**A. 608.** Suppose that the convex polygonal disks \(\displaystyle P_1\), \(\displaystyle P_2\) and \(\displaystyle P_3\), lying in the plane, satisfy the property that for arbitrary points \(\displaystyle A\in P_1\), \(\displaystyle B\in P_2\) and \(\displaystyle C\in
P_3\), the area of the triangle \(\displaystyle ABC\) is no more than a unit. Prove that there are two different ones among the disks \(\displaystyle P_1\), \(\displaystyle P_2\) and \(\displaystyle P_3\) whose area is at most \(\displaystyle 8\) units in total.

Proposed by: *Tamás Fleiner,* Budapest

(5 points)

**A. 609.** Let \(\displaystyle a_1,a_2,\dots,a_n\) and \(\displaystyle b_1,b_2,\dots,b_n\) be complex numbers satisfying \(\displaystyle \mathop{\rm Im} a_j\ge 1\) and \(\displaystyle \mathop{\rm Im} b_j\le -1\) (\(\displaystyle j=1,2,\dots,n\)), and let \(\displaystyle f(z) = \frac{(z-a_1)(z-a_2)\dots (z-a_n)}{(z-b_1)(z-b_2)\dots (z-b_n)}\). Prove that the function \(\displaystyle f'(z)\) has no root in the set \(\displaystyle |\mathop{\rm Im}z|<1\).

(5 points)

**A. 610.** There is given a prime number \(\displaystyle p\) and two positive integers, \(\displaystyle k\) and \(\displaystyle n\). Determine the smallest nonnegative integer \(\displaystyle d\) for which there exists a polynomial \(\displaystyle f(x_1,\dots,x_n)\) on \(\displaystyle n\) variables, with degree \(\displaystyle d\) and having integer coefficients that satisfies the following property: for arbitrary \(\displaystyle a_1,\dots,a_n\in\{0,1\}\), \(\displaystyle p\) divides \(\displaystyle f(a_1,\dots,a_n)\) if and only if \(\displaystyle p^k\) divides \(\displaystyle a_1+\dots+a_n\).

(5 points)

**B. 4607.** The sides of a triangle are *a*, *b*, *c*. A line passes through the centre of the inscribed circle, and intersects side *c* at *P*, and side *b* at *Q*. Let *AP*=*p *and *AQ*=*q*. Prove that .

*F. Kacsó,* (Matlap, Kolozsvár)

(5 points)

**C. 1210.** We have four sacks of flour. By three measurements, the following information is obtained: the first and second sacks together are lighter than the other two sacks, the first and third together have the same mass as the other two, and the first and fourth together are heavier than the other two. Which sack is the heaviest?

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1216.** The ideal carnival doughnut weighs 54 grams when done, and 8% is the oil absorbed during frying. It is shaped like a (centrally symmetrical) spherical segment of diameter 78 mm and height 3 cm. The volume of the dough is doubled in frying, and its mass is only increased by the oil absorbed. Before frying, the whole dough weighs 1 kg, and it is kept in a 2.5-litre bowl. Is it possible to cover the bowl with a flat lid?

(5 points)

This problem is for grade 11 - 12 students only.

**K. 411.** The power *x*^{5} cannot only be calculated by doing the multiplication *x*^{.}*x*^{.}*x*^{.}*x*^{.}*x*, but also with fewer multiplications, by calculating partial results (e.g. *y*=*x*^{.}*x*) and obtaining the final answer by multiplying those: *y*^{.}*y*^{.}*x*, which only makes three multiplications altogether. Express the power *x*^{23} with less than 10 multiplications to carry out.

(6 points)

This problem is for grade 9 students only.