# New exercises and problems in Mathematics

April 2000

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

## New exercises in April 2000 |

**C. 580.** Jane took several exams in math during the
academic year, and once she did quite badly. When repeating the poor
exam, the teacher offered that instead of the two grades, she can have
the average of the two grades. Show that this is advantageous for Jane
if and only if the average of all her other grades is better than the
average of these two.

*G. Kovács,* Budapest

**C. 581.** Four vehicles start at the same time from
*A *and arrive at *B *one after the other, at equal
intervals. The speed of the fastest and the slowest vehicle is
*v*_{1} and *v*_{4}, respectively. Calculate the speed of the two
other vehicles.

**C. 582.** The strip depicted on the figure is 1m
wide. Calculate its area.

*Z. Nagy,* Budapest

**C. 583.** The base of a pyramid is a unit
square. One of its lateral edges is also of unit length, moreover, it
coincides with the altitude of the pyramid. Calculate the largest
dihedral angle of the pyramid.

*M. Salát,* Budapest

**C. 584.** The area or the perimeter of a circle can
be better approximated (that is, with the smaller relative error)
using a regular *n*-gon inscribed the circle?

## New problems in April 2000 |

**B. 3362.** Eighteen students solved the problems proposed
in the April issue of the KöMaL, one problem each. They plan to
discuss their solutions via e-mail. At least how many e-mail messages
have to be sent so that each student knows each solution in the end?
(An e-mail message counts as many times as many recipients it is sent
to.) (5 points)

**B. 3363.** Is there any triangle which can be dissected
into three congruent parts by just two straight cuts? (3 points)

**B. 3364.** Assume that 15 different (positive) prime
numbers are consecutive terms of an arithmetic progression. Prove that
the common difference is larger than 30000. (3 points)

*T. Székelyhidi*, Budapest

**B. 3365.** Assume that *abc*=1 and . Prove that at least one of the
numbers *a*, *b*, *c *equals 1. (3 points)

*N. Gyanta*, Budapest

**B. 3366.** The grashopper of
**F. 3279. **rides again. This time it is jumping to and fro
between the points of a circle and a straight line which is not
passing through the center of the circle. The length of each jump is
*r*, the radius of the circle. Prove that there are at most 8
points for the grasshopper to reach. (5 points)

*A. Hraskó*, Budapest

**B. 3367.** An isosceles triangle shaped piece of paper is
folded such that vertex *A *is moved onto the midpoint of leg
*BC*. One endpoint of the folding edge divides leg *AC *into
the ratio *p*:*q*. In what ratio does the other endpoint of
the folding edge divide side *AB*? (5 points)

**B. 3368.** How many positive integer solutions does the
equation *x*^{7}+*y*^{8}=*z*^{9} have?
(4 points)

*A. Behm*, Nagykanizsa

**B. 3369.** On each edge of a tetrahedron, select one of
the points that trisect the given edge. What can be the ratio between
the volume of the convex polyhedron determined by these points and the
volume of the tetrahedron?(4 points)

**B. 3370.** *A *and *B *are two events. Prove
that |*P*(*A*^{.}*B*)-*P*(*A*)^{.}*P*(*B*)|1/4. (5 points)

**B. 3371.** Two ellipses have a common focus. How many
common tangents can the ellipses have? (5 points)

## New advanced problems in April 2000 |

**A. 227.** (Modified version of the problem proposed in
January 2000.) Is there a positive integer *n *such that for
every digit *i*, different from 0, *i *appears the same
number of times in the decimal form of each of the numbers *n*,
2*n*, 3*n*, ..., 2000*n*?

**A. 236.** In a triangle of area *t*, let
*s*_{a}, *s*_{b} and *s*_{c} denote the medians that correspond to
sides *a*, *b *and *c*, respectively. Assume that
*a*<*b*<*c*. Prove that 4*acs*_{a}*s*_{c}16*t*^{2}+(*b*^{2}-*a*^{2})(*c*^{2}-*b*^{2}).

**A. 237.** Prove that among any *kn*+1 different
diagonals of a convex *n*-gon there are 2*k*+1 which form an
opened polygon that does not intersect itself.

**A. 238.** Let *x *be a positive real number and let
*k *be a positive integer. Assume that *x*^{k}+1/*x*^{k} and *x*^{k+1}+1/*x*^{k+1} are both rational numbers. Prove that
*x*+1/*x*, is also a rational number.

### 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.