# New exercises and problems in Mathematics

March 1999

## New exercises for practice in March 1999 |

**C. 533. **How does the length of the product of two
segments change when the length of the unit segment is doubled?

**C. 534. **Among the first 539 positive integers, some
numbers are chosen so that their sum is not less than one-third the
sum of the original numbers. At least how many numbers are chosen?

**C. 535. **How many (unordered) pairs can be formed from
positive integers such that, in each pair, the two numbers are coprime
and add up to 285?

**C. 536. **The whole surface of a sphere of radius *R
*and the curved surface of a cylinder of radius *R *and height
2*R *are covered with a coat of paint of the same
thickness. Which task requires more paint?

## New exercieses in March 1999 |

**Gy. 3262. **Given any 3x3 magic square, prove that the
sum of the squares of the numbers in the first row equals to the sum
of the squares of the numbers in the third row. (A magic square is an
*n*x*n *array of numbers where the sum of the numbers is the
same in each row, each column, and both diagonals of the
array.)

Proposer: *S. Kiss, *Nyíregyháza

**Gy. 3263. **Consider the whole numbers between 1222 and
1999. Choose, in every possible way, two of them, and take their
product. Determine the total sum of these products.

**Gy. 3264. **Determine those natural numbers *k *for
which 2^{2k+3}+3^{k+2}^{.}7^{k} is divisible by 17.

**Gy. 3265. **There are given a pair of scales and 12 balls
of the same size but of different colours. All of them, apart one, are
made of the same substance, the exceptional one having a different
density. Is it possible to find the ball which is different from the
others based on the result of three measurings? Is it possible to find
out, in addition, if this ball is lighter or heavier than the others?

**Gy. 3266. **In a cyclic trapezium, *a*, *b*,
*c *and *d *denote the common length of the legs, the
lengths of the two bases, and the common length of the diagonals,
respectively. I claim that *d*^{2}-*a*^{2}=*bc*. Am I right?

**Gy. 3267. **Circles *k*_{1} and *k*_{2}
intersect at points *P *and *Q*. Points *A *and *B
*lie on circles *k*_{1} and
*k*_{2}, respectively, such that
point *Q *is inside segment *AB*. Let *F *denote the
midpoint of *AB*. Let furthermore *C *and *D *denote
the second intersection points of line *PF *with the two circles,
respectively. Prove that *F *halves *CD*, too.

**Gy. 3268. **In the cartesian system, consider the graphs
of the functions *y*=sin *x* and *y*=sin^{2}*x*. These curves are not just similar, but
also homothetic. Find the ratio of the similarity. Where can the
centre of the enlargement be?

**Gy. 3269. **From each one of *N *towns, a spy is
sent to one of the nearest towns. How large can *N *be if all the
spies arrive to only two towns?

## New problems in March 1999 |

**F. 3274. **Solve the following system of equations:

*x*+*y*+*z*=0*x*^{3}+*y*^{3}+*z*^{3}=0*x*^{1999}-*y*^{1999}+*z*^{1999}=2^{2000}.

Proposer: *T. Nagy, *Zalaegerszeg

**F. 3275. **Find the fraction with the smallest
denominator in the open interval .

**F. 3276.**Let *A*, *B*, *C *denote
distinct points with integer coordinates in **R**^{2}. Prove that if
(|*AB*|+|*BC*|)^{2}<8^{.}[*ABC*]+1 then *A*,
*B*, *C *are three vertices of a square. Here |*XY*| is
the length of segment *XY *and [*ABC*] is the area of
triangle *ABC*.

*W. L. Putnam
Mathematical Competition, 1998 *

*
*

**F. 3277. **In a regular pentagon, *v*_{1}, *v*_{2},
..., *v*_{5} denote the vectors from
the centre to the vertices of the pentagon, respectively. Given that
integers *k*_{1}, *k*_{2}, *k*_{3},
*k*_{4}, *k*_{5} satisfy *k*_{1}*v*_{1}+*k*_{2}*v*_{2}+...+*k*_{5}*v*_{5}=0,
prove that *k*_{1}=*k*_{2}=...=*k*_{5}.

**F. 3278. **On the points of the curve with equation
*y*=*x*^{3}, an operation * is
defined as follows. Let, for any two points *A *and *B *on
the curve, *A***B *denote the reflection of the third
intersection point of line *AB *with the curve about the
origin. (If some two points involved in the definition coincide, then
the tangent to the curve at the given point is considered instead of
the connecting line.) Prove that operation * is associative.

**F. 3279. **Two straight lines form an angle of
36^{o} in the plane. A grasshopper is
springing to and fro from one line to the other so that the lengths of
its jumps are always the same. Prove that it can arrive to at most 10
different points.

## New advanced problems in March 1999 |

**N. 203. **A quadrilateral *ABCD *is given in the
plane. Circles *BCD *and *ACD *cut lines *AC *and *BC
*at points *E *and *F*, respectively. Prove that

**N. 204. **A fair die is thrown *n *times. Find those
values of *n *for which the probability that the total score is
divisible by 4 is exactly 1/4.

**N. 205. **Given an equilateral triangle, determine all
those points inside the triangle through which at least 3 different
lines, each halving the area of the triangle, can be drawn.

**N. 206. **Determine those continuous functions
*f*:*RR *for which *f*(1999)=1000 and which
satisfy
*f*(*xy*)=*f*(*x*)*f*(*y*)+*f*(-*x*)*f*(-*y*)
and *f*(*x*+*y*)*f*(*x*)+*f*(*y*) for any real
numbers *x *and *y*.