# New exercises and problems in Mathematics

December
2000

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

## New exercises in December 2000 |

**C. 605.** Solve the equation

,

where [*x*] and {*x*} denote the integer and the fractional part
of *x*, respectively.

**C. 606.** A rectangular yard is tiled with identical
square tiles. The tiles are arranged in 20 rows, each row containing 35
tiles. A snail is moving along a diagonal of the rectangle, starting at one
vertex. Find the number of tiles crossed by the snail before it reaches the
opposite vertex.

**C. 607.** A square of side 12 cm is rotated about a
point *P* through an angle of 90^{o}. The two
squares cover an area of 211 cm^{2} altogether. A
third square is obtained rotating the square further about *P*, again
through 90^{o}. Given also that the three squares
cover an area of 287 cm^{2} together, determine
the position of *P*.

Proposed by: *G. Bakonyi,* Budapest

**C. 608.** In the course of a KENO drawing, 20 winning
numbers are drawn from the lot of the first 80 positive integers. Calculate the
probability that none of the winning numbers contain the digit 8.

**C. 609.** Prove that the distance between the line
3*x*-4*y*+4=0 and a lattice point in the plane is always a rational
number.

## New problems in Decober 2000 |

**B. 3412.** Prove that for all but two positive integers
*q* there is an integer *p* such that <*p*/*q*<. (3 points)

**B. 3413.** In the Hungarian Soccer League, last year
still called `professional', one group comprises 8 teams. At each match one of
the participating teams plays away while the other plays at home. During a
season, when each team plays exactly one game with any other team, the draw of
a team is called balanced if it plays alternately at home and away. Is it
possible to arrange that each of the 8 teams has a balanced draw?
(4 points)

**B. 3414.** In a cyclic quadrilateral whose diagonals are
perpendicular, the sum of the squares of the edges is 8. Determine the
circumradius of the quadrilateral. (3 points)

Proposed by: *Á. Besenyei,* Tatabánya

**B. 3415.** Determine all values of the parameter *p* for
which the equation

has exactly one root. (4 points)

**B. 3416.** Each face of a convex polyhedron is a
quadrilateral. The surface area of the polyhedron is *A*, the sum of the
squares of its edges is *Q*. Prove that *Q*2*A*. (4 points)

Proposed by: *Á. Besenyei,* Tatabánya

**B. 3417.** Consider all circles in the plane whose equation
can be written in the form ((*x*-2)^{2}+*y*^{2}-1)+((*x*+2)^{2}+*y*^{2}-1)=0 with suitable real numbers , . Determine the locus
of points which do not lie on any of these circles. (4 points)

**B. 3418.** In a triangle *ABC*, the medians starting at
vertices *A*, *B*, *C* intersect the circumcircle at points
*A*_{1}, *B*_{1}, *C*_{1},
respectively. Prove that *t*_{A1BC}+*t*_{B1CA}+*t*_{C1AB}*t*_{ABC}. (5 points)

Proposed by: *I. Varga,* Békéscsaba

**B. 3419.** Find those points in the plane through which
3 different tangents can be drawn to the curve *y*=*x*^{3}. (4 points)

**B. 3420.** Find a non-constant function *f*:**RR** whose graph can be obtained from the
graph of the square of the function by an enlargement of
ratio 3. (5 points)

Proposed by: *L. Lóczi,* Budapest

**B. 3421.** Assume that (*x*^{2}+3*y*^{2})/(*u*^{2}+3*v*^{2}) is an
integer number for certain integers *x*, *y*, *u* and
*v*. Prove that this fraction can also be written in the form
*a*^{2}+3*b*^{2} with suitable integers *a*
and *b*. (5 points)

Proposed by: *M. Csörnyei,* London

## New advanced problems in December 2000 |

**A. 251.** Given a sphere and a point *P* on it, let
*P*_{Q} denote, for any point *Q*
not on the sphere, the second intersection of line *PQ* with the
sphere. In the degenerate case when *PQ* is tangent to the sphere, let
*P*_{Q}=*P*. Prove that
if points *A,B,C,D* of the sphere are not coplanar, then there are at
most two different points *Q* such that the tetrahedron
*A*_{Q}*B*_{Q}*C*_{Q}*D*_{Q}
is equilateral. (A tetrahedron is called equilateral if all its faces are
congruent.)

**A. 252.** The integers *a*_{1}*,...,a*_{n}
give at least *k*+1 different remainders when they are divided by
*n*+*k*. Prove that some of them add up to an integer divisible by
*n*+*k*.

Proposed by: *Gy. Károlyi* and *K. Podoski,*
Budapest

**A. 253.** Find all functions *f*: (0*,*)(0*,*) for which each
pair of positive numbers *x* and *y* satisfy
*f*(*x*)*f*(*yf*(*x*))=*f*(*x*+*y*).

*International Mathematical Competition for University
Students* 2000, London

### Send your solutions to the following address:

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

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