# New exercises and problems in Mathematics

February
2001

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

## New exercises in February 2001 |

**C. 615.** A piece of music is composed of three
movements; its total playing time is 60 minutes. No movement is any longer then
the other two altogether. The difference between the playing times of any two
movements is at least 3 minutes. Determine the range for the playing time of
the shortest movement.

**C. 616.** Eight people are travelling together
in a railway compartment. Is it possible that the respective numbers of
acquaintances in the compartment are

a) 1, 3, 3, 4, 5, 6, 6, 6;

b) 1, 3, 3, 4, 5, 6, 6, 7?

**C. 617.** The midpoint of the base *AB* of
the isosceles triangle *ABC* is denoted by *D*. Let *P* denote
the foot of the perpendicular dropped from *D* to *CB* and *F*
the midpoint of segment *DP*. Express the area of the quadrilateral
*AFPC* in terms of the length of *CF* and *AP*.

**C. 618.** Take any rectangle *ABCD* such
that *AB>BC*. The point *P* is on *AB* and *Q* is on
*CD*. Show that there is exactly one position of *P* and *Q*
such that *APCQ* is a rhombus. Show also that, by choosing a suitable
rectangle, the ratio of the area of the rhombus to the area of the rectangle
can take any value strictly between 1/2 and 1.

**C. 619.** Determine the number of such integers
*n*, lying between 1 and 100000 so that *n*^{3}+23*n* is a multiple of 24.

## New problems in February 2001 |

**B. 3432.** The array below is constructed in a
similar way as Pascal's triangle. The first row in the array contains the
numbers 1 and 2, every further entry is the sum of the two neighbours above.

1 | 2 | |||||||

1 | 3 | 2 | ||||||

1 | 4 | 5 | 2 | |||||

1 | 5 | 9 | 7 | 2 | ||||

*a) *Determine the sum of the entries in the 100*th* row.

*b) *What do we get if the entries in the 100*th* row are
summed with alternating signs?

*c) *Determine the 47*th* element in the 100*th* row. (4
points)

**B. 3433.** The participants of a geometry
conference got curious sandwiches with the refreshments. On the top of each
cuboid-shaped slice of bread there was a thick slice of cheese in the shape of
a cylinder and also a spherical meat ball. Is it always possible to cut such a
sandwich into two parts with a straight cut such that the two parts contain the
same amount of each ingredient? (4 points)

**B. 3434.** A trapezium has bases *a* and
*b*. A line parallel to its bases cuts the trapezium into two parts such
that a circle can be inscribed into each sub-trapezium. Determine the perimeter
of the original trapezium. (3 points)

**B. 3435.** Find all non-negative solutions of
the following system of equations:

*x*_{1}+*x*_{2}+*x*_{3}+...+*x*_{100}=5050,

*x*_{2}^{2}-*x*_{1}^{2}=3*, ..., x*_{k}^{2}-*x*_{k-1}^{2}=2*k*-1*, ... x*_{100}^{2}-*x*_{99}^{2}=199.

(4 points)

**B. 3436.** Triangle *ABC* has a unit
area. Points *A*_{1}, *B*_{1} and *C*_{1} lie on the
sides *BC*, *CA* and *AB*, respectively, such that
*AC*_{1}/*C*_{1}*B*=1/2, *BA*_{1}/*A*_{1}*C*=1/3,
*CB*_{1}/*B*_{1}*A*=1/4. Determine the area of the triangle enclosed
by the lines *AA*_{1}, *BB*_{1} and *CC*_{1}. (3
points)

**B. 3437.** Points *A*, *B*, *C*,
*D* are, in this order, incident to a line such that *BC*=2*AB*
and *CD*=*AC*. Draw two circles; one passing through points *A*
and *C*, the other incident to *B* and *D*. Prove that the
common chord of the two circles halves the segment *AC*. (4 points)

**B. 3438.** The function *f*(*x,y,z*)
satisfy the equations

*f*(*x*+*t,y*+*t,z*+*t*)=*t*+*f*(*x,y,z*)

*f*(*tx,ty,tz*)=*tf*(*x,y,z*)

*f*(*x,y,z*)=*f*(*y,x,z*)=*f*(*x,z,y*)

for every real number *t*. Determine the value of
*f*(2000*,* 2001*,* 2002). (5 points)

**B. 3439.** Consider the infinite sequence 1,
*p*, 1\over *p*, *p*^{2},
1/*p*^{2}, ..., *p*^{n}, 1/*p*^{n}, ..., where *p* is a positive number. Find
those values of *p* for which every positive number can be arbitrarily
closely approximated with a finite sum of suitable elements of the above
sequence. (5 points)

**B. 3440.** Points *A*_{1}, *B*_{1},
*C*_{1} and *A*_{2}, *B*_{2},
*C*_{2} are obtained by reflecting the points
*A*, *B*, *C* in a line *e* and in a point *P*,
respectively. Assume that . What is the
mirror image of *P* through *e*? (3 points)

**B. 3441.** Pinocchio has to pass 9 obstacles
before he can turn into a real boy. It is not easy: any time he fails he has to
go back to the previous obstacle he just passed and try it once more. If he
ever fails at the first obstacle he will remain a wooden doll forever. Stubborn
Pinocchio does not learn from his failures: the probability of success at the
given obstacles is always the same, namely 1/10, 2/10, ..., 9/10 respectively,
no matter how many times had he tried them before. How should The Fairy With
The Blue Hair arrange the obstacles to maximize the probability that Pinocchio
turns into a real boy? What is this probability? (5 points)

## New advanced problems in February 2001 |

**A. 257.** *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 necessary
to determine the weight of each marble. Prove that
*f*(*n*)>*n*+log _{3}*n*-3.

Based on a problem proposed by *L. Surányi* and *B. Virág
*

**A. 258.** Six points, *A*, *B*,
*C*, *D*, *E* and *F* are on a line *e* such that . Given a ruler, construct a line parallel to
*e*.

**A. 259.** Suppose that *u>*1 and the
positive numbers *x*_{1}, *x*_{2}, ..., *x*_{n}
and *y*_{1}<*y*_{2}<...<*y*_{n} satisfy the inequality

*x*_{1}^{u}+*x*_{2}^{u}+...+*x*_{k}^{u}*y*_{1}^{u}+*y*_{2}^{u}+...+*y*_{k}^{u}

for every 1*kn*. Prove that

*x*_{1}+*x*_{2}+...+*x*_{n}*y*_{1}+*y*_{2}+...*y*_{n}.

### 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.