New exercises and problems in Mathematics
March 2000

New exercises in March 2000 
C.575. ``Well, I want you to answer me this. If two
stagecoaches leave Pozsony for Brassó every day, and as many leave
Brassó for Pozsony, and assuming that the journey takes ten days, then
how many coaches would you meet on the way if you travelled in one
stagecoach from Pozsony to Brassó?''
Proposed by K. Mikszáth, Fogaras
C.576. Find the smallest positive integer which, multiplied
by 1999, yields a product whose last 4 digits are
2,0,0,1.
Proposed by M. Braun
C.577. Deep inside a drawer, there are 3 pairs of socks,
somewhat different from each other. We try it three times to fish one
sock out of the drawer. What is the probability that we end up with
one sock of each pair after the 3 trials?
C.578. Consider the triangle whose vertices in the Cartesian
system are A(2;1), B(3;1) and C(2^{1999};2^{2000}). Calculate the area of the triangle.
C.579. A circular disc is cut into two sectors along two
of its radii. The two pieces are then turned into cone shaped
funnels. Is it true that the largest total volume is obtained from two
half discs?

New problems in March 2000 
B.3352. The number of the crew of a certain ship, multiplied
by one less the number of the crew exceeds, by 15, the product of the
age of the captain and the number two less the number of the
crew. What is the captain's name? (3 points)
Repeta
B.3353. Determine the value of the following expression:
(4 points)
Proposed by M. Ábrány, Ukraine
B.3354. Two intersecting lines and a point P outside
the lines are given in the plane. Construct points X and Y
on the two lines, respectively, such that P is incident to
the segment XY, and PX^{.}PY is minimal.
(5 points)
B.3355. Prove that a triangle has a right angle if and
only if the product of two of its escribed circles equals to the area
of the triangle. (4 points)
Proposed by B. Bíró, Eger
B.3356. Demonstrate that there are an infinite number of
cases when the difference between the fourth powers of two consecutive
integers can be written as the sum of two perfect sqares. (4 points)
B.3357. Is there any perfect square which has the same
number of positive divisors of the form 3k+1 as of the form
3k+2? (4 points)
B.3358. Prove that among any 4 different real numbers
there exist two, a and b such that
(5 points)
Proposed by J. Mezei, Vác
B.3359. The real roots x_{1}, x_{2},
x_{3} of the equation
x^{3}3x1=0 satisfy
x_{1}<x_{2}<x_{3}. Prove that x_{3}^{2}x_{2}^{2}=x_{3}x_{1}. (5
points)
Proposed by S. Mihalovics, Esztergom
B.3360. In a triangle ABC with centroid S, Let
E and F denote the midpoints of sides AB and
AC, respectively. Prove that the quadrilateral AESF is
cyclic if and only if AB^{2}+AC^{2}=2BC^{2}. (4
points)
Proposed by S. Kiss, Szatmárnémeti
B.3361. Two spheres G_{1} and G_{2}
are given. Cubes K_{1} and
K_{2} are inscribed into
G_{1} and G_{2}, respectively. Consider the 64 vectors whose
starting points are the vertices of K_{1} and whose endpoints are the vertices of
K_{2}. Prove that the sum of these
vectors does not depend on the position of the cubes within the
spheres. (3 points)

New advanced problems in March 2000 
A.233. A sequence (a_{n}) is defined as follows:
a_{0}=a_{1}=1, (n+3)a_{n+1}=(2n+3)a_{n}+3na_{n1}. Prove that each term of the sequence
is an integer.
A.234. Any finite sequence of the letters A,B,C is
called a word. A word can be transformed into an other word by
the following two operations:
a) we choose any contiguous part of the word and double it, as in
the example BBCACBBCABCAC;
b) (the reverse of the first operation) if two consecutive
contiguous parts of the word are identical, we may omit one of them,
as in the example ABCABCBC ABCBC.
Prove that any word can be transformed into a word of not more than
8 letters.
A.235. Let a be a fixed complex number. Find the
locus of all complex numbers b for which there exist
nonnegative real numbers x_{1},x_{2},...,x_{n} and complex numbers z_{1},z_{2},...,z_{n} of unit modulus such that
x_{1}+x_{2}+...+x_{n}=1, x_{1}z_{1}+x_{2}z_{2}+...+x_{n}z_{n}=a and x_{1}z_{1}^{2}+x_{2}z_{2}^{2}+...+x_{n}z_{n}^{2}=b.
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok),
Budapest Pf. 47. 1255, Hungary
or by email to: megoldas@komal.elte.hu.
Deadline: 15 April 1999
April's fool!
