# New exercises and problems in Mathematics

December
2004

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

## New exercises for beginnersSolutions can be submitted only by students of grade 9. Maximum score for each exercise (sign "K") is 6 points. |

**K. 19.** Consider a two digit number, multiply its digits and proceed similarly with the result until you get a single digit number. How many two digit numbers are there to start with to arrive to 0 at the end?

**K. 20.** It is an old tradition in Whohascum to set up a Christmas tree on the railway station. The stationmaster has seven Christmas lamps of different colours and they can be switched on and off independently. His taste, however, objects to the light of both the violet and the pink lamps on the tree at the same time. On the 7^{th} of December the stationmaster puts some of his lamps on the tree. How many ways are there to choose them if he would like the lamps to go on giving light until the 6^{th} of January in different combinations every day?

**K. 21.** Using 19 regular dice we assemble a solid that can be obtained by removing the corners of a 3 *x*3 *x*3 cube. What is the least number of spots that can be seen on the surface of this solid? (There are 7 spots on any two opposite faces of a regular die.)

**K. 22.** A square based brick has a face of area 49 cm^{2} and another one of 84 cm^{2}. Determine its volume.

**K. 23.** The figure shows the December page of this year's calendar. Somewhere on the page there is a 3 *x*3 square containing a group of numbers whose sum is equal to 160. What is the smallest one among these numbers?

Mon | Tue | Wed | Thu | Fri | Sat | Sun |

1 | 2 | 3 | 4 | 5 | ||

6 | 7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 | 19 |

20 | 21 | 22 | 23 | 24 | 25 | 26 |

27 | 28 | 29 | 30 | 31 |

**K. 24.** There are two cylindrical tanks next to each other. The diameter of the first one is 4 m and the depth of water inside is 12,5 m. The other one is dry and its diameter is 3 m. A pump whose capacity is 10 m^{3}/min starts drawing water from the first tank into the second one. After how many minutes will the depth of water be equal in the tanks?

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

**C. 785.** Find the smallest positive integer that is divisible by 111 and its last four digits are 2004.

**C. 786.** We want to fix a curtain to rings hanging on the curtain rod. We want the rings to have equal distances between them. We first fix the ends of the curtain to the first and last rings. Then it is easy to achieve the equidistant arrangement of the rings if we always select the middle one among the remaining rings and attach it to the midpoint of the corresponding section of the curtain and proceed similarly in both halves, and so on until all rings are used. How many rings should we hang on the rod to make this procedure possible? (Suggested by *G. Zentai and G. Kiss,* Budapest)

**C. 787.** Prove that if *x* and *y* are positive numbers, then \(\displaystyle \frac{x+y}{\sqrt{xy}}\le\frac{x}{y}+\frac{y}{x}\).

**C. 788.** Represent the solution of the equation *x*^{5}-10*x*^{3}*y*^{2}+5*xy*^{4}=0 in the Cartesian coordinate plane. (Suggested by *A. Hraskó,* Budapest)

**C. 789.** 8 spheres of radius *r* are placed on a horizontal plane. Their centres form a regular octagon, and the neighbouring spheres touch. What is the radius of the sphere that touches the plane and the 8 spheres? (Suggested by *L. Németh,* Fonyód)

## 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. 3772.** How many pairs (*n*,*k*) are there, such that *n*>*k*, and the difference between the interior angles of the regular *n*-sided and *k*-sided polygons is equal to 1^{o}? (*3 points*)

**B. 3773.** Is 20^{2004}+16^{2004}-3^{2004}-1 divisible by 323? (*3 points*)

**B. 3774.** *ABC* is an isosceles right-angled triangle. *K* and *M* are given points on the hypotenuse *AB*. *K* separates *A* and *M*, and \(\displaystyle \measuredangle KCM=45^{\circ}\). Prove that *AK*^{2}+*MB*^{2}=*KM*^{2}. (*3 points*)

**B. 3775.** Solve the equation *y*^{3}=*x*^{3}+8*x*^{2}-6*x*+8 on the set of non-negative integers. (*4 points*)

**B. 3776.** The diagonal *BD* of a cyclic quadrilateral *ABCD* is a diameter of the circumscribed circle. The length of each side of the triangle *ABC* is at least 1. Prove that the area of the quadrilateral is greater than \(\displaystyle \tfrac{1}{2}\). (*E. Klein and T. Tao,* Sidney, Australia) (*4 points*)

**B. 3777.** \(\displaystyle \measuredangle BAC=\alpha\) is the apex angle of the isosceles triangle *ABC*. The base *BC* is divided into *n* equal parts by the points *D*_{1}, *D*_{2},..., *D*_{n-1} in the order of increasing distance from *B*. The point *E* divides the leg *AB* in the ratio 1:(*n*-1). Prove that

\(\displaystyle \measuredangle AD_1E+\measuredangle AD_2E+\dots+\measuredangle AD_{n-1}E=\frac{\alpha}{2}. \)

(*4 points*)

**B. 3778.** The diagonals of the convex quadrilateral *ABCD* intersect at the point *E*. Prove that

\(\displaystyle \big|t(ABE\Delta)-t(CDE\Delta)\big|\le\frac{1}{2}AD\cdot BC. \)

(*4 points*)

**B. 3779.** For every vertex *V* of a 2 unit cube consider the plane through the midpoints of the edges starting from *V*. Find the volume of the convex solid bounded by these planes? (*3 points*)

**B. 3780.** Prove that if *x*, *y*, *z* are positive numbers then

\(\displaystyle \frac{x+y+z}{\sqrt[3]{xyz}}\le\frac{x}{y}+\frac{y}{z}+\frac{z}{x}. \)

(*Elemente der Mathematik,* Basel) (*5 points*)

**B. 3781.** Evaluate the sum \(\displaystyle \sum_{n=1}^{\infty}\mathop{\rm arccot}\,(2n^2)\). (*5 points*)

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

**A. 359.** \(\displaystyle f\colon R\to R\) is a monotonic function. *c*_{1}, *c*_{2}>0 are constants, such that *f*(*x*)+*f*(*y*)-*c*_{1}\(\displaystyle \le\)*f*(*x*+*y*) \(\displaystyle \le\)*f*(*x*)+*f*(*y*)+*c*_{2} for all real numbers *x*, *y*. Prove that there exists a number *k*, such that *f*(*x*)-*kx* is a bounded function.

**A. 360.** 50 senators each vote for exactly one of three alternative proposals. We want to construct a table for evaluating the results, that is to tell which alternative wins. The table is to have 51 rows and 3^{50} columns. The first 50 rows show the votes of each senator and the last row shows the winning alternative in each case. The table is expected to have the following properties:

Prove that it is impossible to meet all these requirements.

**A. 361.** Prove that the following inequality for primes is true if *n*\(\displaystyle \ge\)3 is an integer:

\(\displaystyle \sum_{\substack{p\le n\\p\text{prime}}}\frac{1}{\sqrt{p}}\ge\frac{1}{2}\log n- \log\log n. \)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary

### Deadline:

10 January 2005 for problems K and

15 January 2005 for problems A, B, C.