A. 648. In the acute angled triangle \(\displaystyle ABC\), the midpoints of the sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) are \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\), respectively. The foot of the altitude of the triangle starting from \(\displaystyle C\) is \(\displaystyle T_1\). On some line, passing through point \(\displaystyle C\) but not containing \(\displaystyle T_1\), the feet of the perpendiculars starting from \(\displaystyle A\) and \(\displaystyle B\) are \(\displaystyle T_2\) and \(\displaystyle T_3\), respectively. Prove that the circle \(\displaystyle DEF\) passes through the center of the circle \(\displaystyle T_1T_2T_3\).
Proposed by: Bálint Bíró, Eger
(5 points)
B. 4730. The circles \(\displaystyle k_1\) and \(\displaystyle k_2\) touch at point \(\displaystyle E\). Points \(\displaystyle X_i\) and \(\displaystyle Y_i\) are marked on each circle \(\displaystyle k_i\) (\(\displaystyle i = 1,2\)) such that the two lines \(\displaystyle X_iY_i\) intersect each other on the common interior tangent of the circles. Prove that the line connecting the centres of circles \(\displaystyle X_1X_2E\) and \(\displaystyle Y_1Y_2E\), and the other line connecting the centres of circles \(\displaystyle X_1Y_2E\) and \(\displaystyle X_2Y_1E\) also intersect each other on the common interior tangent of the circles.
Proposed by K. Williams, Szeged
(5 points)
B. 4731. Let \(\displaystyle 0\le a,b,c \le 2\), and \(\displaystyle a+b+c=3\). Determine the largest and smallest values of
\(\displaystyle
\sqrt{a(b+1)} + \sqrt{b(c+1)} + \sqrt{c(a+1)}.
\)
Proposed by K. Williams, Szeged
(6 points)
C. 1307. For what real number q will the numbers \(\displaystyle \big(3\sqrt{5}\,\big)\), \(\displaystyle \Big(\frac{3\sqrt{5}}{5}q\Big)\), \(\displaystyle \Big(0,6\frac{1}{\sqrt{5}}\Big)\) form a geometric progression?
(5 points)
This problem is for grade 11  12 students only.
K. 463. A certain amount of money was divided among four people. The first one got 3000 forints (HUF, Hungarian currency) more than one third of the total, the second one got 6000 forints more than one fourth of the total, the third one got 9000 forints more than one fifth of the total, and the fourth one got 12 000 forints more than one sixth of the total. What was the total?
(6 points)
This problem is for grade 9 students only.
K. 465. A treasure trunk has an electronic lock mechanism controlled by eight switches. Every switch has two settings: on or off. The lock opens if each switch is on. It is possible to change the setting of any switch to the opposite. However, the electronic sensors will detect which switch has been manipulated, and as a result, three other switches will be automatically changed, too. (These automatic changes will not generate further switches changing.) The table below shows which switch induces which further switches to change. (For simplicity, the switches are numbered.)
Number of switch manipulated 
1 
2 
3 
4 
5 
6 
7 
8 
Numbers of further switches changing automatically 
2, 5, 7 
1, 3, 8 
5, 6, 7 
1, 6, 8 
2, 3, 6 
2, 5, 8 
1, 3, 4 
1, 4, 7 

\(\displaystyle a)\) Initially, every switch is off, except for 6 and 7. The trunk can now be opened by manually changing the setting of two appropriate switches. Which two?
\(\displaystyle b)\) Initially, every switch is off, except for 7. Is it possible to open the trunk now by manipulating the appropriate switches?
(6 points)
This problem is for grade 9 students only.