New exercises and problems in Mathematics
October 2000

New exercises in October 2000 
C. 595.
Find all 3digit numbers which are equal to the sum of the factorials of their
respective digits.
C. 596.
Solve the equation
on the set of real numbers. ([a] denotes the integer part of a, that is, the largest integer not exceeding a.)
C. 597.
Prove that, in any right triangle, the distance between the incentre and the
circumcentre is at least
(1) times the radius of the circumcircle.
C. 598.
Is it possible to find 2000 positive integers such that none of them is
divisible by any of the other numbers but the square of each is divisible by
all the others?
Proposed by: M. Ábrány, Ukraine
C. 599.
A right cone of unit height whose circular base has a radius R is truncated in the following way. First, it is cut along a plane whose distance from the base is h. Next, the removed cone is reflected in the cutting plane and its image is removed from the frustum. Calculate the volume of the solid obtained this way.

New problems in Oktober 2000 
B. 3392.
We have 25 vessels whose volumes are 1,2,...,
25 litres, respectively. 10 vessels are to be selected such that from an
unlimited supply of water, exactly 1 litre can be measured with the help of any
two selected vessels. How many different selections can be made? (4 points)
Proposed by: L. Kozma, Székelyudvarhely
B. 3393.
Triangles ABC and A'B'C' are given in the plane
such that points A', B' and C' are the midpoints of
segments CC', AA' and BB', respectively. Determine the
ratio of the areas of the two triangles. (3 points)
B. 3394.
Prove that the positive numbers a, b, c are the lengths of
the sides of some triangle if and only if (a^{2}+b^{2}+c^{2})^{2}>2(a^{4}+b^{4}+c^{4}). (3 points)
B. 3395.
Construct the pentagon A_{1}A_{2}A_{3}A_{4}A_{5}, given the
reflections of point A_{i} through
point A_{i+1} for i=1, 2,
..., 5, respectively, where A_{6}=A_{1}. (3 points)
B. 3396.
Find all positive integer solutions of the equation
xy+yz+zxxyz=2. (4 points)
B. 3397.
An acute triangle ABC and a rectangle KLMN are given such that
N is incident to segment AC and K,L are incident to
segment AB. Translating the rectangle parallel to AC an other
rectangle K'L'M'N' is obtained whose vertex
M' is incident to segment BC. Prove that the line connecting the
intersection point of CL' and AB with the intersection point of
AM and CB is perpendicular to line AB. (4 points)
B. 3398.
Points A, B, C, D are points in the space such that segments AB and CD are of 4 unit length while segments BC and DA are of 5 unit length. Let x and y denote the squares of the lengths of segments AC and BD, respectively. Plot in the Cartesian system all pairs (x,y
) that arise this way. (4 points)
Proposed by: A. Hraskó Budapest
B. 3399.
Two circles e and f are given in the plane, and a point P
outside the circles. A tangent line from P is drawn to each circle,
touching the circles at points E and F, respectively. Prove that
the ratio of the lengths of the two segments that the two circles cut out of
line EF does not depend on the choice of the two tangent lines.(5 points)
B. 3400.
For a given interger d let S_{d}={x
^{2}+dy^{2}:x,y
Z}.
Assume are a
S_{d}, b
S_{d} such that b is a prime and
is an integer. Prove that
S_{d}. (5 points)
Proposed by: M. Csörnyei, London
B. 3401.
In a ware house there are a lot of parcels, each weighing at most 1 (metric)
ton. We have two lorries which can be loaded with weights of 3 and 4 tons,
respectively. A contract is to be made according to which we carry at least
N tons of wares at each turn. What is the largest value of N with
which we may undertake the contract? (5 points)

New advanced problems in October 2000 
A. 245.
We want to make a round trip with our car. There are a few gasstations along the road where the total amount of fuel is enough to make two round trips. Prove that there is a station from where the round trip can be completed in both directions.
A. 246.
Find all quadruples of real numbers x, y, z, w which satisfy x+y+z+w=x^{7}+y^{7}+z^{7}+w^{7}=0. Shay Gueron, Haifa
A. 247.
Prove that there is no point on the lines determined by the sides of the unit square whose distance from each vertex of the square is rational. Á. Kovács, Budapest
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok),
Budapest Pf. 47. 1255, Hungary
or by email to: solutions@komal.elte.hu.
Deadline: 15 November 2000
