Let $ABC$ be a triangle with circumcircle $k$. The tangents at $A$ and $C$ intersect at $T$. The circumcircle of triangle $ABT$ intersects the line $CT$ at $X$ and $Y$ is the midpoint of $CX$. Prove that the lines $AX$ and $BY$ intersect on $k$.

Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.

A pair of words consisting only of the letters $a$ and $b$ (with repetitions) is good if it is $(a,b)$ or of one of the forms $(uv, v)$, $(u, uv)$, where $(u,v)$ is a good pair. Prove that if $(\alpha, \beta)$ is a good pair, then there exists a palindrome $\gamma$ such that $\alpha\beta = a\gamma b$.

Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?

The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

Let $ABC$ be a triangle with incircle $\gamma$. The circle through $A$ and $B$ tangent to $\gamma$ touches it at $C_2$ and the common tangent at $C_2$ intersects $AB$ at $C_1$. Define the points $A_1$, $B_1$, $A_2$, $B_2$ analogously. Prove that: a) the points $A_1$, $B_1$, $C_1$ are collinear; b) the lines $AA_2$, $BB_2$, $CC_2$ are concurrent.