# New exercises and problems in Mathematics

December
2002

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

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

**C. 695.** We decreased a six-digit number
by the sum of its digits, and went on repeating same procedure with
the resulting number. Is it possible that we obtained 2002 in this
way?

**C. 696.** Solve the equation
|*x*+3|+*p*|*x*-2|=5, where *p* is a real
parameter.

**C. 697.** In the quadrilateral
*ABCD*,

*AB*=1, *BC*=2, ,
*ABC*\(\displaystyle \angle\)=120^{o}, and *BCD*=90^{o}.

Find the exact value of the length of side
*AD*.

**C. 698.** The length of side *AB* in
a triangle is 10 cm, the length of side *AC* is 5,1 cm,
and *CAB*\(\displaystyle \angle\)=58^{o}. Determine the measure of
*BCA*\(\displaystyle \angle\) to the nearest hundredth of a degree.

**C. 699.** What is the probability that at
least one of the five numbers drawn (out of 1 to 90) in a lottery was
also drawn last week? (There is one lottery draw per week.)

## 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. 3592.** Santa Claus was watching the
sky anxiously, deep in contemplation. He wanted to travel as far away
as possible the day after to deliver gifts to children. Finally, at
midnight it started snowing. Being an expert on snowfall he saw
immediately that it was the kind of snow that would not stop falling
for at least 24 hours. He also knew that during the first
16 hours the sleigh can travel faster and faster. (The speed can
be increased uniformly.) It would be standing still at the beginning
but by the end of the 16-th hour it would be flying like an
arrow. Then, however, it would be harder and harder to travel in the
thickening snow, and during the remaining 8 hours the top speed
would be uniformly decreasing back to zero. Santa Claus, on the other
hand did not want to exhaust his reindeer by forcing them for more
than 8 hours. When should he depart in order to cover the
greatest distance possible? (*4 points*)

**B. 3593.** Is there an arithmetic
progression of different positive integers in which no term is
divisible by any square number greater than 1?
(*3 points*)

**B. 3594.** Is there a square number in
decimal notation in which the sum of the digits is 2002?
(*4 points*)

**B. 3595.** Solve the following
equation:

2*x*^{4}+2*y*^{4}-4*x*^{3}*y*+6*x*^{2}*y*^{2}-4*xy*^{3}+7*y*^{2}+7*z*^{2}-14*yz*-70*y*+70*z*+175=0.

(*3 points*) (Suggested by
*M. Haragos* and *M. Zsoldos,* Budapest)

**B. 3596.** The circle
*k*_{1} of radius *R* touches the circle
*k*_{2} of radius 2*R* externally at the point
*E*_{3}, and the circles *k*_{1} and
*k*_{2} are also touched from the outside by the circle
*k*_{3} of radius 3*R*. The circles
*k*_{2} and *k*_{3} touch at the point
*E*_{1}, and the circles *k*_{3} and
*k*_{1} at the point *E*_{2}. Prove that the
circumcircle of the triangle
*E*_{1}*E*_{2}*E*_{3} is
congruent to the circle *k*_{1}. (*3 points*)
(Suggested by *B. Bíró,* Eger)

**B. 3597.** Is it true that if there
exists a line parallel to the bases of a given trapezium that halves
both its area and its perimeter then the trapezium is a parallelogram?
(*4 points*)

**B. 3598.** Isosceles triangles with apex
angles of 140^{o} are drawn over the sides *AB*
and *BC* of a given triangle *ABC* (on the outside). The new
vertices obtained are *A*_{1} and
*C*_{1}. Then an isosceles tiangle of apex angle
80^{o} is drawn over the side *AC*
(outside). Determine
*C*_{1}*B*_{1}*A*_{1}?
(*4 points*)

**B. 3599.** A right truncated cone is
circumscribed about a sphere. What is the maximum possible ratio of
the volume of the truncated cone to its surface area?
(*4 points*)

**B. 3600.** Find a cube in the 3D
cartesian system whose edges are not parallel to the coordinate axes
but have an integer length. (*5 points*)

**B. 3601.** Ann and Sophie take turns
rolling a die. The number shown by the die is always added to their
respective scores. Whoever obtains the first score divisible
by 4, wins the game. Given that Anna starts the game, what is the
probability that she will win? (*5 points*)

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

**A. 305.** Prove that if *n* is an
arbitrary positive integer then

\(\displaystyle \sum_{\textstyle{k_1,\dots,k_n\geq0\atop k_1+2k_2+\dots+nk_n=n}} \frac{(k_1+k_2+\dots+k_n)!}{k_1!\cdot\ldots\cdot k_n!}=2^{n-1}.\)

**A. 306.** The orthocentre of the triangle
*ABC* is *M*, and its incircle, centred at *O*, touches
the sides *AC* and *BC* at the points *P* and
*Q*. Prove that if *M* lies on the line *PQ* then the
line *MO* passes through the midpoint of the side *AB*.

**A. 307.** Let *a*_{n}
denote the coefficient of *x*^{n} in the
polynomial (*x*^{2}+*x*+1)^{n}. Prove
that if *p*>3 is a prime number then

*a*_{p}1 (mod
*p*^{2}).

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary