**B. 4796.** Solve the following equation on the set of real numbers:

\(\displaystyle
x^2-6\{x\}+1=0,
\)

where \(\displaystyle \{x\}\) stands for the fractional part of a number \(\displaystyle x\) (that is, the difference obtained when the largest integer not greater than \(\displaystyle x\) is subtracted from \(\displaystyle x\)).

Proposed by *J. Szoldatics,* Budapest

(4 points)

**B. 4797.** In triangle \(\displaystyle ABC\), \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\) are arbitrary interior points of sides \(\displaystyle AB, BC\) and \(\displaystyle CA\), respectively. Let \(\displaystyle G\), \(\displaystyle H\) and \(\displaystyle I\) denote the centroids of triangles \(\displaystyle ADF\), \(\displaystyle BED\) and \(\displaystyle CFE\), respectively. Furthermore, let \(\displaystyle S\), \(\displaystyle K\), \(\displaystyle L\) be the centroids of triangles \(\displaystyle ABC\), \(\displaystyle DEF\) and \(\displaystyle GHI\), respectively. Prove that the points \(\displaystyle K\), \(\displaystyle L\) and \(\displaystyle S\) are collinear.

Proposed by *Sz. Miklós,* Herceghalom

(3 points)

**B. 4800.** \(\displaystyle T\) is a point on line \(\displaystyle BC\), different from the midpoint of line segment \(\displaystyle BC\). Circle \(\displaystyle k\) is centred at \(\displaystyle T\), and \(\displaystyle A\) is its intersection with the perpendicular drawn to \(\displaystyle BC\) at \(\displaystyle T\). The intersections of \(\displaystyle k\) with the lines \(\displaystyle AB\) and \(\displaystyle AC\) are \(\displaystyle K\) and \(\displaystyle L\), respectively. Let \(\displaystyle k\) intersect the circumscribed circle of \(\displaystyle ABC\) again at \(\displaystyle M\). Prove that the lines \(\displaystyle KL\), \(\displaystyle AM\) and \(\displaystyle BC\) are concurrent.

Proposed by *K. Williams,* Szeged

(5 points)

**B. 4801.** Define the sequence \(\displaystyle f_n\) of functions by the following recurrence relation:

\(\displaystyle
f_0(x) = f_1(x) = 1,
\mathrm{~and ~for ~} n\ge 2 \quad f_n(x) = f_{n-1}(x) \cdot 2\cos(2x) -
f_{n-2}(x).
\)

Determine the number of roots of \(\displaystyle f_n(x)\) in the interval \(\displaystyle [0,\pi]\).

Proposed by *L. Bodnár,* Budapest

(5 points)

**B. 4802.** The inscribed sphere of a right circular cone \(\displaystyle \mathcal{K}\) is \(\displaystyle \mathcal{G}\). The centres of the spheres \(\displaystyle g_1, g_2, \dots, g_n\) of radius \(\displaystyle r\) form a regular \(\displaystyle n\)-gon of side \(\displaystyle 2r\). Furthermore, each sphere \(\displaystyle g_i\) is tangent to both the lateral surface and the base of \(\displaystyle \mathcal{K}\), and also tangent to \(\displaystyle \mathcal{G}\). What may be the value of \(\displaystyle n\)?

*Competition problem from the Soviet Union*

(6 points)

**B. 4803.** Is it possible to specify closed intervals of rational endpoints on the number line such that every rational number is the endpoint of exactly one interval, and

\(\displaystyle a)\) one of any pair of closed intervals contains the other;

\(\displaystyle b)\) no pair of two intervals are disjoint, but no interval contains another?

*Based on an idea of M. E. Gáspár*

(6 points)

**C. 1358.** The pentagon \(\displaystyle ABCDE\) has an inscribed circle of centre \(\displaystyle O\) and radius \(\displaystyle r\). Given that the angle at vertex \(\displaystyle A\) is a right angle, \(\displaystyle \angle EOA=60^\circ\), and that triangle \(\displaystyle OCD\) is equilateral, find the area of the pentagon.

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1362.** The volume of a cuboid is 10.9545 cm\(\displaystyle {}^3\), the arithmetic mean of the lengths of the edges is 2.2655 cm, and their harmonic mean is 2.1769 cm. Determine the length of the diagonal of the cuboid, to the nearest thousandth of a centimetre.

(5 points)

This problem is for grade 11 - 12 students only.

**C. 1363.** The parallel sides of a right-angled trapezium are \(\displaystyle 405\) and \(\displaystyle 80\) units long. The length of the right-angled leg is \(\displaystyle 65 \big(\sqrt{3}+\sqrt{2}\,\big)\) units. The trapezium is sliced into eight trapezoids, all similar to one another, with cuts parallel to the bases. What is the height of the largest slice?

Proposed by *Z. G. Szepesi,* Budapest

(5 points)

This problem is for grade 11 - 12 students only.