**A. 491.** In the triangle \(\displaystyle A_1A_2A_3\), for each \(\displaystyle i=1,2,3\), the excircle, which is tangent to the side \(\displaystyle A_{i+1}A_{i+2}\), touches the half lines \(\displaystyle A_iA_{i+1}\) and \(\displaystyle A_iA_{i+2}\) at \(\displaystyle P_i\) and \(\displaystyle Q_i\), respectively. (The indices are considered modulo 3, e.g. \(\displaystyle A_4=A_1\) and \(\displaystyle A_5=A_2\).) The lines \(\displaystyle P_iP_{i+1}\) and \(\displaystyle Q_iQ_{i+2}\) meet at \(\displaystyle R_i\); finally, the lines \(\displaystyle P_{i+1}P_{i+2}\) and \(\displaystyle Q_{i+1}Q_{i+2}\) meet at \(\displaystyle S_i\) (\(\displaystyle i=1,2,3\)). Prove that the lines \(\displaystyle R_1S_1\), \(\displaystyle R_2S_2\) and \(\displaystyle R_3S_3\) are concurrent.

(From the idea of *Bálint Bíró,* Eger)

(5 points)

**A. 492.** For every finite, nonempty set \(\displaystyle H\) of positive integers, denote by \(\displaystyle \mathrm{gcd}(H)\) the greatest common divisor of the elements in \(\displaystyle H\). Show that if \(\displaystyle A\) is a finite, nonempty set of positive integers then the following inequality holds:

\(\displaystyle
\sum_{\emptyset\ne H\subseteq A} {(-2)}^{|H|-1} \mathrm{gcd}(H) > 0.
\)

(5 points)

**A. 493.** Prove that

\(\displaystyle
\sum_{i=1}^n\ \sum_{j=1}^n \frac{a_ia_j}{{(p_i+p_j)}^c}\ge 0
\)

holds for arbitrary reals \(\displaystyle a_1,a_2,\ldots,a_n\), and positive numbers \(\displaystyle c,p_1,p_2,\ldots,p_n\).

(5 points)

**B. 4212.** In a cinema, 80% of the grown-up spectators are men, 40% of the male spectators are children, and 20% of the children are boys. What is the minimum possible number of spectators?

Suggested by *J. Pataki,* Budapest

(3 points)

**B. 4219.** Let \(\displaystyle f\), \(\displaystyle g\), \(\displaystyle h\) be different lines in the space, such that if any line \(\displaystyle e\) in the space that has common points with both \(\displaystyle f\) and \(\displaystyle g\) than also has a common point with \(\displaystyle h\). What can be stated about the mutual position of the lines \(\displaystyle f\), \(\displaystyle g\), \(\displaystyle h\)?

Suggested by *G. Mészáros,* Kemence

(4 points)

**B. 4220.** Solve the following simultaneous equations:

\(\displaystyle 1-\frac{12}{3x+y} =\frac{2}{\sqrt{x}},\)

\(\displaystyle 1+\frac{12}{3x+y} =\frac{6}{\sqrt{y}}.\)

*Lithuanian competition problem*

(4 points)

**K. 223.** A bank pays 6% annual interest on fixed deposits. The actual amount paid by the bank is calculated by finding the interest per day (counting 365 days per year), and multiplying by the number of days for which the deposit is fixed. When the locking period is over, the fixed amount is returned to the bank account and the interest is added. In addition, the bank sends letters to the customer informing them about the beginning and the end of the locking period. The letters cost 75 forints each. What is the minimum deposit it is worth fixing for 30 days in this bank? (The bank pays 0% interest on accounts not fixed.)

(6 points)

This problem is for grade 9 students only.

**K. 225.** Little Pete is playing with six identical square sheets of plastic on the table. He lays a square on the table, then he lays the next one, always making sure that the newly laid square joins at least one of the previous ones along the full length of an edge. (Vertices always meet vertices.) Show that the perimeters of all those figures obtained that may form the net of a cube unfolded in the plane, is the same.

(6 points)

This problem is for grade 9 students only.

**K. 226.** Kate wrote each of the numbers 1, 11, 121, 1331, 14 641 and 161 051 on 10 sheets of paper, and put them in a box. She picked some of the sheets at random, added the numbers on them and got 1 111 111. How many sheets did she pick?

(6 points)

This problem is for grade 9 students only.