# New exercises and problems in Mathematics

February
2004

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

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

**C. 750.** A train left the station on schedule. Having covered 8 kilometres, the engineer looked at his watch to see that the short and long hands of the watch were exactly in the same position. The average speed of the train over the first 8 kilometres was 33 kilometres per hour. When did the train leave the station?

**C. 751.** The sides of a kite are *a* and *b* (*a*\(\displaystyle \ne\)*b*). The sides of different lengths enclose a right angle. Find the radius of the circle touching each line obtained by extending the sides.

**C. 752.** Prove that if the positive numbers *a*, *b*, *c* are consecutive terms of a geometric progression then *a*+*b*+*c*, \(\displaystyle \sqrt{3(ab+bc+ca)}\) and \(\displaystyle \sqrt[3]{27abc}\) are also consecutive terms of a geometric progression.

**C. 753.** Some 1.5-litre mineral water bottles have a narrower part at the middle to provide a more comfortable grip. The normal perimeter of a bottle is 27.5 cm but only 21.6 cm at the narrow part. The two cylindrical surfaces are connected by 2-cm-high conical surfaces both above and below the narrow part. How much taller are such bottles than those of normal perimeter without a narrower part?

**C. 754.** Solve the equation \(\displaystyle \frac{2003x}{2004}=2003^{\log_x2004}\).

## 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. 3702.** Two players play the following game: They take turns taking at least one but at most nine out of 110 counters. No player is allowed to repeat the previous move. The player unable to move is losing the game. Which player has a winning strategy? (*4 points*)

**B. 3703.** In a sequence *a*_{n}, *a*_{1}=1337 and *a*_{2n+1}=*a*_{2n}=*n*-*a*_{n} for all positive integers *n*. Find the value of *a*_{2004}. (*3 points*)

**B. 3704.** Construct a triangle, given the lines of the perpendicular bisectors of the sides and a point lying on the line of a side. (*4 points*)

**B. 3705.** The positive integers *a* and *b* are relative primes but the greatest common factor of the numbers *A*=8*a*+3*b* and *B*=3*a*+2*b* is not 1. What is the greatest common factor of *A* and *B*? (*3 points*)

**B. 3706.** Consider the twelve planes containing one edge of a unit cube, not intersecting the cube and enclosing 45^{o} angles with the faces meeting at that edge. What is the volume of the convex solid bounded by these planes? (*3 points*)

**B. 3707.** The edges of the tetrahedron *ABCD* are *AB*=*c*, *BC*=*a*, *CA*=*b*, *DA*=*a*_{1}, *DB*=*b*_{1}, and finally *DC*=*c*_{1}. Let *h* denote the length of the line segment connecting the vertex *D* of the tetrahedron to the centroid of the opposite face. Prove that \(\displaystyle h^2=\frac{1}{3}
\big(a_1^2+b_1^2+c_1^2\big)-\frac{1}{9}\big(a^2+b^2+c^2\big)\). (*4 points*)

**B. 3708.** A line through a given point *P* is cutting a circle *k* at points *A* and *B* such that *PA*=*AB*=1. The tangents drawn from *P* to *k* touch the circle at the points *C* and *D*. *M* is the intersection of *AB* and *CD*. Determine the distance *PM*. (*4 points*)

**B. 3709.** Given is the quadratic equation *ax*^{2}+*bx*+*c*=0. Its coefficients *a*, *b*, *c* satisfy 2*a*+3*b*+6*c*=0. Prove that the equation has a root *x* such that 0<*x*<1. (*4 points*)

**B. 3710.** *T*_{i} is the foot of the altitude drawn from the vertex *A*_{i} of the acute-angled non-isosceles triangle *A*_{1}*A*_{2}*A*_{3}. Let *B*_{i} denote the intersection of the lines *A*_{j}*A*_{k} and *T*_{j}*T*_{k} (where *i*, *j* and *k* are all different). Show that the points *B*_{1}, *B*_{2}, *B*_{3} are collinear. (*5 points*)

**B. 3711.** The sum of the non-negative real numbers *s*_{1}, *s*_{2},..., *s*_{2004} is 2 and *s*_{1}*s*_{2}+*s*_{2}*s*_{3}+ ...+ *s*_{2003} *s*_{2004}+ *s*_{2004}*s*_{1}=1. Find the largest and smallest possible values of *S*=*s*_{1}^{2}+*s*_{2}^{2}+...+*s*_{2004}^{2}. (*5 points*)

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

**A. 338.** For any positive integer *n* denote the closest integer to \(\displaystyle \sqrt{n}\) by *f*(*n*). Calculate the value of the sum

\(\displaystyle \sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}. \)

**A. 339.** We want to select 4 tuples from a 28-element set with the following properties: *a*) Any two 4-tuple has at most two common elements; *b*) for any element *x* and 4-tuple *A* that is not containing *x* there exists at least one 4-tuple *B* that contains *x*-et and it has exactly two common elements with *A*. Is it possible to select such a system of 4-tuples?

**A. 340.** Is it possible that the length of the intersection of a circular disc of unit radius and a parabola is greater than 4 units?

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary