# New exercises and problems in Mathematics

March
2002

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

## New exercises in March 2002Maximum score for each exercise (sign "C") is 5 points. |

**C.** **665.** Evaluate the following fraction, given
that the numerator and the denominator contain the same number of
digits:

\(\displaystyle {166\dots6\over66\dots64}\)

**C.** **666.** The value of a quadratic function of
integer coefficients is divisible by 3 at every integer. Prove that
all three coefficients are divisible by 3.

**C.** **667.** Let \(\displaystyle a=x+{1\over x}\), ,
\(\displaystyle c=xy+{1\over
xy}\). Prove that the value of the expression
*a*^{2}+*b*^{2}+*c*^{2}-*abc*
is independent of *x* and *y*.

**C.** **668.** Given the triangle *ABC*, find the
locus of the points *P* in the plane of the triangle, such that
*PA*^{2}=*PB*^{2}+*PC*^{2}.

**C.** **669.** Among all circular sectors with a given
perimeter, which one has the greatest area?

## New problems in March 2002The 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.** **3532.** A bug is crawling about on a sheet of
squared paper. In every step, it can move two units to the right, four
units to the left, three units up, or five units down the page. During
its walk, it makes a turn of exactly 90^{o} after each
step. Which are the fields that the bug can visit in this way?
(*3* *points*)

**B.** **3533.** Ann wrote 32 integers on a large sheet
of paper, and covered each number with a card. Then he told Bob that
if he chose 7 cards, she would tell him whether the sum of the
7 covered numbers is odd or even. At least how many times did Bob
have to choose 7 cards in order to find out if the sum of all
32 numbers on the sheet was odd or even?
(*4* *points*)

**B.** **3534.** Charging at the opponent's goal, a
soccer player has run along a polygonal line not intersecting more
than once any straight line parallel to a side of the soccer
field. Show that the soccer player cannot have run more than the sum
of the two sides of the field. (*3* *points*)

**B.** **3535.** Let *U*, *V*, *W* be
points on the lines containing the sides *BC*, *CA* and
*AB* of a triangle *ABC*, respectively. Prove that the
perpendiculars drawn to the sides at *U*, *V*, and *W*
are concurrent if and only if

*AW*^{2}+*BU*^{2}+*CV*^{2}=*AV*^{2}+*CU*^{2}+*BW*^{2}.

(*3* *points*)

**B.** **3536.** Find the minimum value of . (*4* *points*)

**B.** **3537.** The foot of the altitude from vertex
*B* of a triangle *ABC* on side *AC* is *H*, the
foot of the angle bisector from *B* is *D*, and the
inscribed circle touches *AC* at *E*. The midpoint of the
segment *AC* is *F*. Prove that *EF* is the geometric
mean of *FD* and *FH*. (*4* *points*)

**B.** **3538.** Find a positive real number that will
increase by a factor of 2501 if the first and fifth digits after the
decimal point are interchanged. (*5* *points*)

**B.** **3539.** There are 8 breakable objects in a
box. One of them is worth 50,000 forints (HUF), three are worth
30,000 forints each, and the remaining four are worth 20,000
each. The box is accidentally dropped during transportation. If each
object will break with a probability of 1/2 independently of each
other, what is the probability that the damage is no more than
100,000 forints? (*4* *points*)

**B. 3540.** In a triangle, consider the sum of the squares of
the heights and the sum of the squares of the sides. In what interval
does the ratio of the two numbers vary? (*5* *points*)
(based on the idea of *Balogh János,* Kaposvár)

**B.** **3541.** Let \(\displaystyle f(x)={ax+b\over cx+d}\). Prove that if
*f*(*f*(*f*(1)))=1, and *f*(*f*(*f*(2)))=3, then
*f*(1)=1. (*5* *points*)

## New advanced problems in March 2002Maximum score for each advanced problem (sign "A") is 5 points. |

**A.** **287.** Someone has drawn an ellipse on a sheet
of paper, and he has also marked the endpoints of its axes. The major
axis is twice as long as the minor axis. Find a construction method
for dividing any given acute angle into three equal parts, using a
pair of compasses, a straight edge, and the given ellipse only.

**A.** **288.** Let *p* be an *n* degree
polynomial, where *n\(\displaystyle ge\)*1. Prove that there exist at least *n*+1 complex
numbers yielding 0 or 1 as the value of *f*. (IMC 7, London,
2000)

**A.** **289.** Prove that if the function *f* is
defined on the set of positive real numbers, its values are real, and
*f* satisfies the equation

\(\displaystyle f\left({x+y\over2}\right)+f\left({2xy\over x+y}\right)=f(x)+f(y)\)

for all positive *x*,*y*, then

\(\displaystyle 2f\big(\sqrt{xy}\big)=f(x)+f(y)\)

for every positive number pair *x*, *y*. (Miklós
Schweitzer Memorial Competition, 2001)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary