# New exercises and problems in Mathematics

January
2003

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

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

**C. 700.** A quadrilateral is cut out of a
sheet of paper, and each vertex is folded in so that they meet at a
common point. What kind of quadrilateral should that be in order for
the folded parts to cover the rest of the quadrilateral without
overlap?

**C. 701.** Show that
1^{.}2^{.}...^{.}1001+1002^{.}1003^{.}...^{.}2002
is divisible by 2003.

**C. 702.** The acute angles of a right
triangle are 60^{o} and 30^{o}. Two
circles of the same radius are inscribed in the triangle so that they
touch each other, the hypotenuse and one leg each. By what factor is
the smaller leg longer than the radius of the circles?

**C. 703.** Depending on the value of the
real parameter *p*, how many roots does the equation
2*x*^{2}-10*px*+7*p*-1=0 have in the interval
(-1;1)?

**C. 704.** For what natural numbers
*n* is it true that
log_{2}3^{.}log_{3}4^{.}log_{4}5^{.}...^{.}log_{n}(*n*+1)=10?

## 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. 3602.** Jerry is swimming in a square
pool. Tom is watching from the edge of the pool and Jerry wants to get
away from him. Tom cannot swim at all, cannot run as fast as Jerry but
he runs four times as fast as Jerry swims. Can Jerry always escape?
(*5 points*)

**B. 3603.** In the interior of a given
triangle, construct a point such that its ratio of the distances from
the lines of the sides are in a 1:2:3
proportion. (*3 points*)

**B. 3604.** *x*, *y* are real
numbers, and *x*+*y*=1. Determine the maximum value of the
expression

*A*(*x*,*y*)=*x*^{4}*y*+*xy*^{4}+*x*^{3}*y*+*xy*^{3}+*x*^{2}*y*+*xy*^{2}.

(*3 points*)

**B. 3605.** Point *D* lies on the
extension of the side *CA* of a triangle *ABC* beyond the
point *A*, point *E* lies on the extension of *CB*
beyond *B*, and *AB*=*AD*=*BE*. The angle
bisectors from *A* and *B* intersect the opposite sides at
the points *A*_{1} and *B*_{1},
respectively. Find the area of the triangle *ABC*, given that the
area of triangle *DCE* is 9 units and that of triangle
*A*_{1}*CB*_{1} is
4 units. (*3 points*) (Suggested by
*G. Bakonyi,* Budapest)

**B. 3606.** Find suitable integers
*a* and *b*, such that \(\displaystyle 20034 points)

**B. 3607.** The lines containing the
opposite sides of a convex quadrilateral intersect each other. Draw
the interior angle bisectors of the angles formed at the
intersections. Prove that the quadrilateral is cyclic if and only if
the two angle bisectors are perpendicular to each other, and show that
in that case they intersect the sides of the quadrilateral at the
vertices of the rhombus. (*4 points*) (Suggested by
*J. Rácz,* Budapest)

**B. 3608.** The roots of the equation
*x*^{3}+*ax*^{2}+*bx*+*c*=0 are
equal to the fifth powers of the three roots of the equation
*x*^{3}-3*x*+1=0, respectively. Find the numbers
*a*, *b*, *c* in decimal
notation. (*4 points*)

**B. 3609.** Is there a 2003rd-degree
polynomial *f*(*x*) of integer coefficients, such that the
values of *f*(*n*), *f*(*f*(*n*)), are pairwise relative primes
for every integer *n*? (*4 points*)

**B. 3610.** Prove that

sin 25^{o}^{.}sin 35^{o}^{.}sin 60^{o}^{.}sin 85^{o}=sin 20^{o}^{.}sin 40^{o}^{.}sin 75^{o}^{.}sin 80^{o}.

(*5 points*)

**B. 3611.** The infinite series is formed out of the elements
of the sequence defined by the recursion
*x*_{n+1}=*x*_{n}^{2}-*x*_{n}+1. What
is the sum of the series if *a*) *x*_{1}=1/2;
*b*) *x*_{1}=2? (*5 points*)

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

**A. 308.** *A*, *B*, *C*,
*D*, *E* are points in the plane, such that
*AB*=*BC*=*CD*=*DA*=1, and each of *AE*,
*BE*, *CE* and *DE* is at most 1. What is the
maximum possible value of
*AE*+*BE*+*CE*+*DE*+*AC*+*BD*?

**A. 309.** In a simple graph on *n*
points, the orders of the points are 0<*d*_{1}...*d*_{n}. Prove that it is possible to
select at least \(\displaystyle \sum\frac{2}{d_i+1}\) points, such that the subgraph formed by
these points contains no loop.

**A. 310.** Let , for every positive integer *n*, and define
the polynomials *p*_{0},*p*_{1},... by the
following recursion: *p*_{0}(*x*)=1, . Prove that the coefficients of the
polynomial *p*_{n} are all integers.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary