New exercises and problems in Mathematics January
2003
 New exercises Maximum
score for each exercise (sign "C") is 5 points. 
C. 700. A quadrilateral is cut out of a
sheet of paper, and each vertex is folded in so that they meet at a
common point. What kind of quadrilateral should that be in order for
the folded parts to cover the rest of the quadrilateral without
overlap?
C. 701. Show that
1^{.}2^{.}...^{.}1001+1002^{.}1003^{.}...^{.}2002
is divisible by 2003.
C. 702. The acute angles of a right
triangle are 60^{o} and 30^{o}. Two
circles of the same radius are inscribed in the triangle so that they
touch each other, the hypotenuse and one leg each. By what factor is
the smaller leg longer than the radius of the circles?
C. 703. Depending on the value of the
real parameter p, how many roots does the equation
2x^{2}10px+7p1=0 have in the interval
(1;1)?
C. 704. For what natural numbers
n is it true that
log_{2}3^{.}log_{3}4^{.}log_{4}5^{.}...^{.}log_{n}(n+1)=10?
 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. 3602. Jerry is swimming in a square
pool. Tom is watching from the edge of the pool and Jerry wants to get
away from him. Tom cannot swim at all, cannot run as fast as Jerry but
he runs four times as fast as Jerry swims. Can Jerry always escape?
(5 points)
B. 3603. In the interior of a given
triangle, construct a point such that its ratio of the distances from
the lines of the sides are in a 1:2:3
proportion. (3 points)
B. 3604. x, y are real
numbers, and x+y=1. Determine the maximum value of the
expression
A(x,y)=x^{4}y+xy^{4}+x^{3}y+xy^{3}+x^{2}y+xy^{2}.
(3 points)
B. 3605. Point D lies on the
extension of the side CA of a triangle ABC beyond the
point A, point E lies on the extension of CB
beyond B, and AB=AD=BE. The angle
bisectors from A and B intersect the opposite sides at
the points A_{1} and B_{1},
respectively. Find the area of the triangle ABC, given that the
area of triangle DCE is 9 units and that of triangle
A_{1}CB_{1} is
4 units. (3 points) (Suggested by
G. Bakonyi, Budapest)
B. 3606. Find suitable integers
a and b, such that \(\displaystyle 20034 points)
B. 3607. The lines containing the
opposite sides of a convex quadrilateral intersect each other. Draw
the interior angle bisectors of the angles formed at the
intersections. Prove that the quadrilateral is cyclic if and only if
the two angle bisectors are perpendicular to each other, and show that
in that case they intersect the sides of the quadrilateral at the
vertices of the rhombus. (4 points) (Suggested by
J. Rácz, Budapest)
B. 3608. The roots of the equation
x^{3}+ax^{2}+bx+c=0 are
equal to the fifth powers of the three roots of the equation
x^{3}3x+1=0, respectively. Find the numbers
a, b, c in decimal
notation. (4 points)
B. 3609. Is there a 2003rddegree
polynomial f(x) of integer coefficients, such that the
values of f(n), f(f(n)), are pairwise relative primes
for every integer n? (4 points)
B. 3610. Prove that
sin 25^{o}^{.}sin 35^{o}^{.}sin 60^{o}^{.}sin 85^{o}=sin 20^{o}^{.}sin 40^{o}^{.}sin 75^{o}^{.}sin 80^{o}.
(5 points)
B. 3611. The infinite series is formed out of the elements
of the sequence defined by the recursion
x_{n+1}=x_{n}^{2}x_{n}+1. What
is the sum of the series if a) x_{1}=1/2;
b) x_{1}=2? (5 points)
 New advanced problems Maximum score
for each advanced problem (sign "A") is 5 points. 
A. 308. A, B, C,
D, E are points in the plane, such that
AB=BC=CD=DA=1, and each of AE,
BE, CE and DE is at most 1. What is the
maximum possible value of
AE+BE+CE+DE+AC+BD?
A. 309. In a simple graph on n
points, the orders of the points are 0<d_{1}...d_{n}. Prove that it is possible to
select at least \(\displaystyle \sum\frac{2}{d_i+1}\) points, such that the subgraph formed by
these points contains no loop.
A. 310. Let , for every positive integer n, and define
the polynomials p_{0},p_{1},... by the
following recursion: p_{0}(x)=1, . Prove that the coefficients of the
polynomial p_{n} are all integers.
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 February 2003
