# New exercises and problems in Mathematics

November
2002

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

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

**C. 690.** Is it possible for the sum of
the volumes of three cubes of integer edges to be 2002 units?

**C. 691.** Express in mm^{2} the
area of Hungary on a globe of radius 25 cm.

**C. 692.** For the real numbers *x*,
*y*, *z*, (1) *x*+2*y*+4*z*3 and
(2) *y*-3*x*+2*z*\(\displaystyle \ge\)5. Prove that
*y*-*x*+2*z*\(\displaystyle \ge\)3.

**C. 693.** In what interval may the apex
angle of an isosceles triangle vary if a triangle can be constructed
out of its altitudes?

**C. 694.** Evaluate the sum
[log_{2}1]+[log_{2}2]+[log_{2}3]+...+[log_{2}2002].

Suggested by Á. Besenyei, Budapest

## 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. 3582.** Solve the equation
3*xyz*-5*yz*+3*x*+3*z*=5 on the set of natural
numbers. (3 points)

**B. 3583.** The incentre of a triangle
*ABC* is connected to the vertices. One of the resulting three
triangles is similar to the original triangle *ABC*. Find the
angles of the triangle *ABC*. (3 points)

**B. 3584.** Write down all the integers
from 1 to 10^{n-1}, and let *A* denote the number
of digits hence written down. Now write down all the integers from 1
to 10^{n}, and let *B* denote the number of zeros
written down this time. Prove that
*A*=*B*. (4 points)

**B. 3585.** Find the possible values of
the parameter *a*, such that the inequality holds for all
positive *x*. (3 points)

11th International Hungarian Mathematics Competition

**B. 3586.** For what values of the
parameter *a* does the equation log (*ax*) =2log
(*x*+1) have exactly one root? (4 points)

**B. 3587.** A truncated right pyramid
with a square base is circumscribed about a sphere. Find the range for
the ratio of the volume and the surface area of the
solid. (4 points)

**B. 3588.** *M* is a point in the
interior of a given circle. The vertex of a right angle is *M*
and its arms intersect the circle at the points *A* and
*B*. What is the locus of the midpoint of the line segment
*AB* as the right angle is rotated about the point *M*?
(4 points)

**B. 3589.** Prove that there are
infinitely many odd positive integers *n*, such that
2^{n}+*n* is a composite
number. (4 points)

**B. 3590.** The roots of the equation
*x*^{3}-10*x*+11=0 are *u*, *v* and
*w*. Determine the value of arctan *u*
+arctan *v*+arctan *w*. (5 points)

**B. 3591.** The area of the convex
quadrilateral *ABCD* is *T*, and *P* is a point in its
interior. The parallel through the point *P* to the line segment
*BC* intersects the side *BA* at *E*, the parallel to
the line segment *AB* intersects the side *BC* at*F*,
the parallel to *AD* intersects *CD* at *G*, and the
parallel to *CD* intersects *AD* at the point *H*. Let
*t*_{1} denote the area of the quadrilateral *AEPH*,
and let *t*_{2} denote the area of the quadrilateral
*PFCG*. Prove that \(\displaystyle \sqrt{t_1}+\sqrt{t_2}\leq\sqrt{T}\). (5 points)

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

**A. 302.** Given the unit square
*ABCD* and the point *P* on the plane, prove that

\(\displaystyle 3AP+5CP+\sqrt5(BP+DP)\ge6\sqrt2.\)

**A. 303.** *x*, *y* are
non-negative numbers, and
*x*^{3}+*y*^{4}\(\displaystyle \le\)*x*^{2}+*y*^{3}. Prove that
*x*^{3}+*y*^{3}\(\displaystyle \le\)2.

**A. 304.** Find all functions
*R*^{+}\(\displaystyle \mapsto\)*R*^{+}, such that

*f*(*x*+*y*)+*f*(*x*)^{.}*f*(*y*)=*f*(*xy*)+*f*(*x*)+*f*(*y*)?

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary