New exercises and problems in Mathematics December
2001

New exercises in December 2001
Maximum score for each exercise (sign "C") is 5 points.

C. 650. In a picture gallery, the price of the
picture frames is proportional to the price of the paintings in them.
In order to reduce the differences between the prices of the pictures,
the manager interchanges the frames of two pairs of pictures. In one
case, one picture, that was originally five times as expensive as the
other one, is now only three times as expensive. How does the ratio
of the prices of the "Winter Landscape" and the "Naughty Boy" change
if the "Winter Landscape" originally cost nine times as much as the
"Naughty Boy"?
C. 651. The shaded figure in the diagram is
bounded by semicircles. The diameter AB has a segment of
length 1/5 units within the shaded figure. Find the perimeter and
area of the shaded figure.
C. 652. Let s be a positive integer of an
odd number of digits. Let f denote the number that consists of
the digits of f, but in opposite order. Prove that
s+f is divisible by 11 if and only if s is also
divisible.
C. 653. For how many different values of the
parameter p do the following simultaneous equations have
exactly one solution? x^{2}y^{2}=0,
xy+pxpy=p^{2}
C. 654. f_{a},
f_{b} and f_{c} denote the
lengths of the interior angle bisectors in a triangle of sides
a, b, c, and area T . Prove that
Proposed by:: G. Kovács, Budapest

New problems in December 2001
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. 3502. The semicircle of diameter AB
intersects the altitude drawn from vertex C of triangle
ABC at point C_{1}, and the semicircle of
diameter BC intersects the altitude from A at point
A_{1}. Prove that
BC_{1}=BA_{1}. (3 points)
B. 3503. Given are the segments
a,b,c,d and f. Assuming that there
exists a quadrilateral whose sides are
a,b,c,d and the length of the segment
connecting the midpoints of two opposite sides is f construct
the quadrilateral. (4 points)
B. 3504. If S(n) denotes the sum of
the digits of the number n in decimal notation, and
U_{k}=11...1 is the number that consists of
k ones, find the values of k for which
S(U_{k}^{2})=(S(U_{k})^{2}).
(4 points)
B. 3505. A regular octagon is subdivided into
parallelograms. Prove that there is a rectangle among the
parallelograms. (5 points)
B. 3506. f is a polynomial for which
f(x^{2}+1)f(x^{2}1)=4x^{2}+6. Find
the polynomial
f(x^{2}+1)f(x^{2}).
(4 points)
B. 3507. Let f(x) be a polynomial of
integer coefficients, p and q coprime numbers such that
q\(\displaystyle ne\)0.
Prove that if p/q is a root of the polynomial, then
f(k) id divisible by pkq for every
integer k. Is the converse of the statement also true?
(4 points)
B. 3508. The triangles ABC and
A_{1}B_{1}C_{1} are
symmetric about a line. Draw a parallel through A_{1}
to BC, through B_{1} to AC, and finally,
through C_{1} to AB. Prove that the three lines
all pass through a common point. (4 points)
B. 3509. Prove that, if , then (3 points)
Proposed by:: J. Balogh, Kaposvár
B. 3510. Consider the planes through the vertices
of tetrahedron ABCD that are parallel to the opposite faces.
The four planes enclose another tetrahedron. Prove that A,
B, C and D are the centroids of the faces of the
new tetrahedron. (3 points)
B. 3511. For the nonnegative numbers a,
b, c and d, a\(\displaystyle le\)1, a+b5,
a+b+c\(\displaystyle le\)14, a+b+c+d30. Prove that
\(\displaystyle \sqrt a+\sqrt
b+\sqrt c+\sqrt d\leq10\). (5 points)

New advanced problems in December 2001
Maximum score for each advanced problem (sign "A") is 5 points.

A. 278. P is a point on the extension of
the diagonal AC of rectangle ABCD beyond C, such
that BPD\(\displaystyle angle\)=CBP\(\displaystyle angle\). Find the ratio PB:PC.
A. 279. Are there such rational functions f
and g that (f(x))^{3}+(g(x))^{3}=x?
A. 280. For each positive integer n, let
f_{n}(\(\displaystyle vartheta\))=sin\(\displaystyle vartheta\)^{.}sin(2)^{.}sin(4\(\displaystyle vartheta\))^{.}...^{.}sin(2^{n}). For
all real \(\displaystyle vartheta\) and all n, prove that
IMC 8, Prague, 2001
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 January 2002
