:

3/12/2012
Let pp be a prime number. Prove that the determinant of the matrix [xyzxpypzpxp2yp2zp2] \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} is congruent modulo pp to a product of polynomials of the form ax+by+czax+by+cz, where aa, bb, and cc are integers. (We say two integer polynomials are congruent modulo pp if corresponding coefficients are congruent modulo pp.)
Putnamlinear algebramatrixalgebrapolynomialmodular arithmeticbinomial theorem