# New exercises and problems in Mathematics

September 1997

## New exercises for practice in September 1997

**C. 473.** Can it happen for a whole calendar year that no single
Sunday falls on the seventh day of a month?

**C. 474.** A passanger has been walking for 3.5 hours, covering
exactly 5 km in the course of any period of one hour. Is it possible that
his average speed during his walk exceeded 5 km/h?

**C. 475.** In a triangle *ABC*, a point *P* on median
*CC*_{1} is selected such that *CP/PC*_{1}=*m/n*.
Find the ratios in which *P* divides the segments of lines *AP*
resp. *BP* lying inside the triangle.

**C. 476.** In a right circular cone, both the diameter of the base
and the slant height are 20 cm. Find the maximum length of a 2 cm wide
self-adhesive band that can be sticked on the lateral surface of the cone
without any creasing, cutting, or overlapping.

## New exercieses in September 1997

**Gy. 3142.** Find the smallest positive integer that is divisible
by 28, ends in 28 (in decimal system), and the sum of whose digits is 28.

**Gy. 3143.** Draw a chessboard on the plane. Let *A*_{1},
*A*_{2}, ..., *A*_{32} and *B*_{1},
*B*_{2}, ..., *B*_{32} denote the midpoints of
the white and the black fields, respectively. Let furthermore *P*
be any point of the chessboard. Prove that

*A*_{1}*P*^{2}+ *A*_{2}*P*^{2}+
*A*_{32}*P*^{2}= *B*_{1}*P*^{2}+
*B*_{2}*P*^{2}+ *B*_{32}*P*^{2}.

**Gy. 3144.** Decide if 7/17 can be expressed as 1/*a*+1/*b*,
where *a* and *b* are positive integers.

**Gy. 3145.** Solve inequality

**Gy. 3146.** Let *a* and *b* denote the lengths of the
legs of a right triangle, and *r* its inradius. Prove that

**Gy. 3147.** Let *P* be a point inside a regular hexagon, and
*e* a line incident to *P* and parallel to a side of the hexagon.
Draw five additional lines passing through *P* such that among the
six lines, any two consecutive lines form an angle of 30^{o}. These
lines divide the hexagon into 12 regions. Prove that these 12 regions can
be divided into three groups such that the sum of the areas of the regions
is the same in each group.

**Gy. 3148.** Let *e*_{1}, *e*_{1} and
*e*_{1} be three pairwise skew edges of a cube. Select a point
*E _{i}* on each edge

*e*for

_{i}*i*=1,2,3. Find the locus of the centroids of the triangles

*E*

_{1}

*E*

_{2}

*E*

_{3}.

**Gy. 3149.** In a triangle *ABC*, side *AB* is of unit
length. The angles which include this side measure 15^{o} and 60^{o},
respectively. Find the lengths of the two other sides of the triangle without
the help of trigonometric functions.

## New problems in September 1997

**F. 3184.** Prove inequality .

**F. 3185.** Consider a binary operation defined by
on the set of numbers not less than 1. Prove that the operation satisfies
the associative law.

**F. 3186.** The non-negative numbers
satisfy for
. Prove inequality
.

**F. 3187.** Find those triangles which satisfy
*a*^{2}+*b*^{2}+*c*^{2}=8*R*^{2}
where *a*, *b*, *c* and *R* denote the lengths of the
sides and the circumradius of the triangle, respectively.

**F. 3188.** There are given a circle centered at *O*, a point
*A* selected on the circle, and a straight line *d* passing through
*O*. A secant starting at *A* intersects the circle and line
*d* at points *B* and *D*, respectively. Prove that upon
rotating the secant about *A*, in any position, the circle incident
to points *O*, *B* and *D* passes through an other fixed
point different from *O*.

**F. 3189.** Suppose
are the angles of a triangle. Prove inequality .

## New advanced problems in September 1997

**N. 144.** Let *n* denote an arbitrary positive integer. Prove
that 2^{.}(3*n*)! is divisible by *n*!(*n*+1)!(*n*+2)!.

**N. 145.** Does there exist a polynomial *f*(*x*,*y*,*z*)
of real coefficients such that, *f*(*x*,*y*,*z*) is
positive if and only if |*x*|, |*y*| and |*z*| are the sides
of a triangle?

**N. 146.** Prove that any rational number *r* in (0,1) of odd
denominator can be expressed in the form
with suitable integers *x*, *y* and *z*.

**N. 147.** Is it always true that in a tetrahedron, an inner point
of each face can be selected such that they form the vertex set of a regular
tetrahedron?