# New exercises and problems in Mathematics

March 2000

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

## New exercises in March 2000 |

**C.575. **``Well, I want you to answer me this. If two
stage-coaches leave Pozsony for Brassó every day, and as many leave
Brassó for Pozsony, and assuming that the journey takes ten days, then
how many coaches would you meet on the way if you travelled in one
stage-coach from Pozsony to Brassó?''

Proposed by *K. Mikszáth, *Fogaras

**C.576. **Find the smallest positive integer which, multiplied
by 1999, yields a product whose last 4 digits are
2*,*0*,*0*,*1.

Proposed by *M. Braun*

**C.577. **Deep inside a drawer, there are 3 pairs of socks,
somewhat different from each other. We try it three times to fish one
sock out of the drawer. What is the probability that we end up with
one sock of each pair after the 3 trials?

**C.578. **Consider the triangle whose vertices in the Cartesian
system are *A*(2;-1), *B*(3;1) and *C*(2^{1999};2^{2000}). Calculate the area of the triangle.

**C.579. **A circular disc is cut into two sectors along two
of its radii. The two pieces are then turned into cone shaped
funnels. Is it true that the largest total volume is obtained from two
half discs?

## New problems in March 2000 |

**B.3352. **The number of the crew of a certain ship, multiplied
by one less the number of the crew exceeds, by 15, the product of the
age of the captain and the number two less the number of the
crew. What is the captain's name? (3 points)

*Repeta*

**B.3353. **Determine the value of the following expression:

(4 points)

Proposed by *M. Ábrány, *Ukraine

**B.3354. **Two intersecting lines and a point *P *outside
the lines are given in the plane. Construct points *X *and *Y
*on the two lines, respectively, such that *P *is incident to
the segment *XY*, and *PX*^{.}*PY *is minimal.
(5 points)

**B.3355. **Prove that a triangle has a right angle if and
only if the product of two of its escribed circles equals to the area
of the triangle. (4 points)

Proposed by *B. Bíró, *Eger

**B.3356. **Demonstrate that there are an infinite number of
cases when the difference between the fourth powers of two consecutive
integers can be written as the sum of two perfect sqares. (4 points)

**B.3357. **Is there any perfect square which has the same
number of positive divisors of the form 3*k*+1 as of the form
3*k*+2? (4 points)

**B.3358. **Prove that among any 4 different real numbers
there exist two, *a *and *b *such that

(5 points)

Proposed by *J. Mezei*, Vác

**B.3359. **The real roots *x*_{1}, *x*_{2},
*x*_{3} of the equation
*x*^{3}-3*x*-1=0 satisfy
*x*_{1}<*x*_{2}<*x*_{3}. Prove that *x*_{3}^{2}-*x*_{2}^{2}=*x*_{3}-*x*_{1}. (5
points)

Proposed by *S. Mihalovics, *Esztergom

**B.3360. **In a triangle *ABC *with centroid *S*, Let
*E *and *F *denote the midpoints of sides *AB *and
*AC*, respectively. Prove that the quadrilateral *AESF *is
cyclic if and only if *AB*^{2}+*AC*^{2}=2*BC*^{2}. (4
points)

Proposed by *S. Kiss, *Szatmárnémeti

**B.3361. **Two spheres *G*_{1} and *G*_{2}
are given. Cubes *K*_{1} and
*K*_{2} are inscribed into
*G*_{1} and *G*_{2}, respectively. Consider the 64 vectors whose
starting points are the vertices of *K*_{1} and whose endpoints are the vertices of
*K*_{2}. Prove that the sum of these
vectors does not depend on the position of the cubes within the
spheres. (3 points)

## New advanced problems in March 2000 |

**A.233. **A sequence (*a*_{n}) is defined as follows:
*a*_{0}=*a*_{1}=1, (*n*+3)*a*_{n+1}=(2*n*+3)*a*_{n}+3*na*_{n-1}. Prove that each term of the sequence
is an integer.

**A.234. **Any finite sequence of the letters *A,B,C* is
called a *word*. A word can be transformed into an other word by
the following two operations:

a) we choose any contiguous part of the word and double it, as in
the example *B BCACBBCABCAC*;

b) (the reverse of the first operation) if two consecutive
contiguous parts of the word are identical, we may omit one of them,
as in the example * ABCABCBC ABCBC*.

Prove that any word can be transformed into a word of not more than 8 letters.

**A.235. **Let *a *be a fixed complex number. Find the
locus of all complex numbers *b *for which there exist
nonnegative real numbers *x*_{1}*,x*_{2}*,*...*,x*_{n} and complex numbers *z*_{1}*,z*_{2}*,*...*,z*_{n} of unit modulus such that
*x*_{1}+*x*_{2}+...+*x*_{n}=1, *x*_{1}*z*_{1}+*x*_{2}*z*_{2}+...+*x*_{n}*z*_{n}=*a *and *x*_{1}*z*_{1}^{2}+*x*_{2}*z*_{2}^{2}+...+*x*_{n}*z*_{n}^{2}=*b*.

### 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.