New exercises and problems in Mathematics November
2001

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

C. 645. Two players play the following
game. They take turns in taking matches from a heap that initially
contains 7 matches. In each step, a player can take one, two or three
matches. The game continues until there are no matches left. The
winner is the player holding an even number of matches at the
end. Which player has a winning strategy, the one who starts or his
opponent? How should he play in order to win?
Kvant
C. 646. In the sequence obtained by omitting
the squares from the sequence of natural numbers, what is the 2001st
term? Which term of the sequence is the number 2001?
Proposed by: P. Nádor, Pécs
C. 647. We have graphed the function
f(x)=1/x in the coordinate plane. We want to
alter the units on the coordinate axes so that the curve should
represent the function g(x)=2/x. How should the
new unit be set if it is to be the same on both axes?
C. 648. Evaluate
2^{log618}^{.}3^{log63}.
C. 649. The area of the base of a truncated
pyramid is 8 cm^{2}, and the area of the cover is
1 cm^{2}. The pyramid is cut with a plane parallel to the
base into two parts of equal volumes. Find the area of the
intersection.
Proposed by: Á. Besenyei, Tatabánya

New problems in November 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. 3492. The numbers 1, 2, ...,
n^{2} are written in increasing order into an
nxn array:
1 
2 
... 
n 
n+1 
n+2 
... 
2n 
... 
... 

... 
n^{2}n+1 
n^{2}n+2 
... 
n^{2} 

A number is selected from each row, such that no two numbers belong
to the same column. What are the possible values of the sum of the
selected numbers? (3 pont)
IMC 8, Prague, 2001
B. 3493. Two players play the following
game. They take turns in taking matches from a heap that initially
contains an odd number of matches. In each step, a player can take
one, two or three matches. The game continues until there are no
matches left. The winner is the player holding an even number of
matches at the end. Which player has a winning strategy, the one who
starts or his opponent? (5 points)
Kvant
B. 3494. Prove that if the equation
ax^{3}+bx^{2}+cx+d=0 has
three different positive roots, then bc<3ad. (4
points)
Proposed by: G. Bakonyi, Budapest
B. 3495. The sides of rectangle ABCD
are AB=3, BC=2. P is the point on side AB
for which the line PD touches the circle of diameter
BC. Denote the point of tangency by E. The line passing
through E and the centre of the circle intersects the side
AB at Q. Find the area of the triangle PQE. (3
points)
Proposed by: L. Gerőcs, Budapest
B. 3496. P is a point in the interior
of the square ABCD, such that
AP:BP:CP=1:2:3. Find the measure of the angle
APB. (4 points)
B. 3497. For the real number x,
{x}+{x^{1}}=1. Calculate {x^{2001}}+{x^{2001}}. (4 points)
Proposed by: Á. Besenyei, Tatabánya
B. 3498. In triangle ABC, there is a
right angle at vertex C. The bisectors of the acute angles
intersect the opposite sides at the points M and N
respectively. Let P be the intersection of segment MN
and the altitude drawn from C. Prove that the length of
CP is equal to the radius of the inscribed circle. (4 points)
Proposed by: I. Merényi, Budapest
B. 3499. The geometric mean of the positive
numbers x and y is g. Prove that if g3, then ,
and if g2,
then \(\displaystyle {1\over\sqrt{1+x}}+{1\over\sqrt{1+y}}\leq{2\over\sqrt{1+g}}\). (5 points)
B. 3500. A line of one hundred boys and
another line of one hundred girls are standing facing one
another. Every boy chooses a girl (it is allowed for more than one boy
to choose the same girl) and walks up to her along the shortest
path. In doing so, their paths do not cross. Then the boys go back to
their places, and now the girls do the same, making sure that their
paths do not cross as they are walking up to the chosen boy. Prove
that there is a girl and a boy who chose each other. (4 points)
B. 3501. Given the points A(1, 1, 1)
and P(1, 1, 0) in the coordinate space, rotate the point
P about the ray OA in the positive direction through an
angle of 60^{o}. Find the coordinates of the rotated point. (4
points)

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

A. 275. The numbers 1^{2},
2^{2}, 3^{2}, ..., (n^{2})^{2}
are written in increasing order into an nxn array:
1^{2} 
2^{2} 
... 
n^{2} 
(n+1)^{2} 
(n+2)^{2} 
... 
(2n)^{2} 
... 
... 

... 
(n^{2}n+1)^{2} 
(n^{2}n+2)^{2} 
... 
(n^{2})^{2} 

A number is selected from each row, such that no two numbers belong
to the same column. What are the possible values of the sum of the
selected numbers?
From the idea of T. Terpai
A. 276. The product of the positive numbers
x_{1}, ..., x_{n} is ,
where 1 is a real
number. Prove that
\(\displaystyle \sum_{i=1}^n{1\over(x_i+1)^{1/\alpha}}\ge1.\)
A. 277. Let H_{1} be an
nsided polygon. Construct the sequence H_{1},
H_{2}, ..., H_{n} of polygons as
follows. Having constructed the polygon H_{k},
H_{k+1} is obtained by reflecting each vertex of
H_{k} through its kth neighbour in the
counterclockwise direction. Prove that if n is a prime, then
the polygons H_{1} and H_{n} are
similar.
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 December 2001
