# New exercises and problems in Mathematics

May 1999

## New exercises for practice in May 1999 |

**C. 541. **Four figures are to be inserted into a
six-page essay, in a given order. One page may contain at most two
figures. How many different ways are there to assign page numbers to
the figures under these restrictions?

**C. 542. **Find those positive integers *n *for
which the inequality

(*n*-2)(2+2^{2}+...+2^{n})<*n*^{.}2^{n}

holds.

**C. 543. **Construct a triangle, given one of its
angles and two of its altitudes.

**C. 544. **The `frame' of a unit cube consists of two
parallel faces and the edges that connect the two faces. We fix one of
the faces and twist the other one about the axis that connects the
midpoints of the faces through an acute angle , while the length of the connecting edges remains the
same. How does this deformation reduce the distance between the
parallel faces?

## New exercieses in May 1999 |

**Gy. 3278. **In a 3x3 magic square, 3 entries *a*,
*b *and *c *are given according to the following
diagram. Fill the rest of the diagram (so that the sum of the entries
in each row, column, and diagonal are the same).

b | ||

a | c | |

Proposer: *S. Kiss, *Nyíregyháza

**Gy. 3279. **Ten children play `undo-the-knot' the
following way. First, they form a circle, and with their eyes closed,
arms extended, they start moving towards the centre of the
circle. Next, with each hand they grab someone else's hand. Finally,
they open their eyes and start undoing the knot: they bend down and
slip under and step over each other's arm, and so on (each child is
limber enough), but they never lose their grip. In what percent of all
possible cases is it true that, when two children standing next to
each other lose their grip, the ten children find themselves forming
an unbroken chain?

**Gy. 3280. **Let *p *and *q *be positive real
numbers whose reciprocals add up to 1. Prove that

and .

Proposer: *S. Róka, *Nyíregyháza

**Gy. 3281. **In a rectangular box, the lengths of the
three edges starting at the same vertex are prime numbers. It is also
given that the surface area of the box is a power of a prime. Prove
that exactly one of the edge-lengths is a prime number of the form
2^{k}-1.

Proposer: *Á. Kovács, *Budapest

**Gy. 3282. **Let *a*, *b *and *c *denote
the sides of a triangle. Prove that

.

Is it possible that equality holds?

**Gy. 3283. **On two sides of a right triangle, one of
which is the hypotenuse, squares are built externally and a point *M
*is constructed according to Figure 1. Prove that
*M *is incident to the perimeter of the square inscribed into the
triangle (see Figure 2).

Figure 1 | Figure 2 |

**Gy. 3284. **Assume that an arc of a circle halves the
area of another circle *k*. Prove that the length of the arc is
greater than the diameter of *k*.

**Gy. 3285. **What kind of rhombus has the property that
there can be found two different parallelepipeds whose faces are all
congruent to this rhombus?

Proposer: *K. Peták, *Szolnok

## New problems in May 1999 |

**F. 3286. **Find the values of *t *for which the
binary operation, defined on positive numbers by is associative.

**F. 3287. **A *partition *of a set is obtained by any
representation of the set as the union of pairwise disjoint
subsets. Prove that a set of *n *elements has no more than
*n*! partitions.

Proposer: *S. Róka, *Nyíregyháza

**F. 3288. **We have *p*^{k}|*n*! for some positive integers
*n*, *k *and prime number *p*. Prove that *n*! is
divisible by (*p*!)^{k}, too.

**F. 3289. **A point *P *and distances *a*, *b
*and *c *are given. Construct an equilateral triangle such
that the segments connecting *P *with the vertices of the
triangle have length *a*, *b *and *c*, respectively.

**F. 3290. **Prove that the points, given by their
coordinates as *A*(1*, *0*, *2), *B*(4*,
*3*, *-1), *C*(0*, *3*, *-1), *D*(5*,
*-2*, *4), lie in the same plane. Find, moreover, the area of
the convex quadrilateral determined by these four points.

**F. 3291. **Prove that any equilateral lattice polygon has
an even number of edges.

Proposer: *I. Z. Ruzsa, *Budapest

## New advanced problems in May 1999 |

**N. 211. **Count those *n*-term sequences of positive
integers in which the number of terms greater than *i *is at most
*n*-*i*, for each 1*in*.

**N. 212. **Determine those pairs of positive integers *a
*and *n *for which each prime divisor of also divides *n*.

**N. 213. **Let *p*_{n}(*x*)=*a*_{n,0}+*a*_{n,1}*x*+...+*a*_{n,n}*x*^{n} denote the unique polynomial, of degree
at most *n*, whose values at 2^{0},
2^{1}, ..., 2^{n} are 0, 1, ..., *n*,
respectively. Find, for any positive integer *k*, the limit .

**N. 214. **Consider the parabola
*y*=*x*^{2}. Suppose that two
tangents are drawn to the parabola from a point *P *in the
plane. Denote by *Q*_{P} and
*R*_{P}, respectively, the
points of contact, and by *l*(*Q*_{P}*, R*_{P}) the length of the arc of the parabola
connecting *Q*_{P} and
*R*_{P}. Find the locus of
points *P *for which *PQ*_{P}+*PR*_{P}-*l*(*Q*_{P}*, R*_{P}) is constant.