# New exercises and problems in Mathematics

December 1999

## Please read The Conditions of the Problem Solving Competition.

## New exercises in December 1999 |

**C. 560. **A loaf of rye bread has a mass 25% smaller
than a loaf of wheat bread, moreover, it is 20% more expensive. On the
other hand, we always eat the whole loaf of rye bread, while 15% of
the wheat bread becomes dry before we could finish up the whole
loaf. Assuming the same consumption, determine, in percentage, how
much more we spend if we consume rye bread instead of wheat bread.

**C. 561. **Find those prime numbers *p *for
which *p*^{2}+11 has exactly 6
positive divisors.

**C. 562. **Which positive integers *n *satisfy

**C. 563. **In a circle of centre *O *and radius
*r*, let *AB *denote any chord shorter than the diameter of
the circle. Let denote the radius
of the circle inscribed the smaller sector of the the circle bounded
by the radii *OA *and *OB*. Express the length of *AB
*in terms of *r *and .

**C. 564. **The edges of a solid rectangular box are
26, 20 and 8 units long. Attached to each 26x8 face, build a tent roof
whose lateral edges are 13 units long and whose `top' edge, parallel
to its base, is 20 units long. Remove from the box the parts obtained
by reflecting these tent roofs in their respective bases. Determine
the volume of the solid which remains from the box.

## New problems in December 1999 |

**B. 3322. **Find the maximum number of parts into
which the space can be divided by 4 concurrent planes. (3
points)

**B. 3323. **What is the longest interval in which the
fraction of smallest denominator is 19/9999? (5 points)

**B. 3324. **How many ways are there to express
1999^{1999} as the sum of consecutive
natural numbers? (4 points)

Proposed by: *T. Káspári,* Paks

**B. 3325. **Prime numbers *p*_{1}, *p*_{2},
*p*_{3}, *p*_{4} are called `quadruplets' if *p*_{1}<*p*_{2}<*p*_{3}<*p*_{4} and
*p*_{4}-*p*_{1}=8. Prove that if both *p*_{1}, *p*_{2},
*p*_{3}, *p*_{4} and *q*_{1},
*q*_{2}, *q*_{3}, *q*_{4} are
quadruplets such that *p*_{1}*,q*_{1}>5,
then *p*_{1}-*q*_{1} is divisible by 30. Prove, moreover, that
*p*_{1}-*q*_{1} is never equal to 60. (3 points)

**B. 3326. **One face of a regular octahedron is
projected perpendicularly to the plane of the opposite face. What
proportion of the area of the opposite face is covered by the
projection? (4 points)

Proposed by:
*M. Juhász,* Budapest

**B. 3327. **Give an example of a nonzero polynomial
of integral coefficients which admits cos 18^{o} as a root. (4 points)

**B. 3328. **A convex polygon is `enlarged' by
translating the line of each side outwards by 50 mm,
perpendicularly to the given side. Show that the perimeter of the
polygon is increased by more than 300 mm. (3
points)

Proposed by: *Zs. Magyar,*
Budapest

**B. 3329. **A class of 31 students organize a
Christmas party and they decide to present each other. Each of them
draws the name of the one whom (s)he should present from a box. What
is the probability that this drawing must be repeated because someone
has drawn his or her own name? (4 points)

**B. 3330. **There are 1999 unit squares inside a
square whose sides are 105 units long. Prove that one more unit
square, disjoint from each of the others, can be placed inside the big
square. (5 points)

**B. 3331. **In a certain school there are as many
senior boys as girls therefore each senior student can have a partner
for the dance at the graduation ball. Before arranging the couples,
the girls are ranked by each boy, and vice versa. Prove that it is
possible to arrange the pairs in such a way that there is not a boy
and a girl who would rather dance with each other than with their
actual partners. (5 points)

## New advanced problems in December 1999 |

**A. 224. **Prove that arbitrary positive numbers
*a*_{1}, *a*_{2}, ..., *a*_{n} satisfy the following inequality

**A. 225. **Let *S*_{1} and *S*_{2}
denote the sum of the numbers of odd resp. even divisors of the
numbers 1, 2, ..., *n*. Prove that

**A. 226. **Let *k *denote an arbitrary positive
integer, and consider any graph whose vertex set is the set of
positive integers and which does not contain a complete *k*x*k
*bipartite subgraph. Prove that there exist arbitrarily long
arithmetic progressions such that no two elements of the same
progression are joined by an edge in the graph.

Proposed by: *J. Solymosi,* Budapest

### Send your solutions to the following address:

KöMaL Szerkesztőség (KöMaL feladatok), Budapest Pf. 47. 1255, Hungary

or by e-mail to: megoldas@komal.elte.hu.