POTD 2006-06

Table of contents

1 June 2006

1.1 Tuesday, June 27, 2006
1.2 Sunday, June 25, 2006
1.3 Saturday, June 24, 2006
1.4 Friday, June 23, 2006

June 2006

Tuesday, June 27, 2006

from Karlsen`

Show that numbers of the form 4444...44 (2 $n$ 4's in a row) - 88...8 ( $n$ 8's in a row) are alway perfect squares. examples:

44-8 = $62$ ; 4444444444-88888 = $666662$

solution by landen

Sunday, June 25, 2006

from kmh

problem (http://kilian.byethost5.com/mytemp/landen.pdf)

solution (http://www.efnet-math.org/old/solution_potd6_25.pdf) from kmh

Saturday, June 24, 2006

from tommi

Find the minimum value of $f (x, y, z) = x + 2 y + 3 z$ on the curve formed by the intersection of the plane $x - y + z = 1$ and the cylinder $x 2 + y 2 = 1$

Solution (http://www.efnet-math.org/old/prob062406.pdf) from tommi

Solution (http://www.efnet-math.org/old/solution_potd6_24.pdf) with nice graphics from kmh

Friday, June 23, 2006

This problem is very easy. Please do not give away the answer until 23:59 UTC today.

For any positive integer $n\,$ define a sequence $\{n_k\}_{k=0}^\infty$ as follows: Set $n_0 = n\,$ , and for each $k \ge 1\,$ let $n_k\,$ be the sum of the (decimal) digits of $n_{k-1}.\,$

The sequence eventually becomes a constant sequence of single digits. For example, for 5137 it is 5137,16,7,7,7,...

Find the constant limit digit for $n=2^{2006}\,$

Solution (http://www.efnet-math.org/old/solution_potd6_23.pdf) from kmh

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