# New exercises and problems in Mathematics

March
2004

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

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

**C. 755.** In how many ways can we give change for a 1000-forint (HUF) note, using 1, 2 and 5-forint coins only?

**C. 756.** Solve the following inequality on the set of real numbers:

\(\displaystyle \frac{|1-x|}{1-|x|}<\frac{1+|x|}{|1+x|}. \)

**C. 757.** A large cube of edge *n* is put together out of *n*^{3} unit cubes. Is there a value of *n*, such that the diagonals of the large solid intersect half as many small cubes altogether as the number of small cubes not intersected by those diagonals?

**C. 758.** The lengths of the legs of a right-angled triangle are 1 and \(\displaystyle \sqrt2\). The smallest angle of the triangle is \(\displaystyle \alpha\). Find the exact value of cos 8\(\displaystyle \alpha\).

**C. 759.** To every point *P*(*x*,*y*) of the coordinate plane (*x*,*y*), we assign the point *P*'(*x*-*y*,-*y*). Which lines will be mapped onto themselves under this transformation?

## 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. 3712.** Each of the eight participants of a dinner party made new acquaintances during the evening. Upon leaving, each of them wrote down on a sheet of paper the number of new acquaintances he had made. The following list of numbers was produced: 1,2, 3, 3, 5, 6, 6, 6. Prove that there was someone who made a mistake. (Acquaintances are mutual.) (*3 points*)

**B. 3713.** The diagonals of a convex quadrilateral *ABCD* intersect at point *M*. Prove that if *M* lies on the line connecting the centroids of the triangles *ABM* and *CDM* then the quadrilateral is a trapezium. (Suggested by *G. Holló,* Budapest) (*3 points*)

**B. 3714.** Given the perimeter of a triangle, one side and the angle opposite to that side, construct the triangle with ruler and compass. (*4 points*)

**B. 3715.** Let *x* denote a positive number and let *n* be an integer greater than 1. Prove that

\(\displaystyle \frac{x^{n-1}-1}{n-1}\le\frac{x^n-1}{n}. \)

(*4 points*)

**B. 3716.** Solve the following simultaneous equations:

\(\displaystyle x^2+7y+2=2z+4\sqrt{7x-3},\)

\(\displaystyle y^2+7z+2=2x+4\sqrt{7y-3},\)

\(\displaystyle z^2+7x+2=2y+4\sqrt{7z-3}.\)

(*3 points*)

**B. 3717.** Find the rectangle *H* of unit area and minimum perimeter for which there exists a rectangle *H*_{1}, such that its perimeter is 50% shorter than that of *H* and its area is 50% greater than the area of *H*. (*4 points*)

**B. 3718.** Prove that if an arbitrary line passing through the circumcentre of a triangle is reflected about the bimedians of the triangle then the three reflections are concurrent. (Suggested by *Z. Csík,* Budapest) (*5 points*)

**B. 3719.** Given five points on the surface of a sphere, prove that there exists a closed hemisphere that contains at least four of them. (*4 points*)

**B. 3720.** Is there a pair *u*, *v* of real numbers such that a) *u*+*v* is rational and *u*^{n}+*v*^{n} is irrational where *n*\(\displaystyle \ge\)2 is an integer; b) *u*+*v* is irrational and *u*^{n}+*v*^{n} is rational where *n*\(\displaystyle \ge\)2 is an integer? (*5 points*)

**B. 3721.** Consider a set *S* and a binary operation * on it (that is, for all *a*, *b* in *S*, *a***b* also belongs to *S*). Given that (*a***b*)**a*=*b* is true for all *a*, *b* in *S*, prove that *a**(*b***a*)=*b* is also true for all *a*, *b* in *S*. (*4 points*)

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

**A. 341.** For which positive integer values of *n* and *k* is it possible to divide the numbers 1,2,...,*n* into *k* groups in such a way that the sum of the numbers in each group is the same?

**A. 342.** Prove that for any *p* prime of the form 4*k*+1

\(\displaystyle \sum_{n=1}^{p-1}\big[\sqrt{np}\;\big]=\frac{(p-1)(2p-1)}{3}. \)

**A. 343.** The sum of the positive real numbers *a* and *b* is less than 1. Find those continuous \(\displaystyle G\colon\mathbb{R}\to\mathbb{R}\) functions for which

*g*(*g*(*x*))=*ag*(*x*)+*bx*.

for any real number *x*.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary