# New exercises and problems in Mathematics

March 1998

## New exercises for practice in March 1998

**C. 497.** Is there any positive integer *n* such that
1^{.}2^{.}3^{.}...^{.}(n-1)^{.}n,
that is, *n*!, ends with exactly 100 zeros?

**C. 498.** Solve inequation *x*^{3}+1>*x*^{2}+*x*.

**C. 499.** There is given a circle of unit radius in the plane.
A wreath of *n* congruent circles is formed around the given circle
such that each circle in the wreath touches its two neighbours and also
the given circle. Determine, in terms of *n*, the common radius of
the circles. Calculate these radii, up to four digits precision after the
decimal point, for the first four possible values of *n*.

**C. 500.** Is it possible to slip two fine arts albums, each 22 cm
wide, and a 25 cm wide cookbook into 28 cm wide plastic bag? Assume that
each book has a thickness of 1.5 cm.

## New exercieses in March 1998

**Gy. 3190.** We define a number 44...48...89 with the following
procedure. First, we insert the number 48 between the two digits of 49.
Next, 48 is inserted between the digits 4 and 8 of the resulting number.
Finally, we repeat this second step a few times. Is it true that every
number obtained this way is a perfect square?

**Gy. 3191.** Is it true that every integer has at least as many
positive divisors of the form 4*k*+1 as that of the form 4*k*-1?

**Gy. 3192.** Find all pairs of positive integers *A,B* with the
following property: if the decimal representation of *A* is placed in
front of that of *B*, then the resulting number is a perfect square,
equal to twice the product of *A* and *B*.

**Gy. 3193.** A basketful of peanuts is distributed into *n*
piles. We gather the piles and rearrange them into *n*+*k* new
piles (*k*>0). Prove that at least *k*+1 peanuts are transferred
to smaller piles than the respective original piles that contained them.
(**H**)

**Gy. 3194.** In a triangle, two of the altitudes are at least as long
as the corresponding sides of the triangle. Find the angles of the triangle.

**Gy. 3195.** In a right triangle, parallel lines to each side are
drawn through the incentre of the triangle. These lines divide each side of
the triangle into three parts. Prove that the length of the middle part of
the hypotenuse equals the sum of the lengths of the middle parts arising on
the legs of the triangle.

**Gy. 3196.** There are given rays *e*_{0},
*e*_{1}, ..., *e*_{6}, each starting at the same
point *A* of the plane. Each ray forms an angle of 30^{o} with
the one that precedes it in the given order. Let *B*_{0} denote
the point on ray *e*_{0} such that *A* and
*B*_{0} are a unit distance apart. Let *B*_{1}
denote the orthogonal projection of *B*_{0} to
*e*_{1}. Similarly, denote by *B*_{2} the
orthogonal projection of *B*_{1} to *e*_{2}, and
so on. Calculate the length of the polygonal path
*B*_{0}*B*_{1}...*B*_{6} and the area
of polygon *B*_{0}*B*_{1}...*B*_{6}.

**Gy. 3197.** Four balls, each of radius *r*, are placed on a
horizontal plane such that each of them touches two others and their centres
form a square. A fifth ball of the same radius is placed on the top of these
balls, touching all of them. Calculate the distance of the topmost point of
the fifth ball from the given plane.
(**H**)

## New problems in March 1998

**F. 3220.** A rectangular field is divided into 100 congruent
rectangular plots with the help of lines parallel to the edges of the field.
Neighbouring plots share a common edge. Initially, 9 plots of the field are
overgrown with weeds. The weeds overgrow any plot whose (at least) two
neighbours are already overgown. Is it possible that the whole field becomes
overgrown sooner or later?

**F. 3221.** The numbers 1,2,...,*n* are assigned, in some order,
to the vertices of a regular *n*-gon
*A*_{1}*A*_{2}...*A _{n}*.

- a) Determine the maximum possible value of the sum of the absolute values
of the differences between numbers assigned to consecutive vertices.

- b) Find the number of arrangements for which the above sum is maximum.

**F. 3222.** Given that *n* is a prime, prove that there is no positive integer *x* that satisfies equality
(1^{n}+2^{n}+...+*x ^{n}*)+(1

^{n}+2

^{n}+...+(

*n*-1)

^{n})=1

^{n}+2

^{n}+...+(2

*n*-1)

^{n}.

**F. 3223.** A polygon has exactly *n* axes of symmetry. How many
sides may the polygon have?

**F. 3224.** In a certain convex polytope, every face has a central
symmetry. Prove that at least six of the faces are parallelograms.

**F. 3225.** Find a set of points in 3-space which intersects every
plane in at least one but not more than five points.

## New advanced problems in March 1998

**N. 167.** There are given three unit discs in the plane. Let
*t*_{1} and *t*_{2} denote the areas of the two
regions in the plane which the discs cover exactly once resp. exactly twice.
Prove that .

**N. 168.** Prove that equality
holds for every positive integer *n*.

**N. 169.** Let *a _{n}* denote the number of possible
results of a contest with

*n*participants, where ties may occur. For example,

*a*

_{2}=3 and

*a*

_{3}=13. Prove inequality .

**N. 170.** Given that 15*a*+6*b*+4*c*+8*d*=0,
prove that the equation
*ax*^{3}+*bx*^{2}+*cx*+*d*=0
has a positive root.