POTD 2007-01

Table of contents
1 January 2007

1.1 Wednesday, 31th of January
1.2 Thursday, 18th of January
1.3 Wednesday, 17th of January
1.4 Wednesday, 10th of January
1.5 Tuesday, 9th of January
1.6 Saturday, 6th of January
1.7 Tuesday, 2nd of January

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\,

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