# New exercises and problems in Mathematics

May
2004

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

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

**C. 765.** The diameter of a round table is 1 metre. The table consists of two semicircular discs. It can be enlarged by inserting a 1 m *x*0.5 m rectangular part between the discs. Are there two points on the enlarged table whose distance exceeds 150 cm?

**C. 766.** The clock on the kitchen wall is losing 2 seconds per day while grandfather's antique clock in the lounge gains 15 seconds per day. Sunday at noon the kitchen clock reads 1 minute past 12 and the antique clock reads 1 minute to 12. At what time of the week will the sum of the squares of the differences between the true time and the time read by the clocks be a minimum?

**C. 767.** Find all non-negative numbers *a*, *b*, *c*, such that \(\displaystyle \sqrt{a-b+c}=\sqrt{a}-\sqrt{b}+\sqrt{c}\).

**C. 768.** Two unit circles intersect at points *A* and *B*. One of their common tangents touches the circles at points *E* and *F*. What may be the radius of the circle that passes through the points *E*, *F* and *A*?

**C. 769.** A cylinder of 20-cm radius touches level ground along the line *e*. A rod of length 50 cm, perpendicular to the line *e* is leaning against the cylinder so that its endpoint on the ground is 40 cm from the line *e*. What is the height of the other endpoint above the ground?

## 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. 3732.** The centre of the incircle of a triangle *ABC* is *O*. The lines *AO* and *BO* intersect the sides *BC* and *AC* at the points *A*_{1} and *B*_{1}, respectively, such that *OA*_{1} =*OB*_{1}. Does it follow that the triangle is isosceles? (*3 points*)

**B. 3733.** Is there an integer *m* for which the numbers 100+*m*^{.}101 and 101-*m*^{.}100 are not relatively prime? (*4 points*)

**B. 3734.** Is there a right triangle of integer sides that can be rotated about one of its legs to produce a circular cone whose surface area and volume is expressed by the same number of area and volume units, respectively? (*3 points*) (Suggested by *L. Lorántfy,* Dabas)

**B. 3735.** The first two terms of the sequence *x*_{n} are *x*_{1}=1001, *x*_{2}=1003 and if *n* \(\displaystyle \ge\)1, then \(\displaystyle x_{n+2}=\frac{x_{n+1}-2004}{x_n}\). Find the sum of the first 2004 terms of the sequence. (*4 points*)

**B. 3736.** *N* is a given point on the side *CD* of a unit square *ABCD* and *M* is a given point on the side *CB*, such that the perimeter of the triangle *MCN* is 2. Find the measure of \(\displaystyle \angle\)*MAN*. (*4 points*)

**B. 3737.** In a convex quadrilateral, the distance between the midpoints of each pair of opposite sides is equal to the half of the sum of those sides. Prove that the quadrilateral is a rhombus. (*4 points*)

**B. 3738.** Solve the following equation: \(\displaystyle x-2=\sqrt{4-3\sqrt{4-3\sqrt{10-3x}}}\). (*4 points*)

**B. 3739.** *a* is a real number such that *a*^{5}-*a*^{3}+*a*=2. Prove that 3<*a*^{6}<4. (*5 points*)

**B. 3740.** Consider the sequence given by the recursion *a*_{0}=1, \(\displaystyle a_1=\frac{1}{3}\), \(\displaystyle a_{n+1}=
\frac{2a_n}{3}-a_{n-1}\) (*n*\(\displaystyle \ge\)1). Prove that there exists a positive integer *n* such that *a*_{n}>0.9999. (*5 points*)

**B. 3741.** How many planes are there that intersect a regular octahedron in a regular hexagon? (*5 points*)

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

**A. 347.** Assume that triangle *ABC* is not equilateral and let \(\displaystyle 0_{1} and *A*_{2} be on side *AB*, points *B*_{1} and *B*_{2} on side *BC* and points *C*_{1} and *C*_{2} on side *CA* such that

\(\displaystyle \frac{AA_1}{AB}=\frac{A_2B}{AB}=\frac{BB_1}{BC}=\frac{B_2C}{BC}=\frac{CC_1}{CA}= \frac{C_2A}{CA}=t. \)

Prove that the radical line of circles *A*_{1}*B*_{1}*C*_{1} and *A*_{2}*B*_{2}*C*_{2} does not depend on the value of *t*. (The radical line or radical axis of two circles is the locus of points from which the difference between the squares of the distance to the centre and the radius are the same for both circles.) (*András Dőtsch,* Szeged)

**A. 348.** Let \(\displaystyle \sum_{n=1}^\infty a_n\) be a divergent series with nonincreasing positive terms. Prove that the series

\(\displaystyle \sum_{n=1}^\infty\frac{a_n}{1+na_n} \)

diverges. (*Vojtech Jarnik International Mathematical Competition,* Ostrava, 2004)

**A. 349.** The entries of an *nxn* table are real numbers of absolute value at most 1, and the sum of all entries is 0. Prove that there exists a column or a row of the table in which the absolute value of the sum of entries is at most \(\displaystyle \frac{n}{2}\).

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary