# New exercises and problems in Mathematics

April 1998

## New exercises for practice in April 1998

**C. 501.** One morning in the school the integers from 1 through
a certain number were written on the blackboard. A student carefully erased
one of them. There would have been nothing special about it if someone had
not noted that the average of the remaining numbers had been
. Try to figure out which number was erased.

**C. 502.** Let us denote the roots of equation
*x*^{2}-2*bx*+*b*^{2}-*c*^{2}=0
by *x*_{1} and *x*_{2}. Prove that the roots of
equation
*x*^{2}-2*b*(*b*^{2}+3*c*^{2})*x*+(*b*^{2}-*c*^{2})^{3}=0
are and
.

**C. 503.** Given are two isosceles triangles. The inscribed
circles of the triangles touch the legs at their points of trisection
closer to the base of the triangle in one case and further from the base of
the triangle in the other case. In which case covers the inscribed circle a
greater portion of the area of the triangle?

**C. 504.** The points *A*, *B*, *C*, *D*,
*E*, *F* lie in the 3 dimensional space. How are the points
positioned if there exists a plane, in the same distance from each of the
points, which separates points *A*, *B*, *C*
from points *D*, *E*, *F*?

## New exercieses in April 1998

**Gy. 3198.** Do there exist positive integers
*a*, *b*, *c*, *d* such that *ab=cd* and
*a*+*b*+*c*+*d* is a prime number?

**Gy. 3199.** Find all integer solutions to equation
1996*x*+1998*y*+1=*xy*.

**Gy. 3200.** Let *H* be a 1000-element subset of the
set {0, 1, 2, ..., 1998}. Prove that it has two not
necessarily distinct elements *a* and *b* such that
*a*+*b* is a power of 2.

**Gy. 3201.** A 9x9 chessboard is is packed with 1x2 dominoes
such that only one corner of the board is left uncovered. Prove that the
uncovered field of the board can be transferred to any other corner by
shifting the dominoes about on the board.

**Gy. 3202.** A solid cube of edge length *n* cm is divided
into smaller cubes each with egde length 1 cm. Determine those values of
*n* for which a subset of the small cubes can be arranged to form a
cube with edge length 2*n* cm. The latter cube may be hollow.

**Gy. 3203.** The lengths of the diagonals of a rhomboid are
2*a* and 2*b*, respectively. The sides of the rhomboid together
with those tangents of the inscribed circle which are parallel to either
diagonal form an octagon. Express the area of the octagon in terms of
*a* and *b*.

**Gy. 3204.** The segments *AB* and *CD* are parallel.
*P* is an interior point of segment *BC*, the lines *AP* and
*CD* intersect in *E*. For which point *P* is the sum of the
areas of triangles *APB* and *CPE* is minimum?

**Gy. 3205.** Semicircles are drawn externally on each side of
the acute triangle *ABC*. The altitudes drawn from vertices *A,B,C*
intersect the semicircles in points *E,F,G*, respectively. Prove that
the hexagon *AGBECF* can be folded into a pyramid of base *ABC*.

## New problems in April 1998

**F. 3226.** In how many different ways can one colour the
circles depicted on the figure such that each arm contains exactly two
yellow and two blue circles? Two colourings are different if one cannot
be obtained from the other using a rotation or a reflection.

**F. 3227.** Find all values of *m* for which polynomial
*x ^{m}*+

*y*+

^{m}*z*-(

^{m}*x*+

*y*+

*z*)

*is divisible by (*

^{m}*y*+

*z*)(

*z*+

*x*)(

*x*+

*y*).

**F. 3228.** A bus line connects cities *A* and *B*.
The buses leave in every 30 minutes. Each bus has a capacity of 50
passengers. Those who do not fit in the bus wait in the waiting hall, and
those passengers who do not even fit there leave for *B* on foot.
Between two consecutive buses either 0, 25, 50 or 75 passengers arrive, with
probability 1/4, respectively. A new waiting hall is to be built to
replace the old one. It is to be built from modules, each of capacity 25.
At least how many such modules have to be used in order to ensure that the
probability that one has to leave on foot is smaller than 1%?

**F. 3229.** In a triangle of side lengths *a,b,c*,
the exradius belonging to side *c* is the geometric mean of the
exradii belonging to sides *a* and *b*, respectively. Express
*c* in terms of *a* and *b*.

**F. 3230.** Let *S* denote the centroid of triangle
*ABC*. Erect perpendiculars from each vertex of the triangle to the
external and internal bisectors of the angles belonging to the other two
vertices. Prove that the sum of the squares of the lengths of the resulting
perpendicular segments is
6(*SA*^{2}+*SB*^{2}+*SC*^{2}).

**F. 3231.** Given are a sphere *g* and a point *P*
in its interior. Let
*h*_{1}, *h*_{2} és *h*_{3}
be pairwise orthogonal chords passing through *P*. Each of the three
planes determined by the pairs of chords intersect the sphere in a
circle. Prove that the sum of the areas of the three circles is independent of
the position of the chords.

## New advanced problems in April 1998

**N. 171.** There are given lines
*e*_{1}, *e*_{2}, ..., *e _{n}*
and a point

*P*in the plane. Project

*P*perpendicularly on the first line. Next, project its image on the second line, and so on. After projecting on the last line, project again on the first line, and continue the process. Prove that the different images obtained by the procedure form a bounded set.

**N. 172.** An automatic card shuffler is used to shuffle a deck
of 2*n* cards. It can rearrange the deck according to the rule
(1, 2, 3, ..., 2*n*-1, 2*n*)(2*n*, 1, 2*n*-1, 2, ..., *n*+1,
*n*). Prove that the cards will be in their original order after at most
2*n* shuffles.

**N. 173.** Is it possible to partition a
rectangle into a finite number of squares?

**N. 174.** A tour of a tree graph is the following. A figure,
moving along the edges of the graph, visits each vertex such that it travels
along every edge exactly twice. We say that two figures tour the graph
simultaneously if they start at the same time, not necessarily at the same
vertex, and then they always move at the same time during their respective
tours of the graph. They meet if they move to the same vertex or stay on the
same edge of the graph at the same time. Let *f*(*n*) denote the
largest integer for which there exist a tree on *n* vertices which
*f*(*n*) figures can tour simultaneously such that no two of them
meet. Prove that *f*(5*k*+1)=2*k* for every positive integer
*k*.