A Mathematical ‘Hodgepodge’

A collection of problems to be discussed at the November 1999 Teachers’ *SIMMER* meeting

**Problem 1.** Which is greater: 99^{50} + 100^{50} or 101^{50}
?

**Problem 2.** (a) In how many ways can 8
rooks be placed on an 8 x 8 chessboard in such a way that no rook can take another?

(b) In how many ways can 4 rooks be placed on an 8 x 8 chessboard in such a way that no rook can take another?

**Problem 3. **Find the area of the region in the plane, bounded on the right by the
ellipse *x*^{2} + 3*y ^{2}* = 3 and on the left by the
straight line

Hint: this problem can be solved without using integration.

**Problem 4.** Let *X* be a figure in the plane. Assume that *X* is moved to
itself by a rotation about a point *O* by 48^{o}. Does it necessarily follow
that *X* is moved to itself by a rotation about *O* by 90^{o }? by 72^{o}
?

**Problem 5.** Prove that for every integer *n *> 2, (1.2.….*n*)^{2}
> *n ^{n}*.

**Problem 6.** Into how many parts is a plane divided by *n* straight lines, such
that no two lines are parallel and no three lines pass through the same point?

**Problem 7.** Find the last three digits of the sum

1

^{1999}+ 2^{1999}+ 3^{1999}+ 4^{1999}+^{…}+ 999998^{1999}+ 999999^{1999}

**Problem 8.** The monetary unit in the Republic of Oz is called an emerald, and both
three emerald and five emerald bills are in circulation. Prove that any sum greater than 7
emeralds can be paid by three and five emerald notes.

**Problem 9.** Suppose that a plane is divided into parts by *n* straight lines.
Prove that these parts can be coloured with red and white paint, such that any two parts,
having a common side, are coloured with different colours.

**Problem 10.** For a positive integer *x* with at least two digits, let *F*(*x*)
denote the integer obtained from *x* by deleting the first digit. Does there exist *x
*such that

a) x= 58.F(x) ?b) x= 57.F(x) ?

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*This page was last updated: December 14, 1999*