2024 Durer Math Competition Finals

Category E+

Day 1

1

Describe all ordered sets of four real numbers $(a, b, c, d)$ for which the values $a + b, b + c, c + d, d + a$ are all non-zero and \[\frac{a+2b+3c}{c+d}=\frac{b+2c+3d}{d+a}=\frac{c+2d+3a}{a+b}=\frac{d+2a+3b}{b+c}.\]

2

For every subset $\mathcal{P}$ of the plane let $S(\mathcal{P})$ denote the set of circles and lines that intersect $\mathcal{P}$ in at least three points. Find all sets $\mathcal{P}$ consisting of 2024 points such that for any two distinct elements of $S(\mathcal{P}),$ their intersection points all belong to $\mathcal{P}{}.$

3

We have a stick of length $2n{}$ and a machine which cuts sticks of length $k\in\mathbb{N}$ with $k>1$ into two sticks with arbitrary positive integer lengths. What is the smallest number of cuts after which we can always find some sticks whose lengths sum up to $n{}$?

4

Let $\mathcal{H}$ be the set of all lines in the plane. Call a function $f:\mathbb{R}^2\to\mathcal{H}$ polarising, if $P\in f(Q)$ implies $Q\in f(P)$ for any pair of points $P,Q\in\mathbb{R}^2.$ Show that there is no surjective polarising function. Give an example of an injective polarising function. Prove that for every injective polarising function there exists a point $P$ in the plane for which $P\in f(P).$

5

Let $p{}$ be a fixed prime number. Determine the number of ordered $k$-tuples $(a_1,\ldots,a_k)$ of non-negative integers smaller than $p{}$ for which $p\mid a_1^2+\cdots+a_k^2$ where a) $k=3$ and b) $k$ is an arbitrary odd positive integer.

6

On a $1\times n$ board there are $n-1$ separating edges between neighbouring cells. Initially, none of the edges contain matches. During a move of size $0 < k < n$ a player chooses a $1\times k$ sub-board which contains no matches inside, and places a matchstick on all of the separating edges bordering the sub-board that don’t already have one. A move is considered legal if at least one matchstick can be placed and if either $k = 1$ or $k{}$ is divisible by 4. Two players take turns making moves, the player in turn must choose one of the available legal moves of the largest size $0 < k < n$ and play it. If someone does not have a legal move, the game ends and that player loses. Beat the organisers twice in a row in this game! First the organisers determine the value of $n{}$, then you get to choose whether you want to play as the first or the second player.

numbering to be fixed

1

There are 100 merchants who are selling salmon for Durer dollars around the circular shore of the island of Durerland. Since the beginning of times good and bad years have been alternating on the island. (So after a good year, the next year is bad; and after a bad year, the next year is good.) In every good year all merchants set their price as the maximum value between their own selling price from the year before and the selling price of their left-hand neighbour from the year before. In turn, in every bad year they sell it for the minimum between their own price from the year before and their left-hand neighbour’s price from the year before. Paul and Pauline are two merchants on the island. This year Paul is selling salmon for 17 Durer dollars a kilogram. Prove that there will come a year when Pauline will sell salmon for 17 Durer dollars a kilogram. Note: The merchants are immortal, they have been selling salmon on the island for thousands of years and will continue to do so until the end of time.

2

One quadrant of the Cartesian coordinate system is tiled with $1\times 2$ dominoes. The dominoes don’t overlap with each other, they cover the entire quadrant and they all fit in the quadrant. Farringdon the flea is initially sitting at the origin and is allowed to jump from one corner of a domino to the opposite corner any number of times. Is it possible that the dominoes are arranged in such a way that Farringdon is unable to move to a distance greater than 2023 from the origin?

3

A round table is surrounded by $n\geqslant 2$ people, each assigned one of the integers $0, 1,\ldots , n-1$ such that no two people have the same number. In each round, everyone adds their number to their right neighbour’s number, and their new number becomes the remainder of the sum when divided by $n{}.$ We call an initial configuration of integers glorious if everyone’s number remains the same after some finite number of rounds, never changing again. For which integers $n\geqslant 2$ is every initial configuration glorious? For which integers $n\geqslant 2$ is there no glorious initial configuration at all?

4

In a game, two players control an adventurer in a dungeon. The adventurer starts with $H{}$ hit points, an integer greater than one. The dungeon consists of several chambers. There are some passageways in the dungeon, each leading from a chamber to a chamber. These passageways are one-way, and a passageway may return to its starting chamber. Every chamber can be exited through at least one passageway. There are 5 types of chambers: Entrance: the adventurer starts here, no passageway comes in here; Hollow: nothing happens; Spike: the adventurer loses a hit point; Trap: the adventurer gets shot by an arrow; Catacomb: the adventurer loses hit points equal to the total number of times they have been hit by an arrow. The two players control the adventurer alternatively. At a turn, a player can move him through one passageway. A player loses if the adventurer’s hit points fall below zero due to their action (at 0 hit points, the character is alive). Construct a dungeon map, which has at most 20 chambers in total and exactly one entrance, with the following condition: the first player has a winning strategy if $H{}$ is a prime, and the second player has a winning strategy if $H{}$ is composite. Note: If the game doesn’t end after a finite number of moves, neither player wins.

5

For a given triangle $A_1A_2A_3$ and a point $X{}$ inside of it we denote by $X_i$ the intersection of line $A_iX$ with the side opposite to $A_i$ for all $1\leqslant i \leqslant 3.$ Let $P{}$ and $Q{}$ be distinct points inside the triangle. We then say that the two points are isotomic (i.e. they form an isotomic pair) if for all $i{}$ the points $P_i$ and $Q_i$ are symmetric with respect to the midpoint of the side opposite to $A_i.$ Augustus wants to construct isotomic pairs with his favourite app, GeoZebra. He already constructed the vertices and sidelines of a non-isosceles acute triangle when suddenly his computer got infected with a virus. Most tools became unavailable, only a few are usable, some of which even require a fee: Point: select an arbitrary point (with respect to the position of the mouse) on the plane or on a figure (circle or line) - free Intersection: intersection points of two figures (where each figure is a circle or a line) - free Line: line through two points - $5 per use Perpendicular: perpendicular from a point to an already constructed line - $50 per use Circumcircle: circle through three points - $10 per use Agatha selected a point $P{}$ inside the triangle, which is not the centroid of the triangle. Show that Augustus can construct a point $Q{}$ at a cost of at most 1000 dollars such that $P{}$ and $Q{}$ are isotomic. Prove that for any $n\geqslant 1$ Augustus can construct $n{}$ different isotomic pairs at a cost of at most $200 + 10n$ dollars. Note: The parts are unrelated, that is Augustus can’t use his constructions from part a) in part b).