New exercises and problems in Mathematics May
2002

New exercises in May 2002
Maximum score for each exercise (sign "C") is 5 points.

C. 675. Is there a square number in
which the last two digits are both odd?
C. 676. The length of side AB of
the rectangle ABEF is 1 unit, and the length of side
BE is 3 units. The points C and D divide
side BE into three equal parts. Show that BAC+BAD\(\displaystyle \angle\)+BAE=180^{o}.
C. 677. Find the integers a and
b, given that
a^{4}+(a+b)^{4}+b^{4}
is a square number.
C .678. In a triangle ABC,
AC=1, ABC\(\displaystyle \angle\)=30^{o}, BAC=60^{o}, and D denotes the foot of
the altitude from vertex C. Find the distance between the
centres of the inscribed circles of the triangles ACD and
BCD.
C. 679. Given three spheres that
pairwise touch each other and also touch the plane S, determine
the radius of the sphere that has its centre on the plane S,
and that touches all the three spheres.

New problems in May 2002
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. 3552. Each member of the sequence
a_{1}, a_{2}, ...,
a_{2n+1} is either 2, 5 or 9. No two
consecutive members are equal, and
a_{1}=a_{2n+1}. Prove that
a_{1}a_{2}a_{2}a_{3}+a_{3}a_{4}a_{4}a_{5}+...a_{2n}a_{2n+1}=0. (3 points)
B. 3553. In a triangle ABC, the
altitude from vertex A, the angle bisector from vertex
B, and the median from vertex C intersect the opposite
sides at the points A_{1}, B_{1},
C_{1}, respectively. Prove that if the triangle
A_{1}B_{1}C_{1} is
equilateral, then the triangle ABC is also
equilateral. (4 points)
B. 3554. The diagonals of parallelogram
ABCD intersect at point M. The circle passing through
the points A, M, B touches the line
BC. Prove that the circle through the points B,
M, C touches the line CD. (3 points)
B. 3555. A company consists of 2n+1
people. For every group of n members, there exists a member of
the company who does not belong to the group, but who knows every
member of the group. Acquaintances are considered to be mutual. Prove
that there is a member of the company who knows
everybody. (5 points)
B. 3556. Given two sides and the
bisector of the exterior angle at their common vertex, construct the
triangle. (4 points)
B. 3557. Is it possible for a perfect
cube to start with 2002 ones in decimal notation?
(4 points)
B. 3558. Is there a nonconstant
polynomial of integer coefficients, such that its value is a power of
2 at every positive integer? (4 points)
(Inspired by a problem of the National Competition,
2002)
B. 3559. e_{1},
e_{2}, ..., e_{n} are lines in
the plane. Through an arbitrary point P_{1} of line
e_{1}, drop a perpendicular to the line
e_{2}, and denote the foot of the perpendicular by
P_{2}. Let P_{3} be the foot of the
perpendicular from P_{2} onto e_{3}, and
so on. Finally, let P_{n+1} denote the foot of
the perpendicular from P_{n} onto
e_{1}. Prove that there exists a point
P_{1} on the line e_{1}, such that the
point P_{n+1} obtained in this way should
coincide with P_{1}. (4 points)
B. 3560. Prove that no matter how we
leave 89 numbers out of the first 2002 positive integers,
the set of the remaining numbers will contain 20 elements, such
that their sum is also among the remaining
numbers. (5 points)
B. 3561. A convex polyhedron has
exactly three edges meeting at each vertex. Given that all but one of
the faces of the polyhedron are known to have circumscribed circles,
prove that all the faces have circumscribed
circles. (5 points)

New advanced problems in May 2002
Maximum score for each advanced problem (sign "A") is 5 points.

A. 293. Prove that for every integer
m\(\displaystyle \ge\)2 there
exists a pair of positive integers a and b, such that in
basem notation, a and b together contain exactly the
same number of each digit as the number
a^{.}b. (L. Szobonya,
Budapest)
A. 294. Define the sequence
a_{1},a_{2},... with the following
recursion:
\(\displaystyle a_1=1, \quad a_{n+1}={a_1a_n+a_2a_{n1}+\dots+a_na_1\over n+1}.\)
Prove that the sequence is convergent.
A. 295. Positive numbers
x_{1},x_{2}...,x_{n}
satisfy
\(\displaystyle {1\over1+x_1}+{1\over1+x_2}+\dots+{1\over1+x_n}=1.\)
Prove that
\(\displaystyle \sqrt{x_1}+\sqrt{x_2}+\dots+\sqrt{x_n}\ge(n1)
\left({1\over\sqrt{x_1}}+{1\over\sqrt{x_2}}+\dots+
{1\over\sqrt{x_n}}\right).\)
(Vojtech Jarník Mathematical Competition, Ostrava, 2002)
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 June 2002
