**B. 4708.** \(\displaystyle O\) is the centre of the circumscribed circle of triangle \(\displaystyle ABC\), and \(\displaystyle M\) is the orthocentre. Point \(\displaystyle A\) is reflected in the perpendicular bisector of side \(\displaystyle BC\), \(\displaystyle B\) is reflected in the perpendicular bisector of side \(\displaystyle CA\), and finally \(\displaystyle C\) is reflected in the perpendicular bisector of side \(\displaystyle AB\). The reflections are denoted by \(\displaystyle A_1\), \(\displaystyle B_1\), \(\displaystyle C_1\), respectively. Let \(\displaystyle K\) be the centre of the inscribed circle of triangle \(\displaystyle A_1B_1C_1\). Prove that point \(\displaystyle O\) bisects line segment \(\displaystyle MK\).

Suggested by *B. Bíró,* Eger

(5 points)

**B. 4709.** Solve the simultaneous equations

\(\displaystyle x^{2}+y^{2}=13,\)

\(\displaystyle x^{3}+y^{3}=35.\)

Suggested by *J. Szoldatics,* Budapest

(3 points)

**B. 4712.** What percentage of a pencil gets wasted? Assume that a pencil is a cylinder, infinitely long, and the graphite rod inside is also cylindrical. The axes of the two cylinders coincide. When the pencil is sharpened, its point is a perfect cone with an apex angle of 12 degrees. When we write with the pencil, its axis always encloses a 42-degree angle with the plane of the paper. We keep using the pencil until we can no longer write with it since no matter how we rotate it about its axis, the wood will scratch the paper. Then the pencil is sharpened again to the shape of a 12-degree cone, but never longer, that is, the tip of the pencil never changes during sharpening, it only wears in writing. What percentage of the graphite is wasted by scraping it off with the sharpener? Will someone holding the pencil at a 45-degree angle waste more than that or less? If so, by how much?

Suggested by *E. M. Gáspár,* Budapest

(5 points)

**C. 1293.** Alpha & Co. manufacture tennis balls and sell them in packets of four, arranged in a pyramid in a regular tetrahedral box (*figure 1*). Another manufacturer, APHLA, also sells tennis balls in four-packets, arranged in a column in a tall cylindrical box closed at both ends (*figure 2*). What is the difference between the surface areas of the two boxes if the diameter of a tennis ball is 6.50 cm?

(5 points)

This problem is for grade 11 - 12 students only.