# New exercises and problems in Mathematics

February
2002

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

## New exercises in February 2002Maximum score for each exercise (sign "C") is 5 points. |

**C.** **660.** In how many different ways is it
possible to select two fields of a 8x8 chessboard, such that the
midpoint of the segment joining the centres of the two fields should
also be the centre of a field?

**C.** **661.** On what condition will the product of a
number ending in 9 and a number ending in 7 end in 63?

**C.** **662.** The volume of a brick is
8 cm^{3}. If each edge is increased by 1 centimetre,
the volume of the new brick will be 27 cm^{3}. What will
be the volume if each edge is now increased again by
1 centimetre?

**C.** **663.** A sheet of paper in the shape of an
acute triangle has the vertex with the largest angle torn off. On the
remaining piece of paper, construct the radius of the circumscribed
circle of the triangle.

Suggested by *S.* *Róka,* Nyíregyháza

**C.** **664.** Determine the distance between the
tangents drawn to the parabolas *y*=*x*^{2}+1 and
*x*=*y*^{2}+1 parallel to the line
*y*=*x*.

### Correction.

Exercise**C. 658.**in the previous issue contains an error. The correct problem is:

**C.** **658.** Solve the simultaneous equations

\(\displaystyle {1\over x}+{1\over y}={1\over z}\)

\(\displaystyle {1\over x+15}+{1\over y-6}={1\over z}\)

\(\displaystyle {1\over x+24}+{1\over y-15}={1\over z}.\)

The new deadline for this problem is 15 March 2002.

## New problems in February 2002The maximum scores for problems (sign "B") depend on the difficulty. It is allowed to send solutions for any number of problems, but your score will be computed from the 6 largest score in each month. |

**B.** **3522.** Solve the following equation on the set
of integers:
2*x*^{4}+*x*^{2}*y*^{2}+5*y*^{2}=*y*^{4}+10*x*^{2}. (*3* *points*)

**B.** **3523.** Given a semicircle, consider the
triangles in which the diameter of the semicircle is contained in one
side, and the other two sides are tangent to the semicircle. Which of
these triangles has minimum area? (*4* *points*)

**B.** **3524.** The sum of the real numbers *x*
and *y* is 1. Find the maximum value of
*xy*^{4}+*x*^{4}*y*. (*3* *points*)

**B.** **3525.** Prove that there are infinitely many
composite numbers in the sequence
1,31,331,3331,... . (*4* *points*)

**B.** **3526.** *F* is the midpoint of side
*BC* of a rectangle *ABCD*, and *H* divides side
*CD* in the ratio 1:2, *CH* being the shorter part. How
large may the angle *HAF* be? (*4* *points*)

**B.** **3527.** Given a trapezium in which the other
two sides are not parallel, construct, using a straight edge only, a
line whose segment between the non-parallel sides is divided into
three equal parts by the diagonals of the
trapezium. (*5* *points*)

**B.** **3528.** Let \(\displaystyle alpha\) and \(\displaystyle beta\) be acute angles such that
sin^{2}\(\displaystyle alpha\)+sin^{2}\(\displaystyle beta\)<1. Prove that sin^{2}+sin^{2}\(\displaystyle beta\)<sin^{2}(+). (*4* *points*)

**B.** **3529.** The sum of the first few terms in a
geometric progression is 11, the sum of their squares is 341, and the
sum of their cubes is 3641. Find the terms of the
sequence. (*5* *points*)

(Suggested by *Á.* *Besenyei,*
Budapest)

**B.** **3530.** Evaluate .
(*5* *points*)

**B.** **3531.** The inscribed circle of a triangle
*ABC* is centred at *O*, and touches the sides at
*A*_{1}, *B*_{1}, *C*_{1} (in
customary notation). The lines *A*_{1}*O*,
*B*_{1}*O*, *C*_{1}*O* intersect
the segments *B*_{1}*C*_{1},
*C*_{1}*A*_{1},
*A*_{1}*B*_{1} at the points
*A*_{2}, *B*_{2}, *C*_{2},
respectively. Prove that the lines *AA*_{2},
*BB*_{2} és *CC*_{2} are
concurrent. (*5* *points*)

(Suggested by *Á.* *Besenyei,* Budapest)

## New advanced problems in February 2002Maximum score for each advanced problem (sign "A") is 5 points. |

**A.** **284.** Let *f* be a function defined on
the subsets of a finite set *S*. Prove that if
*f*(*S*\*A*)=*f*(*A*) and max(*f*(*A*),*f*(*B*))\(\displaystyle ge\)*f*(*A\(\displaystyle cup\)B*) for all subsets *A*, *B* of *S*, then
*f* assumes at most |*S*| distinct values.

(Miklós Schweitzer Memorial Competition, 2001)

**A.** **285.** Prove that if
*a*^{2}+*ac*-*c*^{2}=*b*^{2}+*bd*-*d*^{2},
for the integers *a*>*b*>*c*>*d*>0,
then *ab*+*cd* is a composite number.

**A.** **286.** Find all continuous functions such
that

\(\displaystyle f\left({x+y\over1+xy}\right)={f(x)f(y)\over|1+xy|},\)

where 1+*xy\(\displaystyle ne\)*0. (based on the idea of *Z.* *Győrfi* and
*G.* *Ligeti*)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary