New exercises and problems in Mathematics January
2002

New exercises in January 2002
Maximum 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 20forint coins,
6 10forint coins, 5 5forint coins, 9 2forint coins,
and 3 1forint coins. What were the 7 numbers drawn?
Note: There exist the following coins in Hungarian currency: 100,
50, 20, 10, 5, 2, and 1forint (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 onedigit 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
\(\displaystyle {1\over x}+{1\over y}={1\over z}\)
\(\displaystyle {1\over x+y}+{1\over y6}={1\over z}\)
\(\displaystyle {1\over x+24}+{1\over y15}={1\over z}.\)
C. 659. For what values of the real parameter
0\(\displaystyle le\)t does the
equation sin (x+t)=1sinx have no solution?

New problems in January 2002
The 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=6a.
(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 20forint (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 2002
Maximum 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 or by email to:
Deadline: 15 February 2002
