# New exercises and problems in Mathematics

November 1998

## New exercises for practice in November 1998 |

**C. 517.**A fair dice is thrown until a total score of 3 or more is obtained. How many times, on the average, has the dice to be thrown?

**C. 518.** Solve the following system of equations:

*x*^{2}+*y**z*=0*,
v*

^{2}+

*yz*=0

*,*

(

*x*+

*v*)

*y*=2

*,*

(

*x*+

*v*)

*z*=-2.

**C. 519.** Given a circle of unit radius, construct a square
such that two consecutive vertices of the square lie on the circle, and the
side connecting the other two vertices is tangent to the circle. Determine the
length of the side of the square.

*Fried Katalin*
(Budapest)

**C. 520.** In three dimensional Euclidean space, consider the
sphere of radius 3 centered at the origin. Find the number of lattice points
(points with integer coordinates) that lie on the sphere.

## New exercieses in November 1998 |

**Gy. 3230.** Is there any sequence *a*_{1}, *a*_{2}, ...
of natural numbers such that, for every natural number *k*, the sequence
*a*_{1}+*k*, *a*_{2}+*k*, *a*_{3}+*k*, ... contains only a finite number of primes?

**Gy. 3231.** Do there exist integer numbers *a*,
*b*, *c* such that each of the equations

*a**x*^{2}+*b**x*+*c*=0
and
(*a*+1)*x*^{2}+(*b*+1)*x*+(*c*+1)=0

has two real solutions?

*Russian contest problem*

**Gy. 3232.** How many hundred-digit natural numbers can be
formed such that only even digits are used and any two consecutive digits
differ by 2?

**Gy. 3233.** Given 100 different positive integers, prove
that some 98 of them can be selected such that their sum is not divisible by
the sum of the two remaining numbers.

**Gy. 3234.** Partition a rectangle into 9 rectangular pieces such
that no two pieces can be arranged to form an other rectangle.

**Gy. 3235.** There are given in the plane two segments
*A**B* and *C**D*. Construct a point *P* such that
triangles *A**P**B* and *C**P**D* are similar,
having equal angles at vertex *P*.

**Gy. 3236.** In an acute-angled triangle *A**B**C*, *T*_{a} denotes the foot of the altitude dropped from vertex *A*. Points *T*_{b} and *T*_{c} are defined in a similar way. Prove that the sum *T*_{a}*C*+*T*_{b}*A*+*T*_{c}*B* is at least as large as the perimeter of triangle *T*_{a}*T*_{b}*T*_{c}.

*Árokszállási Tibor* (Paks)

**Gy. 3237.** Is it possible to intersect a cube by a plane in a) a regular pentagon b) a regular hexagon?

## New problems in November 1998 |

**F. 3250.**Find all functions

*f*:(0

*,*1)

**R**that satisfy

*f*(*x*^{.}*y*)=*x*^{.}*f*(*x*)+*y*^{.}*f*(*y*).

**F. 3251.** At a ping-pong contest, each player played one
game with each of the other players. A player *A* won a prize if, for any
other player *B*, either *A* won against *B*, or *A* beat
someone who won against *B*. Prove that if exactly one contestant won a
prize, then he won against each of the other players.

*Chinese contest problem*

**F. 3252.** How many different ways are there to arrange the
numbers 1,2,...,*n* in a single row such that every number, except the
number which occurs first, is preceded by at least one of its original
neighbours?

**F. 3253.** Given a circle *k* and a point *P* that
lies on the circle, construct the line tangent to *k* at *P*, using
a straight-edge alone.

**F. 3254.** A transformation of the
plane maps the point *P*(*x*;*y*)
onto
if ,
and onto itself if |*x*|=|*y*|. Describe the images of all straight
lines under the above transformation.

**F. 3255.** There are given a sphere of unit radius and four pairwise nonintersecting balls whose centres lie on the sphere. The sum of the radii of the balls equals 2. Prove that at least one quarter of the surface area of the sphere is covered by the balls.

## New advanced problems in November 1998 |

**N. 187.** Four distinct points *A*, *B*, *C*
and *M* are given on a sphere, none of which is opposite to any
other. Prove that if the great circles *A**M* and *B**M*
are orthogonal to great circles *B**C* and *A**C*,
respectively, then also the great circle *C**M* is orthogonal to
great circle *A**B*.

**N. 188.** A tree graph is said to be of type
*p*^{n} if it has *n* levels in
addition to its root, and at every vertex, with the exception of the vertices
of the topmost level, branches out into *p* directions.

a) The leaves
(topmost vertices) of a graph of type 4^{n} are coloured by 3*k* colours. Prove that
the graph has a subgraph of type 2^{n}
whose leaves bear at most *k* different colours.

b) Show, for each
positive integer *k*, a graph of type 4^{n} whose leaves can be coloured by 3*k*+1
colours such that any of its subgraphs of type 2^{n} has *k*+1 leaves which bear different
colours.

*Keleti Tamás* (London)

**N. 189.** Prove that arbitrary positive integers
*a*_{1}, ..., *a*_{n} satisfy the following inequality:

.

**N. 190.** A polynomial *p* has degree at most
. Prove that

.