Anna and Lisa go for a bike ride. Anna's bike breaks down $30$ kilometers before their final destination. The two decide to complete the ride with Lisa's bike as follows: At the beginning, Anna is riding a bike and Lisa leaves. At some point, Anna gets off the bike, parks it on the side of the road and continues by foot. When Lisa gets to the bike, she takes it and rides until she catches up with Anna. After that, they repeat the same procedure. We don't know how many times the procedure is repeated, but they arrive at the final goal at the same time. Anna walks at a speed of $4$ km/h and cycles at a speed of $15$ km/h. Lisa walks at $5$ km/h and cycles with $20$ km/h. How long does it take them to cover the last $30$ km of the road? (Neglect the time it takes to park, lock, unlock the bike, etc.)

A triangular colony area is divided into four fields of varying size as shown in the figure below shows. The only other thing we know is that the distances $AF$, $FD$, $BF$ and $FE$ have the lengths $5$, $2$, $4$ and $2$ respectively (in $10$s of m). When the lots are distributed, Joar gets to choose first. Which lot should he choose to get the one with the largest area?

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be different real numbers, placed one after the other in any order. We say we have a local minimum in one of the numbers if this is less than both of their neighbors. Which is the average number of local minima over all possible ways of ordering the numbers each other?

Let $f$ be a function that associates a positive integer $(x, y)$ with each pair of positive integers $f(x, y)$. Suppose that $f(x, y) \le xy$ for all positive integers $x$, $y$. Show that there are $2023$ different pairs $(x_1, y_1)$,$...$, $ (x_{2023}, y_{2023}$) such that $$f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).$$

(a) Let $x$ and $y$ be integers. Prove that $x = y$ if $x^n \equiv y^n$ mod $n$ for all positive integers $n$. (b) For which pairs of integers $(x, y)$ are there infinitely many positive integers $n$ such that $x^n \equiv y^n$ mod $n$?

Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.