# New exercises and problems in Mathematics

October
2002

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

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

**C. 685.** The surface area of a cylinder is 1000 m^{2} and its height is 1000 km. Find the volume of the cylinder in litres.

**C. 686.** Anna was bored in class, and to kill time she made a list of integers. Starting with a certain number, she obtained the next number by either adding or multiplying the digits of the previous number on the list. She continued writing down numbers hence obtained, and observed that all the numbers were odd. With how many initial values at most six digits is that possible?

**C. 687.** The coordinates of the vertices of a quadrilateral are *A*(0;0), *B*(16;0), *C*(8;8), *D*(0,8). Find the equation of the line parallel to *AC* that halves the area of the quadrilateral.

**C. 688.** Solve the equation [*x*/2]+[*x*/4]=*x*. ([*x*] denotes the greatest integer not greater than the number *x*.)

**C. 689.** Solve the equation

\(\displaystyle x^{\log_2(16x^2)}-4x^{\log_2(4x)+1}-16x^{\log_2(4x)+2}+64 x^3=0. \)

(Suggested by *A. Mosóczi,* Budapest)

## 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. 3572.** Solve the equation [*x*/2]+[*x*/4]=[*x*]. ([*x*] denotes the greatest integer not greater than the number *x*.) (*3 points*)

**B. 3573.** Given the line segment *AB* its midpoint *F* and the point *P*. With the help of a single straight edge, construct a line parallel to *AB* through the point *P*. (*4 points*)

**B. 3574.** The circle *k*_{1} touches the circle *k*_{2} internally at the point *P*. The chord *AB* of the larger circle touches the smaller circle at *C*, *AP* and *BP* intersect the smaller circle at the points *D* and *E*, respectively. Given that *AB*=84, *PD*=11, and *PE*=10. Find the length of the line segment *AC*. (*4 points*)

**B. 3575.** Let *X* denote the set of positive integers that contain different digits in decimal notation. If *n*\(\displaystyle \in\)*X*, let *A*_{n} denote the set of numbers obtained by the permutations of the digits of *n*, and let *d*_{n} be the greatest common factor of the elements of *A*_{n}. What is the maximum value of *d*_{n}? (*4 points*)

**B. 3576.** A random sequence is produced using the digits 0, 1, 2. How long should the sequence be so that the probability that all three digits occur in the sequence is at least 0.61? (*4 points*) (Suggested by *E. Fried,* Budapest)

**B. 3577.** Solve the equation sin 3*x* + 3 cos *x* = 2 sin 2*x* (sin *x*+cos *x*). (*4 points*) (Suggested by *A. Mosóczi,* Budapest)

**B. 3578.** The centre of the face *ABCD* of the cube in the *Figure* is the point *M*. Find the points *P* and *Q* on the lines *AB* and *EM*, respectively, such that the distance *PQ* is equal to the distance of the lines *AB* and *EM*. (*3 points*)

**B. 3579.** Solve the equation

\(\displaystyle x=\sqrt{-3+4\sqrt{-3+4\sqrt{-3+4x}}}. \)

(*5 points*)

**B. 3580.** Out of all obtuse triangles of integer side lengths with the obtuse angle equal to the double of one acute angle, which one has the shortest perimeter? (*4 points*)

**B. 3581.** Find the minimum value of the function

*f*(*x*)=|1001+1000*x*+999*x*^{2}+^{...}+2*x*^{999}+*x*^{1000}|.

(*5 points*)

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

**A. 299.** *P* and *Q* are interior points of the square *ABCD* such that *PAQ*\(\displaystyle \angle\)= *PCQ*\(\displaystyle \angle\)= 45^{o}. Determine the length *PQ* in terms of the lengths *BP* and *DQ*.

**A. 300.** Find all pairs (*a*, *b*) such that *a* and *b* are whole numbers and *a*^{2} + *ab* + *b*^{2}is a multiple of 7^{5}.

**A. 301.** Let *a*_{0},*a*_{1},... a sequence of non negative numbers such that for every *k*, *m* \(\displaystyle \ge\)0, *a*_{k+m} \(\displaystyle \le\)*a*_{k+m+1} + *a*_{k} *a*_{m}. Assume, additionally, that *na*_{n} < 0.2499 holds for sufficiently large *n*. Prove that there exists a number *q* for which 0<*q*<1 and *a*_{n}<*q*^{n} if *n* is large enough.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary