New exercises and problems in Mathematics May
2001

New exercises in May 2001 
C. 630. In a 4digit number, the sum of the first two
digits is equal to that of the last two digits. The sum of the first and last
digits is equal to the third digit. Finally, the sum of the second and fourth
digits is twice the sum of the other two digits. Which number has all these
properties?
C. 631. Determine the value of c so that the area
enclosed by the graphs y=x1+x+1 and y=c
is 30.
C. 632. The sequence 3, 15, 24, 48, ... contains those
multiples of 3, in increasing order, which are 1 less than a perfect
square. Determine the remainder when the 2001st term of the sequence is divided
by 1000.
Proposed by Zs. Magyar, Budapest
C. 633. Each face of a tetrahedron has the same area,
moreover, they all have equal inradii. Prove that the faces are all congruent.
C. 634. A 1 litre measure is in the shape of a frustum of a
cone. Half a litre can be measured at the 2/3 of the altitude of the
frustum. Find the ratio of the diameters of the bases.

New problems in May 2001 
B. 3462. Determine those positive integers n for
which 100 can be expressed in the form \(\displaystyle \pm1\pm2\pm\dots\pm n\). (3 points)
B. 3463. The diameter of a half disc shaped sponge is
20 cm. A corner of a room is wiped with the sponge so that the two
extremal points of its diameter touch the two perpendicular walls all the
time. Determine the area one can wipe this way. (3 points)
Solution
B. 3464. A number has n digits which add up to
9n8. How many such ndigit numbers are there? (4 points)
B. 3465. Let m, n be positive integers, and
set the points E and F on side AB of a unit square
ABCD such that AE=1/n and BF=1/m. Determine
m and n such that the line through B that forms an angle
of 30^{o} with AB touches the semicircle drawn inside the square
with diameter EF. Is it possible that AC touches the semicircle
at the same time? (3 points)
Proposed by B. Bíró, Eger
B. 3466. In the plane a square ABCD is
given. Determine the locus of those points P for which sin(APB)=sin(DPC). (4 points)
B. 3467. Find the maximum of the function
x(1+x)(3x) on the set of positive numbers, without using
calculus. (4 points)
B. 3468. No three diagonals of a convex 10gon intersect at
the same point. A point P is given inside the polygon such that P
is not incident to any diagonal of the polygon. Prove that there are at least
28 ways to select 4 vertices of the polygon such that P is inside the
quadrilateral determined by these points. (5 points)
B. 3469. Find those continuous functions f, for
which f(f(x))=f(x)+x for every real
number x. (5 points)
GillisTurán Mathematics Competition, 2001
B. 3470. Find an equation for the line that touches the
curve y=3x^{4}4x^{3} at two different
points. (4 points)
Solution
B. 3471. An ant is walking inside the region bounded by the
curve whose equation is
x^{2}+y^{2}+xy=6. Its path is formed by
straight segments parallel to the coordinate axes. It starts at an arbitrary
point on the curve and takes off inside the region. When reaching the boundary,
it turns by 90^{o} and continues its walk inside the region. When
arriving at a point on the boundary which it has already visited, or where it
cannot continue its walk according to the given rule, the ant stops. Prove
that, sooner or later, and regardless of the starting point, the ant will
stop. (5 points)
GillisTurán Mathematics Competition, 2000
Solution

New advanced problems in May 2001 
A. 266. Points A_{1}, A_{2},
A_{3}, B_{1}, B_{2},
B_{3} lie on a circle. Let, for any 0<t<1 and
i=1, 2, 3, C_{i}(t) and
D_{i}(t) denote the two points that divide the
segment A_{i}B_{i} in the ratio
t:(1t). Let e(t) denote the radical axis of the
circumcircles of triangles
C_{1}(t)C_{2}(t)C_{3}(t)
and
D_{1}(t)D_{2}(t)D_{3}(t)
(if it exists). Prove that the lines e(t) are either parallel or
concurrent.
A. Dőtsch, Szeged
A. 267. Let m, n be positive integers and let
0\(\displaystyle le\)x\(\displaystyle le\)1. Prove that
(1x^{n})^{m}+(1(1x)^{m})^{n}\(\displaystyle ge\)1.
10th Vojtech Jarnik Mathematical Competition, Ostrava,
2000
A. 268. Let us choose arbitrarily n vertices of a
regular 2ngon and colour them red. The remaining vertices are coloured
blue. We arrange all redred distances into a nondecreasing sequence and do the
same with blueblue distances. Prove that the two sequences hence obtained are
identical.
10th Vojtech Jarnik Mathematical Competition, Ostrava, 2000
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 June 2001
