Found problems: 4
2025 Macedonian Balkan MO TST, 3
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]
for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.
2025 Macedonian Balkan MO TST, 1
A set of $n \ge 2$ light bulbs are arranged around a circle, and are consecutively numbered with
$1, 2, . . . , n$. Each bulb can be in one of two states: either it is [b]on[/b] or [b]off[/b]. In the initial configuration,
at least one bulb is turned on. On each one of $n$ days we change the current on/off configuration as
follows: for $1 \le k \le n$, on the $k$-th day we start from the $k$-th bulb and moving in clockwise direction
along the circle, we change the state of every traversed bulb until we switch on a bulb which was
previously off.
Prove that the final configuration, reached on the $n$-th day, coincides with the initial one.
2025 Macedonian Balkan MO TST, 2
Let $\triangle ABC$ be an acute-angled triangle and $A_1, B_1$, and $C_1$ be the feet of the altitudes from $A, B$, and $C$, respectively. On the rays $AA_1, BB_1$, and $CC_1$, we have points $A_2, B_2$, and $C_2$ respectively, lying outside of $\triangle ABC$, such that
\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]
If the intersections of $B_1C_2$ and $B_2C_1$, $C_1A_2$ and $C_2A_1$, and $A_1B_2$ and $A_2B_1$ are $A', B'$, and $C'$ respectively, prove that $AA', BB'$, and $CC'$ have a common point.
2025 Macedonian Balkan MO TST, 4
Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.