# Problem of the Fortnight

## March 7: Natural Numbers and Their Digits

$Show\; that\; for\; all\; natural\; numbersn,\; at\; least\; one\; of\; two\; numbers,norn+1,\; can\; be\; represented\; in\; the\; following\; form:k+S\left(k\right)for\; some\; integerk,\; whereS(k)\; is\; the\; sum\; of\; all\; digits\; ink.For\; example,21=15+(1+5).$
**Answers are due by 5:00 PM on Friday, March 7. Please show all of
your work. Good Luck!
**

## February 14: Finding the Center of My Heart

A heart shape is formed from a square with sides 2 units long and two
half-circles of radius 1 unit (see figure). Where should the heart be
placed in the Cartesian plane so that the origin is at the center of
the heart, i.e. so there is the same amount of area in each of the four
quadrants?
**Answers are due by 5:00 PM on Friday, February 14. Please show all
of your work. Happy Valintines Day!**