# New exercises and problems in Mathematics

November 2000

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

## New exercises in November 2000 |

**C. 600.** We are travelling in a car by constant
velocity from Budapest to Kosice. Passing by a kilometre mark we notice a
two-digit number on it. After half an hour we reach at another mark which shows
the same two digits in the reverse order. After another 30 minutes of
travelling we notice the same two digits again, but this time there is also a 0
digit on the sign. Determine the velocity of the car.

**C. 601.** Solve the equation

Proposed by: *N. Gyanta,* Budapest

**C. 602.** A large solid cube is built of identical
smaller cubes such that more than half of the small cubes are not visible from
outside. At least how many small cubes are used to build the large cube?

**C. 603.** Determine the range of the function
*f*(*x*)=(*x*^{2}+*x*+1)/(*x*^{2}+1).

**C. 604.** Given a triangle *ABC* whose incenter is
*O*, and its area is *t*. Prove that 2*t*=*AO*^{2}sin +*BO*^{2}sin +*CO*^{2}sin .

Proposed by: *S. Mihalovics,* Esztergom

## New problems in Oktober 2000 |

**B. 3402.** The students of two schools were sitting the
same test. The students attending the first school scored 74 on average; within
this the average scores of the boys and the girls were 71 and 76,
respectively. The corresponding scores in the other school were 84, 81 and 90,
respectively. If the average score of all boys was 79, what did the girls score
on average? (3 points)

**B. 3403.** Given that the real numbers *a*,
*b*, *c* satisfy , prove that
(3 points)

Proposed by: *T. Székelyhidi,* Ercsi

**B. 3404.** In a cube with ten-unit long edges, let
*A*, *B* and *C* denote the endpoints of three edges starting
from the same vertex. Two planes *S*_{1} and
*S*_{2}, parallel to the plane *ABC*,
divide the cube into three parts whose volumes are in the ratio
251:248:251. Calculate the distance between the two planes. (5 points)

**B. 3405.** Five marbles are to be measured with the
help of a one-armed balance. We are allowed to perform nine weighings and each
time we may put either one or two marbles on the scale. It may happen, however,
that one of the results is misread. Is it still possible to determine the
weight of each marble? (4 points)

Proposed by: *L. Surányi,* Budapest

**B. 3406.** For each vertex (*x,y*) of a polygon
*x* and *y* are positive integers such that *x*|2*y*+1 and
*y*|2*x*+1. Find the maximum area of such a polygon. (4 points)

Proposed by: *S. Mihalovics,* Esztergom

**B. 3407.** Arnold, Bob, Chris, Dorothy, Eric and Frank
are to play Dungeons and Dragons again. They have not appointed the Dungeon
Master yet. The following method is used to select the Dungeon Master. They
throw the 10 sided die used in the game one by one in a given order. The one
who throws the first 1 or 2 is the Dungeon Master. If, by the end of the round,
they still do not have the Dungeon Master, they start again in the same order
until someone finally throws 1 or 2. Dorothy has never been a Dungeon Master
yet, and now she is really excited about that. The guys are really polite and
let her choose her position in the order of throws. What would you advise
Dorothy? (3 points)

Proposed by: *Gy. Pogány,* Szolnok

**B. 3408.** An integer is assigned to each vertex of a
regular 2*n*-gon such that numbers assigned to neighbouring vertices
always differ by 1. Numbers that are larger than both of their neighbours are
called *mountains* and the ones which are smaller than their neighbours
are called *valleys*. Prove that the sum of the mountains minus the sum of
the valleys is *n*. (5 points)

Proposed by: *A. Hraskó,* Budapest

**B. 3409.** Reflect the parabola
*y*=*x*^{2} in the point (1,1) and
determine the equation of its image. (3 points)

**B. 3410.** A large painting, three metres in height,
hangs on a vertical wall and the lower edge is one metre above the observer's
eye level. The best view of the painting is assumed to be when the angle
subtended by the painting at the observer's eye is a maximum. Calculate how far
from the wall the observer should stand to maximize this angle. (4 points)

**B. 3411.** A regular polygon approximates the area of
its circumcircle with a relative error less than one thousandth. Find the
minimum number of sides such a polygon should have. (4 points)

## New advanced problems in November 2000 |

**A. 248.** Consider partitions of the power set of an
*n*-element set into two parts *I* and *H* such that the
following conditions are simultaneously satisfied:

*a*)

*abI*and

*abI*for any

*a*,

*bI*;

*b*) *abH* and *abH* for any *a,bH*;

*c*) *abI* and *abH* for any *aI*,
*bH*.

How many such partitions are there?
(The *power set* of a set is the set of all its subsets.)

Proposed by: *E. Csirmaz,* Budapest

**A. 249.** Let *a*, *b*, *c*, *t* be
positive numbers such that *abc*=1. Prove that

**A. 250.** A horizontal board is a support for four
vertical pegs. One of the pegs passes through holes in *n* disks of
increasing size, the largest disk being at the bottom, the smallest one on the
top. We may take disks off a peg, always one at a time, and put each disk on
another peg, but never in such a way that a disk is put on top of a smaller
disk. Prove that the minimum number of transfers one should make to transfer
all the disks to another peg is at least
but less than .

Based on a problem proposed by *B. Énekes*, Tolna

### 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.