New exercises and problems in Mathematics
New exercises for practice in February 1999
C. 529. Find the smallest odd positive integer that has
the same number of divisors as the (even) number 360.
C. 530. Prove that the sine and cosine of an angle are both rational numbers if and only
if tan is either a rational number
or not defined.
Proposer: Z. Hans
C. 531. In the Cartesian system, consider all proper
triangles whose vertices have integer coordinates of absolute value at
most 1. Find a complete list of the distances of the centroids of
these triangles from the origin.
C. 532. Two circles of respective radii r1 and r2
touch each other externally. Consider one of their common external
tangents. Rotate the segment of this line, which lies between the
points of tangency, about the axis that connects the centres of the
circles. Express, in terms of r1 and r2,
the area of the curved surface obtained this way.
New exercieses in February 1999
Gy. 3254. Is there an integer whose cube has a decimal
representation of the form ababab1?
Spanish competition problem
Gy. 3255. All the numbers have fallen down one by one from
the face of an old wall-clock. Prove that if we replace the numbers,
in any order, on the empty face of the clock, there will be three
consecutive numbers among them that add up to 20 at least. Is it
always true that there will also be a sum greater than 20?
Gy. 3256. Find all (not necessarily positive) prime
numbers p, q, r which satisfy
Proposer: Á. Kovács, Budapest
Gy. 3257. We have 2 identical mugs. In a 31-storey
building, we have to determine the highest floor from which, when a
mug is dropped, it still does not break. The experiment we are allowed
to do is to drop a mug from a floor of our choice. How many
experiments are necessary to solve the problem in any case, for sure?
Proposer: B. Marx, Budapest
Gy. 3258. Find the maximum possible area any hexagon
may have whose vertices are the midpoints of the sides of some convex
hexagon of unit area.
Gy. 3259. In a circle with centre O, radius
OA intersects chord BC perpendicularly at a point
M. Let X be an arbitrary point on the longer arc
BC, and denote by Y the intersection point of XA
and BC, and by Z the point of intersection of the
circle with line XM, other than X. Prove that AYMZ.
Gy. 3260. In a rectangular coordinate system a square
is drawn with sides forming angles of 45o with the axes. Is it possible that there are
exactly 7 lattice points inside the square?
Gy. 3261. A convex quadrilateral ABCD is given
in the plane. Prove that the circles with respective diameters AC
and BD, each orthogonal to the plane, intersect each other
if and only if the quadrilateral ABCD is cyclic.
New problems in February 1999
F. 3268. Let p denote a prime number of the form
4k+3. Suppose that x, y are positive integers
such that is divisible by
p. Prove that there exist integers u,v such that
F. 3269. Determine the remainder when the number , written in base s+1, is divided by
F. 3270. Consider those polynomials
p(x)=x4+ax3+bx2+cx+1 whose coefficients are all positive
numbers less than 3, and which do not have any real root. Find the
maximum value of the product abc.
F. 3271. Given a convex n-gon P1P2...Pn, points Q1, Q2,
..., Qn-2 are placed
inside the polygon such a way that each triangle PiPjPk contains exactly one of them in its
interior. Partition the n-gon into regions by drawing all of
its diagonals. Prove that each point Ql lies in a triangular region of the partition.
Based on an idea of F. Sarlós
F. 3272. In 3-space there are given four pairwise unequal
spheres, each lying outside all the others. Each pair of these spheres
admits an internal and an external centre of similarity. Consider the
straight lines determined by these points. Find the maximum possible
number of different straight lines in such an arrangement.
F. 3273. A circle k and a line e containing
one of its diameters are given in the plane. An other circle touches
e such that the distance between its point closest to k
and circle k itself equals to the length of its
radius. Find the locus of the centres of all such circles.
New advanced problems in February 1999
N. 199. We have k identical mugs. In an
n-storey building, we have to determine the highest floor from
which, when a mug is dropped, it still does not break. The experiment
we are allowed to do is to drop a mug from a floor of our choice. How
many experiments are necessary to solve the problem in any case, for
N. 200. Prove that, for infinitely many positive integers
n, there exists a polynomial p of degree n with
real coefficients such that p(1), p(2), ...,
p(n+2) are different whole powers of 2.
N. 201. Four points u1, u2,
v1 and v2 are given on the complex plane. Consider those
two circles, each orthogonal to the plane, whose diagonals are
segments u1u2 and v1v2,
respectively. Prove that these circles intersect each other
perpendicularly if and only if
N. 202. Let p denote a prime number of the form
4k+3 and let x be any positive integer. Prove that the
integer does not have any divisor
of the form kp-1.