# New exercises and problems in Mathematics

March
2001

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

## New exercises in March 2001 |

**C. 620. **A trapezium is called *symmetrical* if the
perpendicular bisectors of its bases coincide. Is it true that every trapezium
which has a symmetry is symmetrical?

**C. 621. **The baker prepares raisin bread. How many raisins
should he add to the dough in order to ensure that every 40 gram slice of the 1
kilogram loaf will contain a raisin with a probability at least 0.99?

**C. 622. **Find all 6-digit numbers , divisible by
7, for which \(\displaystyle \overline{ABC}\) is also divisible by 7. (Here *A*, *B* and
*C* denote different digits.)

**C. 623. **Solve the system of the following equations:
*a*^{2}*b*^{2}-*a*^{2}-*ab*+1=0,
*a*^{2}*c*-*ab*-*a*-*c*=0,
*abc*=-1.

Proposed by *S.* *Kiss,* Szatmárnémeti

**C. 624. **The diameter of a standard table tennis ball is
4 cm. Is it possible to pack 100000 such balls in a box whose dimensions
are 200 cm, 164 cm and 146 cm?

## New problems in March 2001 |

**B. 3442. **Is there any 2001-digit number that can be
obtained as the last 2001 digits of its 2001^{st} power? (4
points)

Proposed by: *T.* *Kiss,* Budapest

**B. 3443. **The sides *CB* and *CD* of a rectangle
*ABCD* are extended beyond *B* and *D*, respectively, and points
*F* and *E* are produced on these lines such that
*BF*=*DE*. Prove that the line connecting *A* with the
intersection point of lines *EB* and *FD* halves the angle at vertex
*A*. (3 points)

**B. 3444. **In a sequence *a*_{1},
*a*_{2}, ... of positive integers, let *f*_{n}
denote the number of times the positive integer *n* occurs in the
sequence, provided that this number is finite. If *f*_{n}
is finite for every *n*, then the sequence *f*_{1},
*f*_{2}, ... is called the *frequency sequence* of
*a*_{1}, *a*_{2}, .... If the sequence
*f*_{1}, *f*_{2}, ... also admits a frequency
sequence, it is then called the 2^{nd} frequency sequence of the
sequence *a*_{1}, *a*_{2}, .... Frequency sequences
of higher order are defined similarly. Is there any sequence which admits a
*k*^{th} frequency sequence for every positive integer
*k*? (4 points)

**B. 3445. **Is there any triangle which can be dissected into
two parts by a line parallel to one of its sides such that the two parts have
equal area and also equal perimeter? (4 points)

JavasoltaProposedBy: *G.* *Reményi,* Budapest

**B. 3446. **Let
*p*(*x*)=*x*^{2}+*x*+1. Determine the polynomial
*q*(*x*) such that
*p*^{2}(*x*)-2*p*(*x*)*q*(*x*)+*q*^{2}(*x*)-4*p*(*x*)+3*q*(*x*)+3
is identically 0. (4 points)

**B. 3447. **Each side of triangle *ABC* is extended, in
clockwise direction, to twice its length. Next, each side of the triangle
*A*_{1}*B*_{1}*C*_{1} thus obtained is
extended, this time in counter-clockwise direction, to twice its length,
yielding triangle
*A*_{2}*B*_{2}*C*_{2}. Prove that this
triangle is similar to triangle *ABC*. Find a similar statement for
quadrilaterals. (4 points)

**B. 3448. **Inscribe a triangle in a given ellipse such that
its centroid coincides with the centre of the ellipse. Prove that the area of
the triangle is independent of the position of the triangle. (4 points)

Proposed by: *S.* *Kiss,* Szatmárnémeti

**B. 3449. **Let *a*>1 be an integer such that the
Diophantine equation *x*^{2}-*ay*^{2}=-1 has a
solution. Prove that it has an infinite number of solutions. (5 points)

Proposed by: *E.* *Fried,* Budapest

**B. 3450. **Given a tetrahedron *T*, consider those
planes parallel to the faces of *T* which cut off 1/3 the volume of
*T*. What proportion of the volume of *T* remains after chopping off
each vertex of *T* with these planes? (5 points)

**B. 3451. **Solve the equation \(\displaystyle \arcsin x+\arcsin\sqrt{15}x=\pi/2\). (4 points)

## New advanced problems in March 2001 |

**A. 260. **Four points, *A*, *B*, *C* and
*D*, are given on a line *e* such that \(\displaystyle \overrightarrow{AB}=\overrightarrow{CD}\). Is it
possible to construct a line parallel to *e* with the help of a single
ruler?

**A. 261. **Alice and Bob assemble equilateral triangles of
unit side to form an equilateral triangle of side *n*. They play the
following game on the triangular grid thus obtained. They move alternately,
Alice moving first. At each move they colour one grid point red, blue or
green. When every point is coloured, they count those triangles of unit side
whose vertices have three different colours. Alice gets a score for each of
those triangles whose red, blue and green vertices are in clockwise order,
while Bob gets a score for each red-blue-green triangle oriented
counter-clockwise. The one who has the higher score, wins. Which player has a
winning strategy?

**A. 262. **Let 0<*a*<*b*<*c* be
positive integers. Prove that there exist integers *x*, *y* and
*z*, not all 0, such that *ax*+*by*+*cz*=0, and |*x*|,
|*y*|, |*z*| are all less than \(\displaystyle {2\over{\sqrt3}}\sqrt c\).

Based on a problem of the 2000 Schweitzer Competition

### 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.