New exercises and problems in Mathematics December
2002
 New exercises Maximum
score for each exercise (sign "C") is 5 points. 
C. 695. We decreased a sixdigit number
by the sum of its digits, and went on repeating same procedure with
the resulting number. Is it possible that we obtained 2002 in this
way?
C. 696. Solve the equation
x+3+px2=5, where p is a real
parameter.
C. 697. In the quadrilateral
ABCD,
AB=1, BC=2, ,
ABC\(\displaystyle \angle\)=120^{o}, and BCD=90^{o}.
Find the exact value of the length of side
AD.
C. 698. The length of side AB in
a triangle is 10 cm, the length of side AC is 5,1 cm,
and CAB\(\displaystyle \angle\)=58^{o}. Determine the measure of
BCA\(\displaystyle \angle\) to the nearest hundredth of a degree.
C. 699. What is the probability that at
least one of the five numbers drawn (out of 1 to 90) in a lottery was
also drawn last week? (There is one lottery draw per week.)
 New problemsThe 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. 3592. Santa Claus was watching the
sky anxiously, deep in contemplation. He wanted to travel as far away
as possible the day after to deliver gifts to children. Finally, at
midnight it started snowing. Being an expert on snowfall he saw
immediately that it was the kind of snow that would not stop falling
for at least 24 hours. He also knew that during the first
16 hours the sleigh can travel faster and faster. (The speed can
be increased uniformly.) It would be standing still at the beginning
but by the end of the 16th hour it would be flying like an
arrow. Then, however, it would be harder and harder to travel in the
thickening snow, and during the remaining 8 hours the top speed
would be uniformly decreasing back to zero. Santa Claus, on the other
hand did not want to exhaust his reindeer by forcing them for more
than 8 hours. When should he depart in order to cover the
greatest distance possible? (4 points)
B. 3593. Is there an arithmetic
progression of different positive integers in which no term is
divisible by any square number greater than 1?
(3 points)
B. 3594. Is there a square number in
decimal notation in which the sum of the digits is 2002?
(4 points)
B. 3595. Solve the following
equation:
2x^{4}+2y^{4}4x^{3}y+6x^{2}y^{2}4xy^{3}+7y^{2}+7z^{2}14yz70y+70z+175=0.
(3 points) (Suggested by
M. Haragos and M. Zsoldos, Budapest)
B. 3596. The circle
k_{1} of radius R touches the circle
k_{2} of radius 2R externally at the point
E_{3}, and the circles k_{1} and
k_{2} are also touched from the outside by the circle
k_{3} of radius 3R. The circles
k_{2} and k_{3} touch at the point
E_{1}, and the circles k_{3} and
k_{1} at the point E_{2}. Prove that the
circumcircle of the triangle
E_{1}E_{2}E_{3} is
congruent to the circle k_{1}. (3 points)
(Suggested by B. Bíró, Eger)
B. 3597. Is it true that if there
exists a line parallel to the bases of a given trapezium that halves
both its area and its perimeter then the trapezium is a parallelogram?
(4 points)
B. 3598. Isosceles triangles with apex
angles of 140^{o} are drawn over the sides AB
and BC of a given triangle ABC (on the outside). The new
vertices obtained are A_{1} and
C_{1}. Then an isosceles tiangle of apex angle
80^{o} is drawn over the side AC
(outside). Determine
C_{1}B_{1}A_{1}?
(4 points)
B. 3599. A right truncated cone is
circumscribed about a sphere. What is the maximum possible ratio of
the volume of the truncated cone to its surface area?
(4 points)
B. 3600. Find a cube in the 3D
cartesian system whose edges are not parallel to the coordinate axes
but have an integer length. (5 points)
B. 3601. Ann and Sophie take turns
rolling a die. The number shown by the die is always added to their
respective scores. Whoever obtains the first score divisible
by 4, wins the game. Given that Anna starts the game, what is the
probability that she will win? (5 points)
 New advanced problems Maximum score
for each advanced problem (sign "A") is 5 points. 
A. 305. Prove that if n is an
arbitrary positive integer then
\(\displaystyle \sum_{\textstyle{k_1,\dots,k_n\geq0\atop k_1+2k_2+\dots+nk_n=n}}
\frac{(k_1+k_2+\dots+k_n)!}{k_1!\cdot\ldots\cdot k_n!}=2^{n1}.\)
A. 306. The orthocentre of the triangle
ABC is M, and its incircle, centred at O, touches
the sides AC and BC at the points P and
Q. Prove that if M lies on the line PQ then the
line MO passes through the midpoint of the side AB.
A. 307. Let a_{n}
denote the coefficient of x^{n} in the
polynomial (x^{2}+x+1)^{n}. Prove
that if p>3 is a prime number then
a_{p}1 (mod
p^{2}).
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 January 2003
