# New exercises and problems in Mathematics

February
2003

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

## New exercisesMaximum score for each exercise (sign "C") is 5 points. |

**C. 705.** The pagination of a book starts on the fifth page. That page carries the number 5. There are two more pages like the first four, with no page number printed on them. The sum of all the page numbers in the book is 23 862. How many pages does the book contain?

**C. 706.** For what natural numbers *a* and *b* is it true that 90< *a*+*b*< 100 and \(\displaystyle 0.9<\frac{a}{b}<0.91\)?

**C. 707.** Parallel to two sides of a triangle, draw the lines that halve the area of a triangle. In what ratio is the area divided by the line drawn through their intersection to the third side of the triangle?

**C. 708.** The angles of an isosceles triangle are \(\displaystyle \alpha\), \(\displaystyle \beta\), \(\displaystyle \gamma\). Find the measures of these angles if sin^{2}\(\displaystyle \alpha\)+sin^{2}\(\displaystyle \beta\)=sin \(\displaystyle \gamma\).

**C. 709.** A well-rounded die is obtained as the intersection of a cube with the sphere touching its edges. Given that the distance between two opposite faces is 2 cm, find the surface area of such a die.

## New problemsThe maximum scores for problems (sign "B") depend on the difficulty. It is allowed to send solutions for any number of problems, but your score will be computed from the 6 largest score in each month. |

**B. 3612.** Given a number in decimal notation, consider the sum of all the different numbers that can be obtained by rearranging its digits. For example, the sum obtained from the number 110 is 110+101+11=222. Find the smallest number that yields the sum 4 933 284. (*4 points*) (Suggested by *G. Bakonyi,* Budapest)

**B. 3613.** \(\displaystyle f\:R\to R\) is a function such that |*f*(*x*)-*f*(*y*)|=|*x*-*y*| for all *x*, *y*. Given that *f*(1)=3 find *f*(2). (*3 points*)

**B. 3614.** *A*_{1},*A*_{2},...,*A*_{n} are distinct points in the plane. The midpoints of all the segments they determine are coloured red. What is the minimum possible number of red points obtained? (*4 points*) (Suggested by *K. Némethy,* Budapest)

**B. 3615.** The base of a parallelepiped of unit edge is *A*_{1}*A*_{2}*A*_{3}*A*_{4}, its top face is *B*_{1}*B*_{2}*B*_{3}*B*_{4} such that the vertices *A*_{i} and *B*_{i} are connected by edges. In what interval does the sum *A*_{1}*B*_{2}^{2}+*A*_{2}*B*_{3}^{2}+*A*_{3}*B*_{4}^{2}+*A*_{4}*B*_{1}^{2} vary? (*3 points*)

**B. 3616.** In the Cartesian coordinate plane, the line *e* passes through the point (1,1) and the line *f* passes through the point (-1,1). Given that the difference of the slopes of the two lines is 2, find the locus of the intersection of the lines *e* and *f*. (*3 points*)

**B. 3617.** For what values of the parameter *t* do the simultaneous equations *x*+*y*+*z*+*v*=0, (*xy*+*yz*+*zv*)+*t*(*xz*+*xv*+*yv*)=0 have a unique solution? (*5 points*)

**B. 3618.** The points *E* and *F* lie on the extension of the side *AB* of a rhombus *ABCD*. The tangents drawn from *E* and *F* to the inscribed circle of the rhombus intersect the line *AD* at the points *E*' and *F*'. Determine the ratio *DE*':*DF*', given that *BE*:*BF*=\(\displaystyle \lambda\):\(\displaystyle \mu\). (*4 points*)

**B. 3619.** Consider three of the four planes that are parallel to a face of a tetrahedron of unit volume and halve that volume. Find the volume of the tetrahedron bounded by those three planes and the fourth face of the original tetrahedron. (*4 points*)

**B. 3620.** Assume that the sequence defined by the recurrence \(\displaystyle a_{n+1}=\frac{1}{1-a_n}-\frac{1}{1+a_n}\) is periodic. Find the first term of this sequence. (*5 points*) (Suggested by *P. Zsíros,* Szombathely)

**B. 3621.** Let *f*(*x*)=*ax*+1, where *a*\(\displaystyle \ne\)0 is a given real number. Find all polynomials *g*(*x*) such that *f*(*g*(*x*))=*g*(*f*(*x*)). (*5 points*)

## New advanced problemsMaximum score for each advanced problem (sign "A") is 5 points. |

**A. 311.** Prove that

\(\displaystyle [nx]\ge\frac{[x]}{1}+\frac{[2x]}{2}+\frac{[3x]}{3}+ \dots+\frac{[nx]}{n} \)

for every positive real number *x*. (American competition problem)

**A. 312.** For what positive integers *n* is it possible to divide the complete graph on *n* points into triangles so that each edge is used exactly once?

**A. 313.** Prove that it is possible to colour any distinct *n* points of the plane with at most 100^{.}ln *n* colours in such a way that every circle containing at least one point contains exactly one of the points of some particular colour. (An adaptation of a Schweitzer Competition problem, 2002)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary