# New exercises and problems in Mathematics

April
2001

## Please read The Conditions of the Problem Solving Competition.

## New exercises in April 2001 |

**C. 625. **At McDonald's you can order Chicken McNuggets in
boxes of 6, 9, and 20. What is the largest number of nuggets that you cannot
get no matter what combination of the above boxes you order?

**C. 626. **Two chords of a disc of radius 5 divide the disc
into three parts of equal areas. How long are these chords?

Proposed by *G.* *Reményi,* Budapest

**C. 627. **Solve the following system of equations:
*a*+*b*=*c*+*d*,
*a*^{3}+*b*^{3}=*c*^{3}+*d*^{3}.

Proposed by *M.* *Ábrány* Beregszász

**C. 628. **A tetrahedron shaped cardboard box is `cut open'
along three edges starting at the same vertex and is developed in the plane
where its four faces form a square with 30 *cm* long sides. Determine the
volume of the tetrahedron.

**C. 629. **Thirteen integers add up to a number divisible by
6. Prove that their thirteenth powers also add up to a multiple of 6.

Proposed by *T.* *Székelyhidi,* Ercsi

## New problems in April 2001 |

**B. 3452. **Two players alternately mark the fields of a 5x5
board. The one who moves first always writes one `X` sign, while the
second player always writes two `O` signs. The one who first completes a
row or a column of the board with her signs, wins the game. How can the second
player win the game? (4 points)

Proposed by *N.* *Gyanta,* Budapest

**B. 3453. **In a triangle *ABC* whose centroid is denoted
by *S*, points *H* and *Q* are the midpoints of segments
*AS* and *AB*, respectively. The line parallel to *BC* through
*H* intersects the segment *AB* at *P* and the line *CQ* at
*R*. Find the ratio between the areas of triangles *PQR* and
*APH*. (3 points)

Proposed by *T.* *Káspári,* Paks

**B. 3454. **We have 27 unit cubes. *a)* Is it
possible to paint the faces of these small cubes with three colours: purple,
white and yellow, in such a way that they can be arranged, in three different
ways, three cubes, each of edge-length 3, whose whole surface is purple, white,
and yellow, respectively? *b)* Find the smallest number such
that no matter how one paints this many faces of the small cubes with purple,
it is possible to form a cube of edge-length 3 whose complete surface is
purple. (4 points)

**B. 3455. **Find infinitely many pairs of integers (*m*,
*n*) such that the numbers *m* and *n* have the same prime
divisors, and also *m*+1 and *n*+1 have identical prime
factors. (5 points)

Proposed by *N.* *Gyanta,* Budapest

**B. 3456. **Each edge of a tetrahedron is labelled by a
positive integer. We may change these numbers such that at each time we choose
one vertex of the tetrahedron and replace the three numbers assigned to the
edges starting at that vertex as follows: To each edge we write either the sum
or the difference of the numbers written on the other two edges, writing the
sum in one case and the difference in the other two cases. When we write the
difference of the numbers *a* and *b* we may freely choose either
*a*-*b* or *b*-*a*. Is it possible to achieve, iterating
this procedure a finite number of times, that 0 is written on every edge of the
tetrahedron in the end? (3 points)

Proposed by *J.* *Mezei,* Vác

**B. 3457. **Determine those positive prime numbers *p*
and *q* for which the equation
*x*^{4}+*p*^{2}*x*+*q*=0 has a repeated
root. (4 points)

Proposed by *Á.* *Besenyei,* Budapest

**B. 3458. **In the plane *n* red points are given. Prove
that it is possible to place 2*n* blue points such that each triangle
formed by the red points contains at least a blue one in its interior. (5
points)

**B. 3459. **Circles of radii *a*, *b*, *c*,
*d* that lie outside each other are drawn around the consecutive vertices
*A*, *B*, *C*, *D* of a parallelogram such that
*a*+*c*=*b*+*d*. Prove that the four lines obtained as
common external tangents to opposite circles form a circumscribed
quadrilateral. (4 points)

**B. 3460. **In a triangle, the point where the incircle
touches the side opposite to the angle \(\displaystyle gamma\) divides the side into segments whose lengths are *x* and
*y*. Prove that the area of the triangle is equal to . (4
points)

Proposed by *Á.* *Besenyei,* Budapest

**B. 3461. **Let *a*>0 and *c\(\displaystyle ne\)*0 be integers. Prove that if the Diophantine
equation *x*^{2}-*ay*^{2}=*c* has at least
4*c*^{2}+1 different solutions, then it has infinitely many
solutions. (5 points)

Proposed by *E.* *Fried,* Budapest

## New advanced problems in April 2001 |

**A. 263. **On a circle there are *n* red and *n*
blue arcs given in such a way that each red arc intersects each blue one. Prove
that some point is contained by at least *n* of the given coloured arcs.

*Schweitzer Competition, 2000*

**A. 264. **The rational numbers in the interval (0,1) are
mapped, by a polynomial *p* of real coefficients, onto the set of all
rational numbers of an other interval. Prove that the degree of *p* is not
greater than 1.

Proposed by *E.* *Fried,* Budapest

**A. 265. **A sequence *a*_{1},
*a*_{2}, ... is defined recursively as follows. Let
*a*_{1}=*a*_{2}=*a*_{3}=1 and for *n\(\displaystyle ge\)*3. Prove
that each term of the sequence is an integer.

### Send your solutions to the following address:

KöMaL Szerkesztőség (KöMaL feladatok), Budapest Pf. 47. 1255, Hungary

or by e-mail to: solutions@komal.elte.hu.