# New exercises and problems in Mathematics

May 2000

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

## New exercises in May 2000 |

**C. 585. **In certain audio cassettes, a
0.0075 mm wide tape is wound on the reels whose common diameter
is 11 mm. The centres of the reels are in distance 42 mm
apart. How long a tape can a usable cassette (in which the tape can be
wound up on each reel without getting stuck at any point) contain?

**C. 586. **Find the maximum possible number of sides
of a convex polygon which does not have two consecutive obtuse angles.

**C. 587. **Let *n *denote a positive
integer. Connect the point of the *x*-axis whose abscissa is *n
*with the points of the *y*-axis whose ordinates are
*n*-1 and *n*+1, respectively. Connect, moreover, the point
of the *y*-axis whose ordinate is *n *with the points of the
*x*-axis whose abscissas are *n*-1 and *n*+1,
respectively. Determine the area enclosed by the four segments
obtained this way.

**C. 588. **Determine the inverse of the function
*f*:(-*,*-2)**R**, *x*2*x*^{2}+8*x*+7.

**C. 589. **A straight stick got stuck in a drainage
ditch of a V-shaped cross-section. The two ends of the stick lean
against the two sides of the ditch, forming equal angles with both
sides. Prove that the two ends of the stick rest in the same distance
from the bottom of the ditch.

## New problems in May 2000 |

**B. 3372. **In a certain right-angled triangle, the
medians can be arranged to form the sides of another right-angled
triangle. Prove that the two triangles are similar. (3 points)

Proposed by: *N. Gyanta,* Budapest

**B. 3373. **Lines *e *and *g*,
respectively, are drawn on each side of a given line *f*, in a
unit distance from *f*. Points *E,G *and *F *lie on the
corresponding lines such that triangle *EFG *has a right angle at
vertex *F*. Determine the length of the altitude of the triangle
starting at *F*. (3 points)

Proposed by: *Gy. Pap,* Debrecen

**B. 3374. **A certain exam consists of a written and
an oral part; the students can obtain an integer mark between 0 and 9
at each part. In the final evaluation student *A *is ranked
higher than *B *if *A*'s mark is not worse than that of *B
*at each part of the exam, and is strictly better than that of *B
*at one of the exams at least. In a group of 20 students no two
students obtained identical results. Prove that there are three
students *A,B,C *in the group such that *A *is ranked higher
than *B *and *B *is ranked higher than *C*. (4 points)

**B. 3375. **Is there any polynomial
*p*(*x*)=*x*^{2}+*ax*+*b *whose integers coefficients
satisfy *a*^{2}-4*b*0 and which assigns a perfect square
value to at least 2000 different integers? (4 points)

**B. 3376. **In a triangle *ABC*, side *AC
*is of unit length. The median starting at vertex *A *divides
the angle at *A *in the ratio 1:2. How long may side *AB
*be? (3 points)

*S. Mihalovics,* Esztergom

**B. 3377. **Let *k *denote an odd integer. Find
the remainder when the sum 2^{n-1}*k*^{2}+2^{n-2}*k*^{4}+2^{n-3}*k*^{8}+...+2*k*^{2n-1}+*k*^{2n} is
divided by 2^{n+2}. (4 points)

Proposed by: *M. Bencze,* Brassó

**B. 3378. **The pupils of a graduating class apply at
five different universities; each university receiving the application
of at least half of the pupils. Prove that there exist two
universities such that at least one fifth of the pupils apply at both
institutions. (5 points)

**B. 3379. **In a triangle *ABC*, the angle
bisectors *AA*_{1},
*BB*_{1}, *CC*_{1} meet at point *O*. Prove that , where *R *and *r*,
respectively, denote the radii of the circumscribed and the inscribed
circle. (5 points)

Proposed by: *M. Bencze,* Brassó

**B. 3380. **Show that every parallelepiped has a
vertex where either none of the 3 plane angles between the edges are
acute or none of the 3 dihedral angles are obtuse. (4 points)

**B. 3381. **A teacher noticed that among the 30
pupils who attend to her class, two celebrate their birthdays on the
very same day. This is not a big surprise: prove that the probability
of two people out of 30 having the same birthday is at least . (5 points)

## New advanced problems in May 2000 |

**A. 239. **Construct a triangle, if the three points
where the extensions of the medians intersect the circumcircle of the
triangle are given.

Proposed by: *J. Rácz,* Budapest

**A. 240. **Let *n *and *m *denote positive
integers. Prove that

where is Euler's totient function.

**A. 241. **Prove that, for every positive integer *n*,

### 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.