# New exercises and problems in Mathematics

October 1997

## New exercises for practice in October 1997

**C. 477.** A game machine accepts two kinds of coins: red and
green ones. For each coin the machine returns 5 coins of the other
kind. One starts playing with 1 green coin. Can it happen that after
a while he has as many green coins as red ones?

**C. 478.** In an arithmetic progression, the sum of the first
*n* terms and the sum of the first 2*n* terms are *A*
and *B*, respectively. Express the sum of the first 3*n*
elements in terms of *A* and *B*.

**C. 479.** The lengths of the bases of a trapezium are *a*
and *c*, respectively. Find the length of the segment parallel to
the bases which halves the area of the trapezium.

**C. 480.** In a tetrahedron, two of the faces are equilateral
triangles of unit side. The two other faces are right isosceles
triangles. Find the volume of the tetrahedron.

## New exercises in October 1997

**Gy. 3150.** In a certain town, each bus line has 3 stops. Any
two bus-stops are connected with a line, and any two lines have a
common bus-stop. How many bus lines may be there in the town?

**Gy. 3151.** For any positive integer *n*, let
*A _{n}* denote the set of positive integers which are not
relatively prime to

*n*. Find all integers

*n*for which implies .

**Gy. 3152.** The real numbers *a*, *b*, *c*,
*d* satisfy and .
Find all possible values of .

**Gy. 3153.** Find the minimum value of expression
, where *a* and *b* are
positive parameters.

**Gy. 3154.** The sides of a triangle *ABC* are denoted by
*a*, *b*, *c*, according to the standard notation. The
lengths of the tangents from vertices *A*, *B*, *C* to
the incircle are *x*, *y* and *z*, respectively.
Assuming , prove inequality .

**Gy. 3155.** Prove that in any triangle, a line passing through
the incenter halves the perimeter of the triangle if and only if it
halves the area of the triangle.

**Gy. 3156.** In a triangle, the segments of the medians lying
inside the inscribed circle are all of the same length. Does this imply
that the triangle is equilateral?

**Gy. 3157.** In a tetrahedron, the segments of the medians lying
inside the inscribed sphere are all of the same length. Does this imply
that the tetrahedron is regular?

## New problems in October 1997

**F. 3190.** Is there any real number *x*>1 which is not an
integer and still satisfies inequality ?

**F. 3191.** The positive integers *n* and *p* satisfy
and . Colour *p* vertices
of a regular *n*-gon with red, colouring the remaining vertices with
blue. Prove that there exist two congruent polygons of at least
vertices such that each vertex of the first
polygon is red and each vertex of the second one is blue.

**F. 3192.** Is it possible to move a knight on a 5x5 chessboard so
that it returns to its original position after having visited each field
of the board exactly once?

**F. 3193.** May a non-planar quadrilateral have only right angles?

**F. 3194.** In a right triangle *ABC*, construct the point
*D* on the hypotenuse *AB* such that triangles *DCA* and
*DCB* have equal inradii.

**F. 3195.** The whole surface of a cube shaped cake (including
its bottom) is covered with chocolate. The cake is to be distributed
among *K* people so that each of them receives the same amount of
cake, and also the same amount of chocolate cover at the same time. In
order to accomplish this task, we divide the cake into
*N*x*N*x*N* alike cube shaped pieces, and give everybody
the same number of pieces, taking care that the total surface area of
the chocolate covered sides of the pieces one gets is also the same
for everybody. Is it possible to realize this for an arbitrary
*K*? At least how many pieces are necessary for *K*=1997?

## New advanced problems in October 1997

**N. 148.** There are given pairwise
distinct points in the plane. Prove that there are three distinct
points *A*, *B* and *C* among them which satisfy
inequality .

**N. 149.** The sequence (*a _{n}*) is defined by the
recursion

*a*

_{0}=

*a*

_{1}=1, (

*n*+1)

*a*

_{n+1}=(2

*n*+1)

*a*+3

_{n}*na*

_{n-1}. Prove that the sequence consists of integer numbers.

**N. 150.** There are given in the plane a parabola and points
*P* and *Q* outside the parabola such that line *PQ*
passes through the focus of the parabola. Draw two tangents to the
parabola from each of points *P* and *Q*. Prove that the
four points of intersection obtained this way lie on the same circle.

**N. 151.** The sequence
*a*_{2}, *a*_{3}, ...
consists of positive real numbers such that
converges. Prove that series
is also convergent.