**B. 4844.** Find all ordered triples \(\displaystyle (p,q,n)\) (where \(\displaystyle n\ge 2\) is a positive integer) such that there exist exactly \(\displaystyle n\) real numbers, not necessarily different, whose product is \(\displaystyle p\) and whose sum is \(\displaystyle q\).

(Proposed by *I. Porupsánszki,* Miskolc)

(5 points)

**B. 4845.** Prove that each term of the sequence

\(\displaystyle a_1 =1,\)

\(\displaystyle a_n =\frac{4a_{n-1}+\sqrt{7a^2_{n-1}-3}}{3};\quad n\ge2\)

is rational.

(Proposed by *J. Szoldatics,* Budapest)

(5 points)

**C. 1392.** The number \(\displaystyle 2017-(2+0+1+7)\) is divisible by the number \(\displaystyle (2+20+201)\), that is, 2017 has the following property: if the sum of the digits of the number is subtracted from it, the result will be a four-digit number that is divisible by the sum of the one-digit number formed by the first digit, the two-digit number formed by the first two digits, and the three-digit number formed by the first three digits. How many four-digit numbers have this property?

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1397.** Prove that the function

\(\displaystyle
f\colon \mathbb{R}\to \mathbb{R},\quad x\mapsto
\begin{cases}
2x+2-6k, & \text{if \ } x\in \left[6k; 6k+2\right[ ,\\
\frac{x-2+6k}{2}, & \text{if \ } x\in \left[6k+2; 6k+6\right[
\end{cases}
\quad (k\in \mathbb{Z})
\)

(where \(\displaystyle k\) takes every integer value) is one-to-one and equal to its own inverse.

(5 points)

This problem is for grade 11 - 12 students only.

**C. 1398.** A cube is inscribed in a right circular cone: one face lies on the base of the cone, and the remaining four vertices are on the lateral surface of the cone. What is the surface area of the cube if the base radius of the cone is \(\displaystyle \frac{\sqrt{3}}{2}\,\), and its slant height is three times as large?

(5 points)

This problem is for grade 11 - 12 students only.

**K. 529.** Four circles are arranged as shown in the *figure.* The numbers \(\displaystyle 1, 2, 3,
\ldots, 10\) are to be written in the ten regions formed by the circles such that the sum of the numbers within each circle is the same. What is the largest possible value of that sum?

(6 points)

This problem is for grade 9 students only.

**K. 531.** Let us consider a \(\displaystyle 4\times 4\) chessboard. In each step, the colour of every square in a chosen \(\displaystyle 2\times 2\) block is changed (black to white, and white to black).

\(\displaystyle a)\) By performing some of these steps, is it possible to obtain a completely black board?

\(\displaystyle b)\) Is this possible in the case of a \(\displaystyle 5\times 5\) board, provided that the corners are initially white?

(6 points)

This problem is for grade 9 students only.

**K. 532.** Greg charges the battery of his cell phone every night, so he can start every new day with a 100% charged battery. If he only uses the phone for talking, the battery lasts 30% longer than when he plays games with it, and it lasts 60% longer than when he is surfing the net. Today, he has talked for 20 minutes on the phone, played for 50 minutes, and spent 80 minutes on the net. The battery is now flat. How long may Greg use the internet with his phone if it is fully charged, and he does not use it for anything else?

(6 points)

This problem is for grade 9 students only.

**K. 534.** Kate has a round table, it is a circle of area 4 m\(\displaystyle {}^{2}\). She also has a circular tablecloth of exactly the same size to go with it. Joe has a square table, two metres on a side, and a square tablecloth of the very same size. One day, the two of them switched the tablecloths. Each of them laid the other's tablecloth on their own table, with the centre of the cloth carefully positioned on the centre of the table. Each observed that the tablecloth overhung the edge of the table at some places, while the table remained partly uncovered. In which case was the area of the uncovered part greater?

(6 points)

This problem is for grade 9 students only.