New exercises and problems in Mathematics December
2000

New exercises in December 2000 
C. 605. Solve the equation
,
where [x] and {x} denote the integer and the fractional part
of x, respectively.
C. 606. A rectangular yard is tiled with identical
square tiles. The tiles are arranged in 20 rows, each row containing 35
tiles. A snail is moving along a diagonal of the rectangle, starting at one
vertex. Find the number of tiles crossed by the snail before it reaches the
opposite vertex.
C. 607. A square of side 12 cm is rotated about a
point P through an angle of 90^{o}. The two
squares cover an area of 211 cm^{2} altogether. A
third square is obtained rotating the square further about P, again
through 90^{o}. Given also that the three squares
cover an area of 287 cm^{2} together, determine
the position of P.
Proposed by: G. Bakonyi, Budapest
Solution
C. 608. In the course of a KENO drawing, 20 winning
numbers are drawn from the lot of the first 80 positive integers. Calculate the
probability that none of the winning numbers contain the digit 8.
C. 609. Prove that the distance between the line
3x4y+4=0 and a lattice point in the plane is always a rational
number.

New problems in Decober 2000 
B. 3412. Prove that for all but two positive integers
q there is an integer p such that <p/q<. (3 points)
NRICH Online
Maths Club
B. 3413. In the Hungarian Soccer League, last year
still called `professional', one group comprises 8 teams. At each match one of
the participating teams plays away while the other plays at home. During a
season, when each team plays exactly one game with any other team, the draw of
a team is called balanced if it plays alternately at home and away. Is it
possible to arrange that each of the 8 teams has a balanced draw?
(4 points)
B. 3414. In a cyclic quadrilateral whose diagonals are
perpendicular, the sum of the squares of the edges is 8. Determine the
circumradius of the quadrilateral. (3 points)
Proposed by: Á. Besenyei, Tatabánya
B. 3415. Determine all values of the parameter p for
which the equation
has exactly one root. (4 points)
B. 3416. Each face of a convex polyhedron is a
quadrilateral. The surface area of the polyhedron is A, the sum of the
squares of its edges is Q. Prove that Q2A. (4 points)
Proposed by: Á. Besenyei, Tatabánya
Solution
B. 3417. Consider all circles in the plane whose equation
can be written in the form ((x2)^{2}+y^{2}1)+((x+2)^{2}+y^{2}1)=0 with suitable real numbers , . Determine the locus
of points which do not lie on any of these circles. (4 points)
B. 3418. In a triangle ABC, the medians starting at
vertices A, B, C intersect the circumcircle at points
A_{1}, B_{1}, C_{1},
respectively. Prove that t_{A1BC}+t_{B1CA}+t_{C1AB}t_{ABC}. (5 points)
Proposed by: I. Varga, Békéscsaba
B. 3419. Find those points in the plane through which
3 different tangents can be drawn to the curve y=x^{3}. (4 points)
B. 3420. Find a nonconstant function f:RR whose graph can be obtained from the
graph of the square of the function by an enlargement of
ratio 3. (5 points)
Proposed by: L. Lóczi, Budapest
B. 3421. Assume that (x^{2}+3y^{2})/(u^{2}+3v^{2}) is an
integer number for certain integers x, y, u and
v. Prove that this fraction can also be written in the form
a^{2}+3b^{2} with suitable integers a
and b. (5 points)
Proposed by: M. Csörnyei, London

New advanced problems in December 2000 
A. 251. Given a sphere and a point P on it, let
P_{Q} denote, for any point Q
not on the sphere, the second intersection of line PQ with the
sphere. In the degenerate case when PQ is tangent to the sphere, let
P_{Q}=P. Prove that
if points A,B,C,D of the sphere are not coplanar, then there are at
most two different points Q such that the tetrahedron
A_{Q}B_{Q}C_{Q}D_{Q}
is equilateral. (A tetrahedron is called equilateral if all its faces are
congruent.)
A. 252. The integers a_{1},...,a_{n}
give at least k+1 different remainders when they are divided by
n+k. Prove that some of them add up to an integer divisible by
n+k.
Proposed by: Gy. Károlyi and K. Podoski,
Budapest
A. 253. Find all functions f: (0,)(0,) for which each
pair of positive numbers x and y satisfy
f(x)f(yf(x))=f(x+y).
International Mathematical Competition for University
Students 2000, London
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 January 2001
