# New exercises and problems in Mathematics

January 2000

## Please read The Conditions of the Problem Solving Competition.

## New exercises in January 2000 |

**C. 565. **Ten eggs are broken up into a bowl, one
after the other. Two of them are bad, but this turns out only when
they are being broken up, so they spoil those eggs already in the
bowl. Thus, when a bad egg is found, the bowl is washed up, and the
procedure is continued with the remaining eggs. What proportion, on
the average, of the good eggs get wasted?

**C. 566. **Solve the following equation:

**C. 567. **The lengths of the edges of a rectangular
box are all integers. Its volume, half of its surface area, and the
lengths of its edges starting at the same vertex add up to
2000. Determine the lengths of the edges of the box.

**C. 568. **How many different ways are there to
select 5 out of the first 90 positive integers so that they form a) an
arithmetic b) a geometric progression?

**C. 569. **Determine those points of the line
*y*=*x *from which one can draw two perpendicular tangents
to the parabola *y*=*x*^{2}.

## New problems in January 2000 |

**B. 3332. **The first *n *positive integers are
to be arranged around a circle in such a way that the sum of any two
consecutive numbers is divisible by the next one in clockwise
order. Find those values of *n *for which this can be
done. (3 points)

Tournament of Towns, 1999.

**B. 3333. **How many faces of an icosahedron can be
chosen without any two of them sharing a common edge? (3 points)

**B. 3334. **Prove that there is exactly one number
system in which there exists a 3-digit number twice as large as the
number it represents in the decimal system. (4 points)

JavasoltaProposedBy: *N. Gyanta,* Budapest

**B. 3335. **A pentagon *K *and another polygon
*L *are given in the plane such that neither of them has a vertex
on any line that contains a side of the other. Find the maximum number
of intersection points their sides can form, assuming that a) *L
*is a triangle b) *L *is a quadrilateral. (5 points)

**B. 3336. **An equilateral triangle is rotated around
its center, in counterclockwise direction, by 3^{o}. Next, it is rotated further by another
9^{o}, then by 27^{o}, and so on, by 3^{k} degrees in the *k*th step. How many
different positions are occupied by the triangle during such a
procedure? (3 points)

**B. 3337. **Given the focus and two points of a
parabola, construct its directrix. (3 points)

**B. 3338. **Solve the following system of equations:

(4points)

JavasoltaProposedBy: M. Ábrány,Ukrajna

**B. 3339. **Prove that, in any spherical triangle,
the three medians meet at a common point. (4 points)

**B. 3340. **When rolling a die 12 times, which of the
following two is more probable: a) at least one of the first 6 scores
is 6, b) at least two of all the 12 scores are 6? (4 points)

**B. 3341. **In a triangle *ABC*, *D
*denotes the foot of the perpendicular dropped from *A*. The
pairwise distinct points *D*, *E *and *F *lie on the
same line, moreover, *AE *and *AF *are perpendicular to
*BE *and *CF*, respectively. The midpoints of segments *BC
*and *EF *are denoted by *M *and *N*. Prove that
*AN *is perpendicular to *MN*. (5 points)

Problem of the 10th Asian Mathematical Olympiad

## New advanced problems in January 2000 |

**A. 227. **Is there a positive integer *n *such
that, every digit, different from 0, appears the same number of times
in the decimal form of each of the numbers *n*, 2*n*,
3*n*, ..., 2000*n*?

**A. 228. **Let *Q *denote the set of rational
numbers. Suppose that the functions *f*, *g*: Q*Q *are strictly monotone
increasing functions which attain every rational value. Is it
necessarily true that the range of values of the function
*f*+*g *is also the whole set *Q*?

*E. Fried,* Budapest

**A. 229. **Two pentagons are given in the plane such
that neither of them has a vertex on any line that contains a side of
the other. Find the maximum number of intersection points their sides
can form.

### Send your solutions to the following address:

KöMaL Szerkesztőség (KöMaL feladatok), Budapest Pf. 47. 1255, Hungary

or by e-mail to: megoldas@komal.elte.hu.