New exercises and problems in Mathematics November
2002
 New exercises Maximum
score for each exercise (sign "C") is 5 points. 
C. 690. Is it possible for the sum of
the volumes of three cubes of integer edges to be 2002 units?
C. 691. Express in mm^{2} the
area of Hungary on a globe of radius 25 cm.
C. 692. For the real numbers x,
y, z, (1) x+2y+4z3 and
(2) y3x+2z\(\displaystyle \ge\)5. Prove that
yx+2z\(\displaystyle \ge\)3.
C. 693. In what interval may the apex
angle of an isosceles triangle vary if a triangle can be constructed
out of its altitudes?
C. 694. Evaluate the sum
[log_{2}1]+[log_{2}2]+[log_{2}3]+...+[log_{2}2002].
Suggested by Á. Besenyei, Budapest
 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. 3582. Solve the equation
3xyz5yz+3x+3z=5 on the set of natural
numbers. (3 points)
B. 3583. The incentre of a triangle
ABC is connected to the vertices. One of the resulting three
triangles is similar to the original triangle ABC. Find the
angles of the triangle ABC. (3 points)
B. 3584. Write down all the integers
from 1 to 10^{n1}, and let A denote the number
of digits hence written down. Now write down all the integers from 1
to 10^{n}, and let B denote the number of zeros
written down this time. Prove that
A=B. (4 points)
B. 3585. Find the possible values of
the parameter a, such that the inequality holds for all
positive x. (3 points)
11th International Hungarian Mathematics
Competition
B. 3586. For what values of the
parameter a does the equation log (ax) =2log
(x+1) have exactly one root? (4 points)
B. 3587. A truncated right pyramid
with a square base is circumscribed about a sphere. Find the range for
the ratio of the volume and the surface area of the
solid. (4 points)
B. 3588. M is a point in the
interior of a given circle. The vertex of a right angle is M
and its arms intersect the circle at the points A and
B. What is the locus of the midpoint of the line segment
AB as the right angle is rotated about the point M?
(4 points)
B. 3589. Prove that there are
infinitely many odd positive integers n, such that
2^{n}+n is a composite
number. (4 points)
B. 3590. The roots of the equation
x^{3}10x+11=0 are u, v and
w. Determine the value of arctan u
+arctan v+arctan w. (5 points)
B. 3591. The area of the convex
quadrilateral ABCD is T, and P is a point in its
interior. The parallel through the point P to the line segment
BC intersects the side BA at E, the parallel to
the line segment AB intersects the side BC atF,
the parallel to AD intersects CD at G, and the
parallel to CD intersects AD at the point H. Let
t_{1} denote the area of the quadrilateral AEPH,
and let t_{2} denote the area of the quadrilateral
PFCG. Prove that \(\displaystyle \sqrt{t_1}+\sqrt{t_2}\leq\sqrt{T}\). (5 points)
 New advanced problems Maximum score
for each advanced problem (sign "A") is 5 points. 
A. 302. Given the unit square
ABCD and the point P on the plane, prove that
\(\displaystyle 3AP+5CP+\sqrt5(BP+DP)\ge6\sqrt2.\)
A. 303. x, y are
nonnegative numbers, and
x^{3}+y^{4}\(\displaystyle \le\)x^{2}+y^{3}. Prove that
x^{3}+y^{3}\(\displaystyle \le\)2.
A. 304. Find all functions
R^{+}\(\displaystyle \mapsto\)R^{+}, such that
f(x+y)+f(x)^{.}f(y)=f(xy)+f(x)+f(y)?
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 December 2002
