# New exercises and problems in Mathematics

April 1999

## New exercises for practice in April 1999 |

**C. 537. **Working on his essay during class, Sam
glanced at his watch and noticed that the time that had already passed
was five times more than that still was left. After *M *minutes
had passed the ratio was already 8. Find out what the ratio was after
another *M *minutes had passed.

**C. 538. **Find those (not necessarily positive)
prime numbers *p *for which each of the following expressions is
a prime: 2*p*+1, 4*p*+1 and 6*p*+1.

Proposer: *Á. Kovács, *Budapest

**C. 539. **Prove that in any acute isosceles triangle
*AM*=*a*/tan holds,
where, conventionally, *a *denotes the base of the triangle, *A
*and , respectively, denote the
vertex and the angle opposite to it, and *M *is the orthocentre.

**C. 540. **A half ball of radius 10 cm emerges from
level plane (for the great annoyance of the motorists). A cylinder
rolls on the plane until its lateral surface hits the half ball. Let
denote the angle the common
tangent plane of the two objects forms with the horizontal
plane. Given that 30^{o}, find the minimum
possible radius of the cylinder.

## New exercieses in April 1999 |

**Gy. 3270. **Let *n*>2 be an even number. Prove
that is not a prime in any number
system.

Proposer: *Á. Kovács, *Budapest

**Gy. 3271. **Which fraction is the larger: or ?

Proposer: *J. Bíró, *Budapest

**Gy. 3272. **The edges *a*, *b*, *c
*of a certain cuboid are integers, moreover, its surface area and
its volume measure the same. Given that *c*=*ab*/2,
determine the edges of the cuboid.

Proposer: *Á. Kovács, *Budapest

**Gy. 3273. **Prove that 5^{12}+2^{10} is a
composite number.

St. Petersburg, Mathematical Olimpiad

**Gy. 3274. **We have three congruent right-angled triangle
shaped pieces of paper. We may cut any of these into two parts along
its altitude. We may repeat this procedure for the new set of pieces,
and so on. Provided that no matter how we choose the procedure there
always will be two congruent triangles, determine the ratio between
the legs of the triangles.

**Gy. 3275. **In a certain quadrilateral, each
diagonal halves the area of the quadrilateral. Prove that they
together partition the quadrilateral into four parts of equal area.

**Gy. 3276. **Centre *O *of a circle
*k*_{2} lies on the circumference of
circle *k*_{1}, of radius
*r*. The circles intersect at points *A *and *B*,
moreover, a point *S *lies inside *k*_{1}. Line *BS *intersects *k*_{1} at point *T*, different from
*B*. Given that the triangle *AOS *is equilateral, prove
that *TS*=*r*.

**Gy. 3277. **Let *a *and *b *denote
positive integers. Given a segment of length , describe how to construct an other segment of length
.

Proposer: *Á. Kovács, *Budapest

## New problems in April 1999 |

**F. 3280. **Prove that, if *x*_{1}, *x*_{2},
..., *x*_{n} are arbitrary
positive numbers, then

.

**F. 3281. **A piece of chocolate is divided into
smaller pieces. In each step the largest piece - or one of them if
this piece is not unique - is broken into smaller parts in such a way
that none of the new parts exceed the half of the piece just
broken. Prove that after the *k*th step each piece is smaller
than the 2/(*k*+1)th part of the whole chocolate.

*Lovász László, *Budapest

**F. 3282. **Find the range of the expression
*x*^{2}+2*xy*, provided that
*x*^{2}+*y*^{2}=1.

**F. 3283. **In a set of 2000 segments, each segment
is at least 1 unit long. It is not possible, moreover, to construct a
closed polygon from any subset of this set of segments. Prove that the
total length of the segments is at least 2^{1999}.

**F. 3284. **In a triangle *ABC*, a point *P
*satisfies *PAB*=*PBC*=*PCA*=. If the
angles of the triangle are denoted by , and , prove that

.

**F. 3285. **A circle *k *and a point *P
*outside the circle are given in the plane. Determine the locus of
the centre of a circle which is obtained as the intersection of a
sphere that contains *k *and a right circular cone with apex
*P*.

## New advanced problems in April 1999 |

**N. 207. **A continuous function
*f*:[0*,*1][0*,*1]
has the following property: for any real number *x*[0*,*1], the sequence
*f*(*x*), *f*(*f*(*x*)),
*f*(*f*(*f*(*x*))), ... contains 0. Does it follow
that, for *n *large enough, is
identically 0?

**N. 208. **Assume that *p *is a prime number
greater than 2. Prove that there exists a prime *q*<*p *such
that *q*^{p-1}-1 is divisible
by *p*^{2}.

**N. 209. **Find 6 points in the plane such that any two of
them are at an integer distance apart, no three of them are collinear,
and no four lie on the same circle.

**N. 210. **Prove that any closed space curve contains
four coplanar points. Is it always true that it contains five coplanar
points, too?

Proposer: *S. Róka, *Nyíregyháza