# New exercises and problems in Mathematics

December 1997

## New exercises for practice in December 1997

**C. 485.** We are given a 100x100 array of numbers such that, for
*k*=1,2,...,100, its *k*th row is an arithmetic progression whose first term is 1, and whose common difference is *k*. Find the largest entry along the diagonal that connects the lower left corner of the array to its upper right corner.

**C. 486.** How many different ways can some 1, 2, 5, 10 and 20 forint coins make 25 forints?

**C. 487.** Given that ,
prove inequality .

**C. 488.** The lateral surface of a frustum of a right circular cone
is coloured with two colours; it is blue below the middle of its slant height
and is red above that. Find the ratio between the radii of the bases of the
frustum, given that the surface area of the blue part is twice as large as
that of the red part.

## New exercieses in December 1997

**Gy. 3166.** Compare the following two numbers: 1997^{1999}
1999^{1997}

**Gy. 3167.** Prove that two positive integers *a* and *b*
have the same parity if and only if there exist positive integers *c*
and *d* such that
*a*^{2}+*b*^{2}+*c*^{2}+1=*d*^{2}.

**Gy. 3168.** In a set of size 15, find 15 subsets, each of size 7,
such that any two of them have exactly 3 elements in common.

**Gy. 3169.** Find all triangles with integer side-lengths in which one
of the angles is twice as large as an other one.

**Gy. 3170.** A triangle *ABC* has a right angle at vertex
*C*. The bisector of the angle at *B* intersects side *AC* and
the circumcircle at points *P* and *Q*, respectively. Find the
measures of the angles of the triangle, given that *BP*=2*PQ*.

**Gy. 3171.** The lengths of the sides of a certain triangle are
integers, one of them being 1. Is it possible that the inradius of the
triangle is a rational number?

**Gy. 3172.** A right circular cylinder shaped vessel, whose axis is
vertical, is filled up with water. The radius of the cylinder is 10 cm, its
height is 25 cm. Find the amount of water which flows out of the vessel when
it is tilted by 30^{o}. Construct the angle by which the vessel
should be tilted from its original position if we want to pour out half of
the water it contains.

**Gy. 3173.** Given three straight lines in the space, each incident to
a given point *P*, construct a plane through *P* which forms the
same angle with each of the given lines.

## New problems in December 1997

**F. 3202.** Let be a positive
integer written in canonical form. Let *r*_{1},
*r*_{2}, ..., *r _{m}* denote those positive integers
which are smaller than

*n*and are relatively prime to

*n*. Derive equality .

**F. 3203.** Prove that any set of size 15 contains 15 subsets, each
of size 6, such that any two of them have either 1 or 3 common elements.

**F. 3204.** There is given a heap of 923^{k} pebbles.
Two players play the following game. They move alternately, removing at each
turn either 9, 2 or 3 pebbles from the heap. The one who cannot move, loses
the game. Which player has a winning strategy?

**F. 3205.** In an arbitrary triangle, let *a, b, c* denote the
lengths of the sides of the triangle, and let *m _{a}*,

*m*,

_{b}*m*, respectively, denote the lengths of the corresponding altitudes. Prove inequality .

_{c}**F. 3206.** We are given a circle *k*_{1}, and
points *H*_{1}, *H*_{2} and *P* outside the
circle. Construct a circle *k*_{2} passing through *P*
such that, for *i*=1,2, the lengths of the tangents from
*H _{i}* to

*k*

_{2}are the same as the lengths of the tangents from

*H*to

_{i}*k*

_{1}.

**F. 3207.** A regular tetrahedron, whose edges are 2 units long, is
inscribed into a right circular cylinder in such a way that it has two
vertices on the perimeter of each base of the cylinder. Find the volume of the
cylinder.

## New advanced problems in December 1997

**N. 155.** In an infinite sequence of positive integers, each term has
the same number of divisors. Prove that there is an infinite subsequence in
which any two terms have the same greatest common divisor.

**N. 156.** Suppose that *a*_{1},*a*_{2},...
and *b*_{1},*b*_{2},... are integer sequences such
that *a*_{1}=*b*_{1}=0, and
*a _{n}*=

*nb*+

_{n}*a*

_{1}

*b*

_{n-1}+

*a*

_{2}

*b*

_{n-2}+

*a*

_{3}

*b*

_{n-3}+...+

*a*

_{n-1}

*b*

_{1}holds for . Prove that, for any prime number

*p*,

*a*is divisible by

_{p}*p*.

**N. 157.** An *n*x*n* array consists of real numbers such
that the entries are monotone decreasing in each row and each column of the
array. Whenever we are given a number *a*, our task is to decide if
the array contains that number or not. For that purpose, we may compare
*a* with any entry in the array. How many comparisons do we have to
make to solve the problem for sure, regardless of the specific value of
*a*?

**N. 158.** There are given 9 convex subsets of (3-dimensional) space
such that any 8 of them have a lattice point in common. Prove that there
exists a lattice point which is contained in each set.