**A. 679.** Let \(\displaystyle n = 2^{128}\), \(\displaystyle M = \{1, 2, 3, 4\}\), and let \(\displaystyle M^n\) denote the set of sequences of length \(\displaystyle n\) and consisting of elements of \(\displaystyle M\). Decide whether there exist functions \(\displaystyle f_1,\ldots,f_n\) and \(\displaystyle g_1,\ldots,g_n\colon M^n\to M\) such that for every pair of sequences

\(\displaystyle
(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in M^n,
\)

at least one of the following statements holds:

\(\displaystyle \bullet\) \(\displaystyle f_i(y_1,\ldots,y_n) = x_i\) for some index \(\displaystyle i\) with \(\displaystyle 1\le i\le n\);

\(\displaystyle \bullet\) \(\displaystyle g_j(x_1,\ldots,x_n) = y_j\) for some index \(\displaystyle j\) with \(\displaystyle 1\le j\le n\).

Proposed by: *Gerhard Woeginger,* Eindhoven

(5 points)

**B. 4815.** The binary operation \(\displaystyle \circ\) is defined on the set of integers. Given that

\(\displaystyle
a\circ(b+c)=b\circ a+ c\circ a.
\)

for all integers \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\), show that there exists an integer \(\displaystyle k\) such that \(\displaystyle a\circ b= k\cdot a \cdot b\) for all \(\displaystyle a\) and \(\displaystyle b\).

(*Italian problem*)

(5 points)

**B. 4817.** Solve the following equation in the set of real numbers:

\(\displaystyle x + y + z = xyz = 8,\)

\(\displaystyle \frac 1x- \frac 1y- \frac 1z = \frac 18.\)

Proposed by *B. Kovács,* Szatmárnémeti

(4 points)

**C. 1373.** Let \(\displaystyle a\) and \(\displaystyle b\) denote positive integers. Given that three line segments of lengths \(\displaystyle a\), \(\displaystyle 1/a\), \(\displaystyle b\) can form a triangle, and three line segments of lengths \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle 1/b\) can also form a triangle, prove that both triangles are isosceles.

(5 points)

**K. 511.** A merchant regularly buys apricot jam from a canning factory. One day, he bought a certain quantity, and paid a total of \(\displaystyle 180\,000\) forints (HUF, Hungarian currency) for it. In the next month, he saw that the price of apricot jam was significantly reduced: they were selling 5 jars of jam for the price of 3 jars. The merchant was so happy that he immediately bought four times the quantity he had bought in the previous month. How much did he pay altogether for the jam with reduced price?

(6 points)

This problem is for grade 9 students only.

**K. 512.** Patrick, Peter and Paul are triplets. One day, they were walking with their mom, and they saw a chewing gum vending machine. It contained a mixture of gums of three different colours (red, yellow and green). 100 forints (HUF, Hungarian currency) need to be inserted in the machine to get a chewing gum. The triplets insisted on getting gums of the same colour. Their mom kept throwing coins in the machine until she was able to fulfil the children's wish. They walked the same route on four consecutive days, and bought chewing gum in the same way every day. What is the maximum amount of money the mom may have spent on chewing gum, if she did all she could to economize?

(6 points)

This problem is for grade 9 students only.

**K. 513.** The *diagram* shows a chessboard with chessmen on it. There is a bishop on the field marked X, and there are pawns on the other fields marked with letters. One by one, eliminate the pawns with the bishop, so that it captures a pawn in every move. Find a possible order by specifying the corresponding letters of the pawns.

(The task can be found on `www.sakkpalanta.hu`, along with several other puzzles of this kind.)

(6 points)

This problem is for grade 9 students only.

**K. 516.** Consider the sets \(\displaystyle A = \{a, 2a+1, a^{2}+1\}\), \(\displaystyle B =\{b+3, 10, b-1\}\). Find appropriate positive integers \(\displaystyle a\) and \(\displaystyle b\) such that the two sets have

\(\displaystyle a)\) no element in common;

\(\displaystyle b)\) exactly 1 element in common;

\(\displaystyle c)\) exactly 2 elements in common;

\(\displaystyle d)\) the same elements.

(6 points)

This problem is for grade 9 students only.