# New exercises and problems in Mathematics

January
2002

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

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

**C.** **655.** Whenever there is a lottery draw,
Barnie's grandmother sets aside some change in a piggy bank for her
grandson. Grandma is a lady of firm discipline; she always observes
the following rules:

1) She only puts coins in the piggy bank.

2) When a number is drawn, she immediately places that number of forints (HUF) in the piggy, using the lowest possible number of coins.

3) When the lottery draw is over, she counts how many of each kind of coin she has put in the piggy.

One day, when 7 numbers were drawn out of the first 35 positive integers, she counted 3 20-forint coins, 6 10-forint coins, 5 5-forint coins, 9 2-forint coins, and 3 1-forint coins. What were the 7 numbers drawn?

Note: There exist the following coins in Hungarian currency: 100, 50, 20, 10, 5, 2, and 1-forint (HUF) coins.

**C.** **656.** A jacket with an original price of
21,250 forints (HUF) was put on sale. Then, with the start of the
great Christmas sale, its price was reduced again, this time to 19,176
forints. By what percentage was the price reduced each time, given
that both percentages of reduction were one-digit numbers?

**C.** **657.** A cone and a cylinder have equal heights
and equal volumes. Find the apex angle of the cone if the areas of the
lateral surfaces are also equal!

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

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

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

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

## New problems in January 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.** **3512.** Find the lowest multiple of 81 in which
*a*) all digits are ones. *b*) all digits are
ones and zeros. (4 points)

**B.** **3513.** Prove that if two circles are not
coplanar and they have exactly two common points, then there exists a
sphere that contains both circles on its surface.
(3 points)

**B. 3514.** The altitudes of triangle *ABC* are
*AA*_{1}, *BB*_{1} and
*CC*_{1}, its area is *a*. Prove that the triangle
*ABC* is equilateral if and only if
*AA*_{1}^{.}*AB*+*BB*_{1}^{.}*BC*+*CC*_{1}^{.}*CA*=6*a*.
(3 points)

**B.** **3515.** Prove that there exist infinitely many
powers of 2 among the numbers of the form . (4 points)

**B.** **3516.** We have drawn a closed polygon of
length 5 cm on a sheet of paper. Prove that it is possible to
cover it with a 20-forint (HUF) coin. (The diameter of the coin is
more than 2.5 cm.) (3 points)

**B.** **3517.** Prove that 71 is a factor of 61!+1.
(4 points)

Suggested by *S. Róka,* Nyíregyháza

**B.** **3518.** Is it true that if the area of each
face of a tetrahedron is equal to the area of a corresponding face of
another tetrahedron then the volumes of the two tetrahedra are equal?
(4 points)

**B.** **3519.** *B*_{i} and
*C*_{i} are interior points on the edges
*OA*_{i} of a tetrahedron
*OA*_{1}*A*_{2}*A*_{3}, such
that the lines *A*_{1}*A*_{2},
*B*_{1}*B*_{2} and
*C*_{1}*C*_{2} are concurrent, and the lines
*A*_{1}*A*_{3},
*B*_{1}*B*_{3} and
*C*_{1}*C*_{3} are also concurrent. Prove
that the lines *A*_{2}*A*_{3},
*B*_{2}*B*_{3} and
*C*_{2}*C*_{3} are either parallel or also
concurrent. (4 points)

**B.** **3520.** We are painting certain real numbers
red according to the following principle: If a number *x* is red,
then the numbers *x*+1 and \(\displaystyle {x\over x+1}\) are also painted red. What
numbers will be red if 1 is the only red number at start?
(5 points)

**B.** **3521.** Every individual bacterium in a colony
will either die or divide (into two) when it is one hour old. If the
probability of division is *p*, what is the probability that the
offspring of a particular bacterium will never die out?
(5 points)

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

**A.** **281.** Given a finite number of points in the
plane, no three of which are collinear, prove that it is possible to
colour the points with two colours (red and blue) so that every half
plane that contains at least three points should contain both a red
point and a blue point.

*L. Soukup,* Budapest

**A.** **282.** Are there rational functions *f*,
*g* and *h* of rational coefficients such that

(*f*(*x*))^{3}+(*g*(*x*))^{3}+(*h*(*x*))^{3}=*x*?

**A.** **283.** Let *n* be an integer. Prove that
if the equation
*x*^{2}+*xy*+*y*^{2}=*n* has a
rational solution, then it also has an integer solution.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary