# New exercises and problems in Mathematics

January
2001

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

## New exercises in January 2001 |

**C. 610.** The target on the diagram consists of
three concentric circles and two perpendicular lines crossing at the common
centre of the circles. Given that the respective areas of the twelve regions
are equal find the ratios of the radii of the circles.

**C. 611.** Find the decimal digits *a*,
*b*, *c*, *d*, *e* if the two numbers formed by these
digits satisfy the equality .

**C. 612.** Triangle *ABC* is right angled
at *C*. The foot of the altitude from *C* is *D* and the feet of
the perpendiculars from *D* on the legs are *P* and *Q*
respectively. Show that *DP*+*DQ* does not exceed the harmonic mean
of the legs.
(The harmonic mean of two numbers *a* and *b* is the reciprocal of
the average of their reciprocals, i.e.,
*H*(*a,b*)=2*ab*/(*a*+*b*).)

*I. Varga,* Nagyszalonta

**C. 613.** The line parallel to the side
*BC* of the triangle *ABC* intersects the lines *AB* and
*AC* at the points *D* and *E* respectively. Let *M* be an
arbitrary point interior to the side *BC*. Given that the areas of the
triangle *ABC* and *ADE* are *T* and *t* respectively, find
the area of the quadrilateral *ADME*.

**C. 614.** Find those values of the real
parameter *m* for which the following equation has no solution.

*m*sin ^{2}*x*+(*m*-1)sin*x*+*m*-2=0.

*G. Reményi,* Budapest

## New problems in Decober 2000 |

**B. 3422.** Show that 7^{2001}-3^{3335} is divisible by
100. (3 points)

**B. 3423.** Is it possible that the points of
tangency of the incircle of a triangle form an obtuse triangle? (3 points)

**B. 3424.** A square is inscribed in an
equlateral triangle in such a way that its vertices are on the perimeter of the
triangle. One side of the square is the side of a smaller equilateral triangle
where there is a smaller square inscribed similarly to the first
one. Proceeding endlessly with smaller and smaller squares what is the ratio of
the total area covered by the squares and that of the triangle. (3 points)

**B. 3425.** Consider the following equalities:
3+4+5+6=3^{.}6; 15+16+...+34+35=15^{.}35. They provide examples
for sequences of consecutive integers whose sum is equal to the product of the
first and the last terms. Prove that there are infinitely many finite sequences
of this property. (5 points)

*S. Kiss,* Nyíregyháza

**B. 3426.** Find the remainder when the
polynomial *x*^{2001} is divided by
(*x*+1)^{2}. (4 points)

**B. 3427.** Reflect each side of the acute
angled triangle *ABC* through the two altitudes that are not perpendicular
to the respective sides. Three of the six mirror images thus produced form a
triangle *A*_{1}*B*_{1}*C*_{1} in the interior
of the triangle *ABC* while the remaining three lines form the triangle
*A*_{2}*B*_{2}*C*_{2}. The measures
of two angles of the latter triangle are 20^{o}
and 70^{o} respectively. Find the angles of the
triangle *A*_{1}*B*_{1}*C*_{1}. (4 points)

*G. Bakonyi,* Budapest

**B. 3428.** Show that the following relation
holds in any triangle:

(*R* is the circumradius, *r* is the inradius and
*r*_{a}, *r*_{b} and *r*_{c} denote the radii of the excircles respectively.)
(4 points)

*P. Tőricht* and *T. Árokszállási,* Paks

**B. 3429.** Let and let *f*:**NN** be a function satisfying

for every *n***N**. Prove that
*f*(*f*(*n*))=*f*(*n*)+*n*. (4 points)

**B. 3430.** Given are the points
*P*_{1}, *P*_{2}, ..., *P*_{n}
in the plane. Subscript *i* is called wrong if the corresponding point
*P*_{i} is not incident to the line
connecting its neighbours *P*_{i-1}
and *P*_{i+1} and, additionally, the
orientation of the triple *P*_{i-1},
*P*_{i}, *P*_{i+1} in this order is negative. For any wrong
subscript *i* the point *P*_{i+1}
can be replaced by its mirror image through the midpoint of the segment
*P*_{i}*P*_{i+2}. This operation can be performed as long as
there are wrong subscripts. (The subscripts are used in the usual cyclic
manner, i.e. *P*_{n+1}=*P*_{1},
*P*_{0}=*P*_{n}, etc.) Show that no matter how the points are
given the above procedure terminates. (5 points)

**B. 3431.** The supersafe of *TweedleBank*
is furnished with several locks. There are *n* tellers working in the Bank
and each of them is holding some keys for the safe. Any teller can have more
than one keys and there might also be locks for which there are more than one
tellers are provided with a key. Given that any *k* tellers are able to
open the safe together but there are no *k*-1 of them who can do this find
the minimum number of locks on the safe. (5 points)

*V. Vígh,* Szeged

## New advanced problems in January 2001 |

**A. 254.** Given are the points
*P*_{1}, *P*_{2}, ..., *P*_{n}
in the plane. Subscript *i* is called wrong if the corresponding point
*P*_{i} is not incident to the line
connecting its neighbours *P*_{i-1}
and *P*_{i+1} and, additionally, the
orientation of the triple *P*_{i-1},
*P*_{i}, *P*_{i+1} in this order is negative. For any wrong
subscript *i* the point *P*_{i+1}
can be replaced by its mirror image through the segment *P*_{i}*P*_{i+2}. This operation can be performed as long as
there are wrong subscripts. (The subscripts are used in the usual cyclic
manner, i.e. *P*_{n+1}=*P*_{1},
*P*_{0}=*P*_{n}, etc.) Show that no matter how the points are
given the above procedure terminates.

**A. 255.** Let and define the function *f* on the set of positive
integers as follows.

Start with *f*(1)=2 and let *n>*1.

If the values *f*(1), ..., *f*(*n*-1) have
already been defined and *n* is not among these values then set
*f*(*n*)=*f*(*n*-1)+1;

otherwise consider *k*, the smallest positive integer,
for which *f*(*k*)=*n* and let
*f*(*n*)=*n*+*k*.

Prove that

for any positive integer *n*.

**A. 256.** *n* marbles are to be measured
with the help of a one-armed balance. We are allowed to measure one or two
marbles at a time. It may happen, however, that one of the results is
misread. Denote by *f*(*n*) the minimum number of weighings
neccessary to determine the weight of each marble. Prove that

*f*(*n*)<*n*+log_{3}*n*+3.

### 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.