# New exercises and problems in Mathematics

February 2000

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

## New exercises in February 2000 |

**C. 570. **Find those 3-digit prime numbers whose
digits multiply up to 189.

**C. 571. **The side of a square based doghouse is
1.2 m long. A dog on a 3 m long leash is tied to the
doghouse such a way that the end of the leash is fixed to the entrance
side of the house, in a 0.3 m distance from its corner. Calculate the
area of the region the dog can reach.

**C. 572. **A square and a regular hexagon are
inscribed in a circle whose radius is 6 cm such that one side of the
hexagon is parallel to a side of the square. Calculate the area of a
segment of the disc which lies between two such parallel sides and
does not contain the centre of the circle.

**C. 573. **Find those positive integers *n *for
which 1^{2}+2^{2}+...+*n*^{2}=1+2+...+(2*n*-1)+2*n*.

**C. 574. **What relations must the edges of a
rectangular box satisfy if it can be cut into two congruent
rectangular boxes, each similar to the original one.

## New problems in February 2000 |

**B. 3342. **Find those positive integers *n *for
which *n*^{2}-19*n*-99 is a
perfect square. (3 points)

*American problem*

**B. 3343. **In a triangle *ABC*, the midpoint of
side *AB *is denoted by *F*. Points *S*_{1} and *S*_{2}
are the centroids of triangles *AFC *and *BFC*. Lines
*AS*_{1} and *BS*_{2} intersect sides *BC *and *AC*,
respectively, at points *P*_{1} and
*P*_{2}. Prove that the
quadrilateral *S*_{1}*S*_{2}*P*_{1}*P*_{2} is a
parallelogram. (3 points)

**B. 3344. **There are certain positive integers the
number of whose divisors can be obtained by subtracting the product of
the indices of their prime divisors from the product of the prime
divisors. Such integers are, for example, 25 and 600. Prove that there
are infinitely many such integers. (4 points)

**B. 3345. **9 teams were participating in a
(one-round) soccer tournament. The Devonshire Bulls won 5 games and
lost the other 3 games. Is it possible that they finished at the 7th
place? (At such a tournament, the winner of a game scores 3 points,
while in case of a draw game both teams score 1 point. If two teams
have the same total number of points in the end, they are ranked
according to the difference of the goals, scored and got by the
respective teams.) (4 points)

**B. 3346. **100 weights, measuring
1*,*2*, ..., *100 grams, respectively, are placed in the
two pans of a scale such that the scale is balanced. Prove that two
weights can be removed from each pan such that the equilibrium is not
broken. (5 points)

*Tournament of Towns, 1999.*

**B. 3347. **Consider a quadrilateral as

a) the quadruple of its four vertices;

b)) a homogeneous lamina.

In both cases one can determine the centroid of the quadrilateral. Find those quadrilaterals, for which the two centroids coincide.

Proposed by: *Z. Bogdán,* Cegléd

**B. 3348. **Find all integer solutions of the equation
(4 points)

Proposed by: *E. Fried,* Budapest

**B. 3349. **Prove that the equation
*p*^{p}=*q*^{q} admits an infinite number of positive
rational solutions with *p
q*.(5 points)

Proposed by: *L. Lóczi,* Budapest

**B. 3350. **A point *P *is given inside a
rectangular box. Reflect *P *in each face of the box. Describe
that position of *P *for which the convex polyhedron determined
by the six mirror images has maximum volume.(3 points)

**B. 3351. **At least four circles are given in the
3-space such that no 4 of their centres lie in the same
plane. Moreover, any two circles lie on a common sphere. Is it
necessarily true that all circles lie on the same sphere? (5 points)

## New advanced problems in February 2000 |

**A. 230. **Find all integers *a *for which is a perfect square.

**A. 231. **In a connected simple graph every vertex
has a degree at least 3. Prove that the graph contains a cycle such
that the graph remains connected when the edges of this cycle are
deleted.

Proposed by: *T. Kiss,* Budapest

**A. 232. **Construct a convex polyhedron with at
least 2000 vertices whose edges are not shorter than 1/2, but it can
be placed in a sphere of unit radius.

### Send your solutions to the following address:

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

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