# New exercises and problems in Mathematics

November 1997

## New exercises for practice in November 1997

**C. 481.**
There is a company sitting at a round-table. They all leave the table
for a while, and after returning to the table they all find that their
neighbours differ from those they had previously. How many people may be
sitting around the table?

**C. 482.**
Prove that if *y*^{3}*x*+1<*x*+*y*^{3}
holds for some real numbers *x* and *y*, then they also satisfy
inequality *x*^{3}*y*+1<*y*+*x*^{3}.

**C. 483.**
The following question was presented at a Jeopardy show. Which plane figure
has the property that its area is half the product of its diameters?
According to te official answer, this is `the kite'. Are kites the only
plane figures with the above property?

**C. 484.** In a right circular cone, the angle formed by the axis
and a generator measures . Consider the
ball inscribed in the cone. Find the ratio of the volume of the ball and
that of the cone.

## New exercieses in November 1997

**Gy. 3158.** The number
(as written in the decimal system) is divisible by 7. Prove that 7 can
be cancelled out of the fraction .

**Gy. 3159.** Find the smallest positive integer *n* for which
every real number *x* satisfies inequality
.

**Gy. 3160.** In an infinite arithmetic progression of distinct
positive integers, we replace each term by the sum of its digits. Can
it happen that the new sequence is again an arithmetic progression?

**Gy. 3161.** At a meeting there are *n* guest, including
Mr. Smith. There is also a journalist there who is looking for Mr. Smith.
He is aware of the fact that no one at the meeting knows Mr. Smith, who
nevertheless knows everybody there. The journalist may approach any guest,
point at someone, and ask the guest if s/he knows that specific person.

- a) Can the journalist find Mr. Smith for sure with less than

*n*questions?

- b) What is the minimum number of questions he has to ask before he can
identify Mr. Smith?

**Gy. 3162.** The lengths of the legs of a right triangle are *a*
and *b*, respectively. Draw a circle of radius *a* centered at one
endpoint of the hypotenuse, and also a circle of radius *b* centered at
the other endpoint. Prove that the segment of the hypotenuse lying in the
intersection of these two discs is as long as the diameter of the incircle
of the triangle.

**Gy. 3163.** Find the minimum number of different lines determined
by *n* non-collinear points in the plane.

**Gy. 3164.** In a triangle, let *S*, *M*, *O* and
*K* denote respectively the centroid, the orthocentre, the incentre
and the circumcentre of the triangle. Suppose that some two of these points
coincide. In which of the six possible cases does this imply that the
triangle is equilateral?

**Gy. 3165.** In a teterahedron it can happen that an altitude goes
outside the polytope. How many such altitudes may a tetrahedron have?

## New problems in November 1997

**F. 3196.** Define a sequence *a _{n}* by

*a*

_{0}=1,

*a*

_{n+1}=sin

*a*(

_{n}*n*=0,1,2,...). Prove that the sequence

*na*

_{n}^{2}is bounded.

**F. 3197.** Some pieces are placed on an 8x8 chessboard such that
there are exactly 4 pieces in each raw and coloumn of the board. Prove
that there can be found 8 pieces among them such that no two of them are
in the same row or coloumn.

**F. 3198.** The numbers
*a*_{1}, *a*_{1}, ..., *a _{n}*
are pairwise distinct positive integers. Prove inequality
.

**F. 3199.** In a triangle *ABC*, the lines which divide the
angle at vertex *C* into *n* equal parts intersect *AB*
at points
*C*_{1}, *C*_{2}, ..., *C*_{n-1}
in this order. Prove that the value of fraction
does not depend on *n*.

**F. 3200.** We are given two segments *AB* and *CD*.
Fix *AB* and move *CD* around, in a position parallel to
*AB*, such that quadrilateral *ABCD* is a trapezoid in which a
circle can be inscribed. Determine the locus of the midpoints of the
segments *CD*.

**F. 3201.** Let *M* be an arbitrary interior point of a
tetrahedron *ABCD*. Let *N*, *P*, *Q* and
*A*_{1}, respectively, denote the intersection points of
planes *BCM*, *CDM*, *BDM*, and *BCD* with lines
*AD*, *AB*, *AC* and *AM*. Let finally
*D*_{1}, *B*_{1} and *C*_{1}
denote the points where lines *A*_{1}*N*,
*A*_{1}*P* and *A*_{1}*Q* intersect
the plane incident to *A* and parallel to *BCD*. Prove that
*A* is the centroid of triangle
*B*_{1}*C*_{1}*D*_{1}.

## New advanced problems in November 1997

ATTENTION! We correct and submit **N. 146.** again. (See KöMaL 1997/6.):

**N. 146.** Prove that any rational number 0<*r*<1 of odd
denominator can be expressed as the fraction part of a number of the form
with suitable integers *x*,
*y* and *z*.

**N. 152.** There are given finitely many discs of unit radius in
the plane such that the centres of any two of them are at least 10 units
apart. Is it true that there exists a (not necessarily closed) polygon
whose vertices are the centres of the given discs such that the polygon
contains the centre of each disc, and each segment of the polygon
intersects only those discs which are centered at its endpoints.

**N. 153.** The functions
are defined by ,
.

- a) Prove that

*f-g*is bounded.

- b) Does the left-hand limit
exist?

**N. 154.** Colour each point of the plane according to the following
rule: the colour of the point (*x*; *y*) is white if
[*x*]+[*y*] is even, and is black otherwise. For arbitrary
positive integers *a* and *b*, consider the triangle with
vertices (0; 0), (0; *a*) and (*b*; 0); and denote by
*f*(*a*,*b*) the absolute value of the difference between
the areas of the white resp. black regions. Prove inequality
, where *u _{n}*
denotes the

*n*th term of the Fibonacci sequence, that is,

*u*

_{1}=1,

*u*

_{2}=1,

*u*

_{n+2}=

*u*

_{n}+

*u*

_{n+1}.