# New exercises and problems in Mathematics

April
2003

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

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

**C. 715.** A device for measuring distance covered uses a wheel with a perimeter of 1 metre. The wheel is rolled down the road, and a counter displays the number of revolutions of the wheel. What will the counter read if the wheel is used in an irregular way and pushed up 100 stairs, 30 cm wide and 20 cm high each? (The wheel rolls without friction.)

**C. 716.** The given points *K*, *L* and *M* do not lie on a given line *e*. Let *N* be an arbitrary point of line *e*. Let *P* and *Q* denote the midpoints of *KL* and *MN*, respectively. Finally, let *R* be the midpoint of *PQ*. What is the locus of the points *R*?

**C. 717.** There are 58 slices of cake on a tray, walnut and poppyseed. The number of possible ways to select three slices of walnut cake equals the number of ways to select two slices of poppyseed cake and one slice of walnut. How many slices of poppyseed cake are there on the tray?

**C. 718.** Solve the equation (9-3*x*)^{.}3^{x}-(*x*-2)(*x*^{2}-5*x*+6)=0 on the set of real numbers.

**C. 719.** Solve the equation

\(\displaystyle \frac{1}{\log_{\frac{1}{2}}x}+ \frac{1}{\log_{\frac{2}{3}}x}+\dots+ \frac{1}{\log_{\frac{9}{10}}x}=1 \)

on the set of real numbers. (*From the mathematics competition of Hajdú-Bihar County, 2002/2003.*)

## 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. 3632.** Solve the equation *x*^{2003}- [*x*]^{2003}=(*x*-[*x*])^{2003} on the set of real numbers. (*3 points*)

**B. 3633.** The diagonals of a convex quadrilateral divide it into four triangles of integer areas. Is it possible for three of these areas to be 2001, 2002 and 2003? (*3 points*) (*From the mathematics competition of Hajdú-Bihar County, 2002/2003.*)

**B. 3634.** Let *k*(*n*) be the greatest odd factor of *n*, and let \(\displaystyle f(n)=\sum_{i=1}^nk(i)\). Prove that *f*(2*n*)-*f*(*n*)=*n*^{2}. (*4 points*)

**B. 3635.** A convex polyhedron has *c* vertices. Show that the sum of all interior angles of the faces of the polyhedron is (*c*-2)^{.}360^{o}. (*4 points*)

**B. 3636.** On what condition is there a 6-sided closed polygon of sides *a*, *b*, *c*, *d*, *e*, *f* in this order, such that opposite sides are pairwise perpendicular? (*5 points*)

**B. 3637.** Is it possible to select a point on each edge of a cube, such that the convex hull of the 12 points has exactly half the volume of the cube? (*4 points*)

**B. 3638.** A line \(\displaystyle \ell\) of the plane is said to touch the point set *H* if it contains exactly one point of *H*. Find a set of points in the plane that has exactly one tangent at each of its points, and has at least one point on each line of the plane. (*5 points*)

**B. 3639.** Mark a point *C* on the extension of the line segment *AB* beyond *A*. Let *e* denote the line perpendicular to *AB* at the point *C*, and let *D* be an arbitrary point of *e*. Erect a perpendicular to *AD* at the point *A*, and denote the intersection of this perpendicular with the line *DB* by *P*. What is the locus of the points *P*? (*4 points*) (Suggested by *I. Blahota,* Nyíregyháza.)

**B. 3640.** Let *a*_{1}=1 and \(\displaystyle a_{n+1}=a_n+\frac{1}{s_n}\), ahol *s*_{n}=*a*_{1}+*a*_{2}+...+*a*_{n}. Is the sequence *a*_{n} bounded? (*5 points*)

**B. 3641.** Is there an infinite sequence *p*_{1}(*x*), *p*_{2}(*x*), ...,*p*_{n}(*x*),... of polynomials, such that the degree of *p*_{k}(*x*) is exactly *k*, *p*_{i}(*p*_{j}(*x*))= *p*_{j} (*p*_{i}(*x*)) for all pairs (*i*,*j*) and *a*) *p*_{2}(*x*)=*x*^{2}-2, *b*) *p*_{2}(*x*)=*x*^{2}-3? (*5 points*)

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

**A. 317.** The function *f*:*A*^{n}\(\displaystyle \to\)*A* defined on the set *A*={yes,no} is said to be a decision function if

(*a*) changing all the arguments changes the value of the function, too, and

(*b*) the substitution of the value of the function for an arbitrary argument does not change the value of the function.

A function *h*: *A*^{n}\(\displaystyle \to\)*A* is said to be an authority function if there is an index *i*, such that the value of the function always equals the *i*th argument. If the value of a function *m*:*A*^{3}\(\displaystyle \to\)*A* is always equal to the argument that occurs at least twice, it is said to be a democratic function. Prove that every decision function is obtained as a composition of authority functions and democratic functions. (*Miklós Schweitzer Mathematics Competition, 2002.*)

**A. 318.** Let *n* be an arbitrary positive integer. *a*) Construct a polynomial *p* of degree *n* such that for all *x*\(\displaystyle \in\)[0,1/2],

\(\displaystyle \left|p(x)-\frac{1}{1-x}\right|<\frac{4}{\big(1+\sqrt2\,\big)^{2n+2}}. \)

*b*) Prove that for each polynomial *q* of degree *n*, there exists a real number *x*\(\displaystyle \in\)[0,1/2] such that

\(\displaystyle \left|q(x)-\frac{1}{1-x}\right\)\frac{1}{\big(1+\sqrt2\,\big)^{2n+2}}. ">

**A. 319.** Does there exist a sequence *a*_{1},*a*_{2}... of real numbers such that for an arbitrary positive integer *k*, the series \(\displaystyle \sum_{n=1}^\infty a_n^{4k-3}\) converges, but the series \(\displaystyle \sum_{n=1}^\infty a_n^{4k-1}\) diverges?

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary