Determine all positive, composite integers $n$ that satisfy the following property: if the positive divisors of $n$ are $1=d_{1}\lt d_{2}\lt \cdots\lt d_{k}=n$, then $d_{i}$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$.
Let $ABC$ be an acute-angled triangle with $AB \lt AC$. Denote its circumcircle by $\Omega$ and denote the midpoint of $\operatorname{arc} CAB$ by $S$. Let the perpendicular from $A$ to $BC$ meet $BS$ and $\Omega$ at $D$ and $E \neq A$ respectively. Let the line through $D$ parallel to $BC$ meet line $BE$ at $L$ and denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$.
Prove that the line tangent to $\omega$ at $P$, and line $BS$ intersect on the internal bisector of $\angle BAC$.
Let $k \geqslant 2$ be an integer. Determine all sequences of positive integers $a_{1}, a_{2}, \ldots$ for which there exists a monic polynomial $P$ of degree $k$ with non-negative integer coefficients such that
$$
P\left(a_{n}\right)=a_{n+1} a_{n+2} \cdots a_{n+k}
$$
for every integer $n \geqslant 1$.
Let $x_{1}, x_{2}, \ldots, x_{2023}$ be distinct real positive numbers such that
$$
a_{n}=\sqrt{\left(x_{1}+x_{2}+\cdots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\right)}
$$
is an integer for every $n=1,2, \ldots, 2023$. Prove that $a_{2023} \geqslant 3034$.
Let $n$ be a positive integer. A Japanese triangle consists of $1+2+\cdots+n$ circles arranged in an equilateral triangular shape such that for each $i=1,2,\ldots,n$, the $i$th row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n=6$, along with a ninja path in that triangle containing two red circles.
In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.
Let $ABC$ be an equilateral triangle. Points $A_1$, $B_1$, $C_1$ lie inside triangle $ABC$ such that triangle $A_1B_1C_1$ is scalene, $BA_1 = A_1C$, $CB_1 = B_1A$, $AC_1 = C_1B$ and
$$
\angle BA_1C + \angle CB_1A + \angle AC_1B = 480^\circ.
$$
Lines $BC_1$ and $CB_1$ intersect at $A_2$; lines $CA_1$ and $AC_1$ intersect at $B_2$; and lines $AB_1$ and $BA_1$ intersect at $C_2$.
Prove that the circumcircles of triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ have two common points.
