6. Find all nonnegative integer solutions of the equation

$$
\left(2^{2015}+1\right)^{x}+2^{2015}=2^{y}+1
$$

(Bojan Bašić)

Time allowed: 270 minutes.

Each problem is worth 7 points.
6. The only solutions with $x \leqslant 1$ are $(0,2015)$ and $(1,2016)$.

Assume that $x>1$. Since $2^{2015}+1$ is divisible by 3 , we have $\left(2^{2015}+1\right)^{x}+2^{2015} \equiv$ $2^{2015} \equiv 5(\bmod 9)$, so $2^{y} \equiv 4(\bmod 9)$, which gives us $y=6 k+2$ for some $k \in \mathbb{N}$. Now modulo 13 we have $2^{y}+1=\left(2^{6}\right)^{k} \cdot 2^{2} \equiv \pm 4$ and $2^{2015} \equiv 7(\bmod 13)$, so $8^{x}+7 \equiv \pm 4(\bmod 13)$. This is impossible, as $8^{x}$ always gives one od the remainders $1,5,8,12$ modulo 13 .

Second solution. For $x>1$ we have $2^{y}=\left(2^{2015}+1\right)^{x}+2^{2015}-1=(x+1) 2^{2015}+$ $\sum_{i=2}^{x}\binom{x}{i} 2^{2015 i} \equiv(x+1) 2^{2015}\left(\bmod 2^{2019}\right)$, but $y>2019$, so we infer $16 \mid x+1$. Now reducing the equation modulo 17 and using $2^{2015} \equiv 9(\bmod 17)$ yields $2^{y} \equiv$ $10^{x}+8 \equiv 10^{15}+8 \equiv 3(\bmod 17)$, which is impossible.

The 32-nd Balkan Mathematical Olympiad was held from May 3 to May 8 in Athens in Greece. The results of the Serbian contestants are shown below:

|  | 1 | 2 | 3 | 4 | Total |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Marijana Vujadinović | 10 | 10 | 10 | 0 | 30 | Silver medal |
| Ognjen Tošić | 5 | 10 | 0 | 0 | 15 | Bronze medal |
| Ivan Damnjanović | 4 | 0 | 0 | 0 | 4 |  |
| Aleksa Milojević | 0 | 10 | 0 | 0 | 10 | H. mention |
| Aleksa Konstantinov | 0 | 10 | 0 | 0 | 10 | H. mention |
| Andjela Sarković | 0 | 10 | 0 | 0 | 10 | H. mention |

To put it mildly, this is obviously not as good as we hoped. A part of the reason could be the simple inequality in problem 1 - seemingly easy for everyone except for the Serbian team. Problems 3 and 4 looked standard but nevertheless turned out to be harder. At the end, 7 contestants ( 6 official +1 guest) with $31-40$ points were awarded gold medals, $15(13+2)$ with $24-30$ points were awarded silver medals, and 55 (22+33) with 12 - 23 points were awarded bronze medals.

Here is the (unofficial) team ranking:

| Member Countries | Guest Teams |  |  |
| :--- | ---: | :--- | ---: |
| 1. Turkey | 177 | Turkmenistan | 127 |
| 2. Romania | 171 | Kazakhstan | 118 |
| 3. Bulgaria | 156 | Greece B | 97 |
| 4. Greece | 133 | Tajikistan | 97 |
| 5. Moldova | 106 | Italy | 90 |
| 6. Bosnia and Herzegovina | 105 | Saudi Arabia | 79 |
| 7. Macedonia (FYR) | 79 | UK and Ireland | 68 |
| 7. Serbia | 79 | Azerbaijan | 61 |
| 9. Cyprus | 45 | Kyrgyzstan | 14 |
| 10. Albania | 38 | Iran (independent) | 8 |
| 11. Montenegro | 3 |  |  |