# New exercises and problems in Mathematics

April
2004

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

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

**C. 760.** We paid with a 1000-forint (HUF) note in the grocery shop. The bill showed the amount to be payed and the amount we got back. We observed that these two numbers had the same digits but in different orders. What was the sum of the digits?

**C. 761.** Given the lengths of two sides of a triangle, and given that the corresponding medians are perpendicular, calculate the length of the third side.

**C. 762.** The surface area of the circumscribed sphere of a cube *K*_{1} is twice the surface area of the inscribed sphere of another cube *K*_{2}. Let *V*_{1} denote the volume of the inscribed sphere of the cube *K*_{1} and let *V*_{2} denote the volume of the circumscribed sphere of the cube *K*_{2}. Determine the ratio \(\displaystyle \frac{V_1}{V_2}\).

**C. 763.** There are three 30 cm *x*40 cm shelves on a stand in the corner of a room. The distances between them are equal. There were three spiders sitting at the point where the middle shelf and both walls meet. One of them crawled diagonally up one wall to the corner of the upper shelf. Another spider crawled diagonally down the other wall to the corner of the lower shelf. The third one remained where it was and observed that he could see his fellows at an angular distance of 120^{o} from each other. Find the (equal) separation between the shelves.

**C. 764.** Given the real number *s*, solve the inequality \(\displaystyle \log_{\frac{1}{s}}\log_s\)\log_s
\log_sx">.

## 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. 3722.** The secant passing through the intersections of a circle of 4-cm radius centred at *O*_{1} and a circle of 6-cm radius centred at *O*_{2} intersects the line segment *O*_{1}*O*_{2} at the point *T*. The length of *O*_{1}*O*_{2} is not smaller than 6 cm. The larger circle intersects the line segment *O*_{1}*O*_{2} at *A*, the smaller circle intersects it at *B*, and *AT*:*BT* =1:2. Find the length of the line segment *O*_{1}*O*_{2}. (*3 points*) (Suggested by *Sz. Békéssy,* Budapest)

**B. 3723.** We have prepared a convex solid out of pentagons and hexagons. Three faces meet at every vertex. Each pentagon is attached to five hexagons along its edges, and each hexagon has common edges with three pentagons. How many faces does the solid have? (*4 points*)

**B. 3724.** Find all polynomials *p*(*x*) for which the polynomials *p*(*x*)^{.}*p*(*x*+1) and *p*(*x*+*p*(*x*)) are identically equal. (*4 points*)

**B. 3725.** Prove that if *a* and *b* are positive numbers then \(\displaystyle 2\sqrt a+3\sqrt[3]{b}\ge5\sqrt[5]{ab}\). (*4 points*)

**B. 3726.** What is the digit preceding the decimal point in the number \(\displaystyle \big(3+\sqrt{7}\,\big)^{2004}\)? (*4 points*)

**B. 3727.** In a convex quadrilateral, the square of the distance between the midpoints of each pair of opposite sides is equal to one half the sum of the squares of those two sides. Prove that the quadrilateral is a rhombus. (*4 points*)

**B. 3728.** Consider the sequence with *a*_{0}=1 where *a*_{2n+1}=*a*_{n} and *a*_{2n+2}=*a*_{n}+*a*_{n+1} for all integers *n*\(\displaystyle \ge\)0. Prove that every positive rational number occurs among the elements of the set

\(\displaystyle \left\{\frac{a_{n+1}}{a_n}\colon n\ge1\right\}=\left\{\frac{1}{1},\frac{1}{2}, \frac{2}{1},\frac{1}{3},\frac{3}{2},\dots\right\}. \)

(*5 points*)

**B. 3729.** The triangle *ABC* has unit area. *E*, *F* and *G* are points on the sides *BC*, *CA* and *AB*, respectively, such that *AE* bisects *BF* at the point *R*, *BF* bisects *CG* at the point *S*, and finally *CG* bisects *AE* at the point *T*. Find the area of the triangle *RST*. (*5 points*)

**B. 3730.** The common centre of a circle *k*_{a} of radius *a* and a circle *k*_{b} of radius *b* (*a*>*b*) is *O*. One of the tangents drawn from a point *A* lying on *k*_{a} to the circle *k*_{b} touches *k*_{b} at *E*. The perpendicular drawn to the radius *OE* at an arbitrary point *P* of it intersects *k*_{b} at the points *Q* and *R*, and intersects the line *AO* at the point *T*. The points *S*_{1} and *S*_{2} are obtained by marking off those points of the perpendicular drawn to *AO* at *T* whose distance from *T* are equal to *PQ*. What is the locus of the points *S*_{i} as *P* moves along the radius *OE*? (*4 points*) (Suggested by *M. Kárpáti,* Bük)

**B. 3731.** How many positive integers \(\displaystyle \overline{a_1a_2\dots a_{2n}}\) are there, such that none of the digits are 0 and the sum *a*_{1}*a*_{2}+*a*_{3}*a*_{4}+...+*a*_{2n-1}*a*_{2n} is even? (*5 points*)

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

**A. 344.** Does there exist such a lattice rectangle which can be decomposed into lattice pentagons congruent to the one shown in the *Figure?*

**A. 345.** Let *r* and *s* be arbitrary positive integers and let *t* be the minimum of \(\displaystyle 2^k\big(\lceil r/2^k\rceil+\lceil s/2^k\rceil-1\big)\) on the nonnegative integer values of *k*. Prove that (*a*) \(\displaystyle \binom{t}{j}\) is even whenever *t*-*s*<*j*<*r*; (*b*) *t* is the smallest integer with this property. (\(\displaystyle \lceil x\rceil\) denotes the smallest integer not smaller than *x*.)

**A. 346.** Determine all functions \(\displaystyle f\:\mathbb{Q}\to\mathbb{R}\) satisfying *f*(*xy*)=*f*(*x*)*f*(*y*) and *f*(*x*+*y*)\(\displaystyle \le\)max (*f*(*x*),*f*(*y*)) for all *x*, *y* real numbers.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary