## English Issue, December 2002 | ||||

Previous page | Contents | Next page | ORDER FORM |

## New problems - competition B

(3582-3591.)

**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 \(\displaystyle \sqrt{1+x^4\)1-2ax+x^2"> 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

\(\displaystyle \arctan u+\arctan v+\arctan w. \)

(*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*)