POTD 2007-01

Table of contents

January 2007

Wednesday, 31th of January

Crito Find all real polynomials that p(x + p(x)) = p(x) + p(p(x))

Solutions by int-e and landen.

Thursday, 18th of January

from oishdf

Find the limit \lim_{n\rightarrow \infty} \frac{1\cdot 3 \cdot 5 \cdots 2n-1}{2\cdot 4 \cdot 6 \cdots 2n}

Solution by landen using methods from the famous Honours Pre-Calc III course.

Another solution by Polytope.

Yet another solution by landen

Wednesday, 17th of January

from Mathica

If n > 0 is a natural and x_{-n} < \dots < x_0 < \dots < x_n are real, prove that the function M\mapsto \sum_{i=-n}^{n} |x_i - M| is minimized on \mathbb{R} when M = x0.

Wednesday, 10th of January

from Zabrien

Show that given a natural number n, the following two statements are equivalent: "neither 2 nor 5 divides n" and "there exists a number a, which can be written using only the digit 3, such that n divides a"

Tuesday, 9th of January

created by brett1479

(a) Let f(x) = sin(x) / x3 on [2\pi,\infty). The region enclosed by f and the positive x-axis is rotated about the y- axis. Compute the volume of the solid swept out by this process. The famous #math integral committee thinks this must be computed numerically.

(b) The region enclosed by f(x) = x, the line x = 4, and the positive x-axis is rotated about the line y = 3x. What is the volume of the solid swept out by this process?

(c) Suppose X and Y are independent random variables with standard normal distributions. In other words, their pdfs are given by

      f_X(x) = f_Y(x) = \frac{e^{-x^2/2}}{\sqrt{2\pi}},\qquad -\infty<x<\infty.

Compute the probability that X^{\,2}+Y^{\,2}<1.

Solution part (b) by i_c-Y.
Solution part (c) by i_c-Y.

Saturday, 6th of January

edited by landen

Find all natural numbers a,b,c\, greater than one such that (a-1)(b-1)(c-1)\, divides abc-1\, exactly.

Inelegant solution done by landen

Tuesday, 2nd of January

edited by landen

Find all natural numbers n\, such that n^2 \,| \,2^n+1\,