# New exercises and problems in Mathematics

October 1999

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

## New exercises in October 1999 |

**C. 550. **An angler went fishing one day. When he returned he
put the three largest fish, which comprised 35% of the total mass of
the fish he caught, into the freezer. The three smallest fish that
formed 5/13 of the remaining part were taken away by the cat, and
the rest was cooked for dinner. How many fish did the angler catch
that day?

Proposed by *S. Dobos, *Budapest

**C. 551. **Find all solutions of the following system of
equations: *x*^{2}+*y*^{2}=*x*, 2*xy*=*y*.

Proposed by *E. Fried, *Budapest

**C. 552. **The circles depicted on the figure have radii
3 cm and 4 cm, respectively, their centres are 5 cm
apart. Find the difference between the areas of the two shaded
regions.

**C. 553. **A cylinder of diameter 10 cm and height
20 cm went under the following deformation. Its bases and its
height remained the same while its generators increased by 1 mm,
and thus it took the shape of two congruent frustum of cones joined at
their bases. In what proportion did the volume of the body increase?

**C. 554. **Prove that there exists a number *a *such that
log_{2}*x*+log_{3}*x*=log_{a}*x *holds for every positive number
*x*.

## New problems in October 1999 |

**B. 3302. **Eight people are waiting in a queue at a box
office, in some order. Each of them wants to buy a 10 dollar
ticket. Four of them have only 10 dollar notes while the others have
20 dollar notes only. Consider all possible orders of the eight
people. Assuming that there is no cash at the box office in the
beginning, determine the proportion of those orders in which the
cashier can issue the tickets without getting to any problem. (3
points)

**B. 3303. **A strip of paper, 5 cm wide, is folded without
creasing according to the figure.

How small can the shaded area, which the strip covers twice, be? (4 points)

**B. 3304. **Is it possible to assign the numbers 1, 2, ..., 11,
12 to the edges of a cube such that the numbers assigned to the edges
starting at a vertex is the same for each vertex of the cube? (4 points)

Proposed by *G. Reményi, *Budapest

**B. 3305. **Determine the value of the real parameter *N *such that the equation

2*x*^{2}+4*xy*+7*y*^{2}-12*x*-2*y*+*N*=0

has exactly one solution (*x,y*). (4 points)

**B. 3306. **In a triangle *ABC *of unit area, *D *and
*E *denote the points which trisect side *AB*, the midpoint
of side *AC *is denoted by *F*. Lines *FE *and *FD
*intersect line *CB *in *G *and *H*,
respectively. Determine the area of triangle *FGH*. (3 points)

**B. 3307.** The Figure shows a
square. *PA*=*AB*=*BC*=*CD*=*DE*. What is the
sum of angles *MAN*, *MBN*, *MCN*, *MDN*,
*MEN* shown in the diagram? (3 points)

*Russian competition problem*

**B. 3308. **Two prime numbers which differ by 2 are called twin
primes. Consider twin primes whose sum is a power of a prime. How many
such pairs can be found? (4 points)

Proposed by *Á. Kovács*, Budapest

**B. 3309. **On each face of a cube, consider those points which
divide a midline of the face in a ratio 1:3. How does the volume of
the polyhedron determined by these points relate to the volume of the
cube? (4 points)

**B. 3310. **Prove that the curve of equation
(*x*^{2}+*y*^{2})^{3}=27*x*^{2}*y*^{2} is
contained in a square of side 4. (5 points)

**B. 3311. **If , prove that

.

(5 points)

Based on a problem proposed by *Á. Kovács*, Budapest

## New advanced problems in October 1999 |

**A. 218. **Find all continuous functions which satisfy

.

**A. 219. **A set *H *consists of 2*n *points in the
plane, no three collinear. Let *P *and *Q *denote any two
distinct points of *H*. Segment *PQ *is called a *halving
segment *if the same number of points of *H *lie on each side
of line *PQ*. Assuming that *H *has exactly *n *halving
segments, prove that any two such segments intersect.

Proposed by *J. Solymosi, *Budapest

**A. 220. **Given any positive integer *n *in decimal
system, denote by *S*(*n*) the sum of its digits. Prove that

.

### 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.