# New exercises and problems in Mathematics

November 1999

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

## New exercises in November 1999 |

**C.555. **Find the smallest integer which can be expressed as
the sum of two distinct positive squares in two different ways.

**C.556. **A running competition is organized in five
rounds. There are 50 contestants whose times are recorded in each
round. The overall result is then determined by adding up the results
obtained in the five rounds. Suppose that Bob finishes 10th in each
round. Is it possible that he will have

- (a) the best

(b) the worst overall result?

**C.557. **Prove that there is no sequence consisting of at least two consecutive odd positive integers whose sum is a prime number.

Proposed by *Sz. Vajda, *Kolozsvár

**C.558. **In an *n*x*n *square grid, determine the number of such squares whose vertices are grid points and whose edges are parallel to the lines of the grid.

**C.559. **A cone shaped container, with its apex pointing downwards, is partly filled with water such that the surface of the water is 100 mm above the bottom of the container. When the container is closed and turned upside down, the water inside is only 20 mm deep. Determine the altitude of the cone.

Proposed by *L. Koncz, *Budapest

## New problems in November 1999 |

**B.3312. **Adding up the digits of his year of birth, Sylvester
noticed that the result is the very number formed by the last two
digits of the year of birth of his grandfather. Moreover, the last two
digits of Sylvester's year of birth form his grandfather's present
age. How old is Sylvester? (3 points)

Proposed by *B. Nyul, *Debrecen

**B.3313. **A rhombus *ABCD *is divided into two
equilateral triangles by its diagonal *BD*. Points *P *and
*Q *lie on segments *AD *and *CD*, respectively, such
that *PBQ*=60^{o}. Find the measure of the remaining two angles
of triangle *PBQ*. (3 points)

**B.3314. **Is it possible to assign the numbers
1*,*2*,*3*,...,*11*,*12 to the edges of a regular
octahedron such that the numbers assigned to the edges starting at a
vertex is the same for each vertex of the octahedron? (3 points)

Proposed by *G. Reményi, *Budapest

**B.3315. **Find a function *f *of the form
*f*(*x*)=*ax *which is a `good' approximation for the
function *g*(*x*)=*x*^{2}
at the places *x*=0.1*,*0.2*,*0.3*,*0.4*,*0.5
in the following sense: the largest of the values
|*f*(*x*)-*g*(*x*)| calculated at these places
should be as small as possible. (4 points)

Proposed by *G. Bakonyi, *Budapest

**B.3316. ***P *denotes any point inside a triangle
*ABC*. Lines *AP *and *BP *intersect the opposite sides
of the triangle at points *D *and *E*, respectively. Prove
1/*AP*=1/*AD*+1/*AC*, provided that *AP*=*BP
*and *BE*=*CE*. (4 points)

Proposed by *H. Lee, *Korea

**B.3317. **Solve the equation

(4 points)

Proposed by *Z. Hans, *Nagykanizsa

**B.3318. **Circles *k*_{1}
and *k*_{2}, each of unit radius,
touch at a point *P*. A line tangent to each circle, not passing
through *P*, is denoted by *e*. Let, for *i*>2,
*k*_{i} denote the circle,
different from *k*_{i-2},
which is tangent to each *k*_{1}*,k*_{i-1} and *e*. Determine the radius of
*k*_{1999}. (5 points)

**B.3319. **It is known that of all polygons determined by four
points lying in a given circle, the inscribed squares possess the
largest area. Is it also true that among all polyhedra determined by
eight points on a given sphere, the largest volume belong to the
inscribed cubes? (4 points)

**B.3320. **Solve the equation

*x*^{x1/2}=1/2.

(5 points)

Proposed by *L. Lovrics, *Budapest

**B.3321. **There is given a convex polygon *K *in the
plane. Prove that six congruent copies of *K *can be arranged
around *K *such that each shares a common boundary point with
*K*, but the interiors of the seven polygons are mutually
disjoint. (5 points)

## New advanced problems in November 1999 |

**A.221. **Consider a square *N *whose vertices have
coordinates (1999; 1999). Let *H *denote the triangle with vertices
(-1; 0), (0; -1) and (1; 1). How many non-overlapping translated
copies of *H *can be packed into *N*?

Proposed by *Gy. Pap, *Debrecen

**A.222. **Integers *a,b,c *have highest common factor
1. The triplet *a,b,c *may be altered into another triplet such
that in each step one of the numbers in the actual triplet is
increased or decreased by an integer multiple of another element of
the triplet. Is it true that, no matter how the numbers *a,b,c
*are given, the triplet (1*,*0*,*0) can be obtained in at
most 10 steps?

Proposed by *M. Abért, *Budapest

**A.223. **Determine all real functions *f *which satisfy

*f*(*x*-*f*(*y*))=*f*(*x*+*y*^{1999})+*f*(*f*(*y*)+*y*^{1999})+1

for arbitrary real numbers *x,y*.

### 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.