# New exercises and problems in Mathematics

March
2003

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

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

**C. 710.** The average age of the students in a school is 16 years, the average age of the teachers is 38 years, and the average age of all students and teachers together is 17 years. The teachers teach 21 hours a week on average, and students have an average of 29 class periods per week. There are the same number of students in each class. Find the total number of students.

**C. 711.** The middle one of every three consecutive terms of a sequence of positive numbers equals to the product of the other two terms. The product of the first five terms and the product of the next five terms are both 2. Determine the first ten terms of the sequence.

**C. 712.** In a triangle *ABC*, \(\displaystyle \gamma\)=3\(\displaystyle \alpha\), *a*=27, *c*=48, in conventional notation. Find the length of the side *b*.

**C. 713.** How many solutions does the equation sin 2002*x*=sin 2003*x* have in the interval [0,2\(\displaystyle \pi\)]?

**C. 714.** Somebody left a ball floating in a pond. The water froze, and the ball was taken out of the ice, leaving a 8-cm-deep hollow of diameter 24 cm. What was the radius of the ball in centimetres?

## 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. 3622.** There are 1280 pine trees of 1 m diameter each in the forest where Smurfs live. The dimensions of the forest are 1001*x*945 metres. In the forest, the Smurfs would like to lay out seven tennis courts 20*x*34 metres each. Is that possible without cutting down any pine tree? (*4 points*)

**B. 3623.** Prove that

\(\displaystyle \root3\of{\sqrt{5}+2}-\root3\of{\sqrt{5}-2} \)

is a rational number. (*4 points*)

**B. 3624.** *a* and *b* are positive numbers such that

\(\displaystyle \frac{{(a+b)}^n-{(a-b)}^n}{{(a+b)}^n+{(a-b)}^n}=\frac{a}{b}, \)

where *n* is a given positive integer. Prove that *a*=*b*. (*4 points*)

**B. 3625.** Find the radius of the sphere that touches three faces of a regular tetrahedron of unit edge, and also touches the three sides of the fourth face. (*3 points*)

**B. 3626.** The first two terms of the sequence *x*_{0}, *x*_{1}, *x*_{2},... are positive, and \(\displaystyle x_{n+2}=\frac{x_{n+1}+1}{x_n}\). Express the 2003rd term in terms of *x*_{0} and *x*_{1}. (*3 points*)

**B. 3627.** Prove that if the product of the positive numbers *a*, *b*, *c*, *d* is 1 then

\(\displaystyle a^3+b^3+c^3+d^3\ge\max\left\{a+b+c+d;\frac{1}{a}+ \frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right\}. \)

(*4 points*) (Suggested by *K. Némethy,* Budapest.)

**B. 3628.** Four brothers inherited a convex quadrilateral lot of land. They divided the lot into four parts by joining the midpoints of the opposite sides. The areas of the shares of the first three brothers are 360, 720 and 900 square metres. What is the area of the fourth brother's share? (*4 points*)

**B. 3629.** Prove that if *ABCD* is a convex quadrilateral then

(*AB*+*CD*)^{2}+ (*BC*+*DA*)^{2}\(\displaystyle \ge\)(*AC*+*BD*)^{2}.

(*4 points*)

**B. 3630.** Given three points on a circle, construct a fourth point on the circle, such that the four points form a quadrilateral that has an inscribed circle. (*5 points*)

**B. 3631.** Let *f*(*x*) be an arbitrary second-degree polynomial. Prove that if the polynomials *p*(*x*) and *q*(*x*) of at least degree one each commute with the polynomial *f*(*x*) under composition, then they also commute with each other.

(See problem **B. 3621.** of KöMaL, February 2003, page 105.)

(*5 points*)

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

**A. 314.** Find the maximum possible length of a polygon lying in a unit square, given that any line drawn through a vertex of the polygon intersects at most seven of its sides. (Based on the idea of *L. Soukup.*)

**A. 315.** We have selected *n*+1 vertices of a regular (*n*^{2}+*n*+1)-sided polygon. Prove that if the number *n* is of the form 12*k*-2, then the distances between the points selected cannot be all different.

**A. 316.** Given a positive integer *n*, consider the sets *A*\(\displaystyle \subset\){1,2,...,*n*} in which the congruence *x*+*y*\(\displaystyle \equiv\)*u*+*v* (mod *n*) has no other solutions but the trivial *x*=*u*, *y*=*v*, and *x*=*v*, *y*=*u*. Let *f*(*n*) denote the maximum number of elements in such sets. *a*) Prove that \(\displaystyle f(n)<\sqrt{n}+1\). *b*) Give an example for infinitely many *n*, such that \(\displaystyle f(n\)\sqrt{n}-1">. (Based on problem 4 of the Schweitzer Competition of 2002.)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary