SMO 2026 Senior Round 2

In square $ABCD$, $E,F$ are points on the sides $AB$ and $BC$ respectively. Let $N$ be the foot of the perpendicular from $B$ onto $CE$. The circular arc from $C$ to $A$ centred at $B$ with radius $BC$ intersects the segment $DN$ at $K$. If $\angle FND=90^\circ$ and $KF$ is parallel to $NB$, find the value of $\dfrac{BE+BF}{AB}$.

Let $P$ be a configuration of $n\ge3$ points in the plane, no three collinear and let $K$ be a maximal set of pairwise nonparallel lines, each joining two of these points. Let $M(P)$ be the largest possible size of $K$. What is the minimum value of $M(P)$ over all possible configurations $P$?

Determine all integers $a,b,c$ satisfying
$$a^2+b^2+c^2=13(ab+bc+ca).$$

Fix a positive integer $N$. Let $a_k,b_k$ $(0\le k\le N)$ and $c_k$ $(1\le k\le N)$ be sequences of real numbers such that for all $1\le k\le N$, we have
$$a_k=\frac{4a_{k-1}+3b_{k-1}}5+c_k,$$
$$b_k=\frac{4b_{k-1}-3a_{k-1}}5.$$
If $a_0=b_0=0$ and $a_N\ge1$, determine the smallest possible value of
$$|c_1|+|c_2|+\cdots+|c_N|.$$

Let $(x_n)$ be a sequence of positive numbers. Prove that there are infinitely many $n$'s such that
$$\frac{x_1+\cdots+x_{n+1}}{x_n}>3.99.$$