$ABC$ is an isosceles triangle. The base $BC$ is $1$ cm long, and legs $AB$ and $AC$ are $2$ cm long. Let the midpoint of $AB$ be $F$, and the midpoint of $AC$ be $G$. Additionally, $k$ is a circle, that is tangent to $AB$ and A$C$, and it’s points of tangency are $F$ and $G$ accordingly. Prove, that the intersection of $CF$ and $BG$ falls on the circle $k$.
2023 Durer Math Competition Finals
Category E
Day 1
When Andris entered the room, there were the numbers $3$ and $24$ on the board. In one step, if there are the (not necessarily different) numbers $k$ and $n$ on the board already, then Andris can write the number$ kn + k + n$ on the board, too. a) Can Andris write the number $9999999$ on the board after a few moves? b) What if he wants to get $99999999$? c) And what about $48999999$?
a) Four merchants want to travel from Athens to Rome by cart. On the same day, but different times they leave Athens and arrive on another day to Rome, but in reverse order. Every day, when the evening comes, each merchant enters the next inn on the way. When some merchants sleep in the same inn at night, then on the following day at dawn they leave in reverse order of arrival, because they can only park this way on the narrow streets next to the inns. They cannot overtake each other, their order only changes after a night spent together in the same inn. Eventually each merchant arrives in Rome while they sleep with every other merchant in the same inn exactly once. Is it possible, that the number of the inns they sleep in is even every night? b) Is it possible if there are $8$ merchants instead of $4$ and every other condition is the same?
Prove that for all $n \ge 3$ there are an infinite number of $n$-sided polygonal numbers which are also the sum of two other (not necessarily different) $n$-sided polygonal numbers! The first $n$-sided polygonal number is $1$. The kth n-sided polygonal number for $k \ge 2$ is the number of different points in a figure that consists of all of the regular $n$-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are $\ell$ cm long where $1 \le \ell \le k - 1$ cm and $\ell$ is an integer. In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly $1$ cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure.
At the end of the first quarter of the Greece-Egypt basketball game, the score was $26-25$. During the first quarter, Áron wrote down the total number of points of the Greeks after every Greek basket, while Benedek wrote down the total number of points of the Egyptians after every Egyptian basket. In the break they noticed that there is no number that both of them wrote down. In how many ways could they have written down the numbers, if there were $21$ baskets, and every basket was a $2$-pointer or a $3$-pointer? Two options are different if at least one of them wrote down different numbers. (original wording)
Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.
Day 2
Csenge and Eszter ate a whole basket of cherries. Csenge ate a quarter of all cherries while Eszter ate four-sevenths of all cherries and forty more. How many cherries were in the basket in total?
Timi was born in $1999$. Ever since her birth how many times has it happened that you could write that day’s date using only the digits $0$, $1$ and $2$? For example, $2022.02.21$. is such a date.
Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?
Benedek wrote down the following numbers: $1$ piece of one, $2$ pieces of twos, $3$ pieces of threes, $... $, $50$ piecies of fifties. How many digits did Benedek write down?
We are given a triangle $ABC$ and two circles ($k_1$ and $k_2$) so the diameter of $k_1$ is $AB$ and the diameter of $k_2$ is $AC$. Let the intersection of $BC$ line segment and $k_1$ (that isn’t $B$) be $P,$ and the intersection of $BC$ line segment and $k_2$ (that isn’t $B$) be $Q$. We know, that $AB = 3003$ and $AC = 4004$ and $BC = 5005$. What is the distance between $P$ and $Q$?
In Eldorado a year has $20$ months, and each month has $20$ days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the $19$th day of the $4$th month, she would say $4 \cdot 19 = 76$.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?
The area of a rectangle is $64$ cm$^2$, and the radius of its circumscribed circle is $7$ cm. What is the perimeter of the rectangle in centimetres?
Zoli wants to fill the given $4 \times 4$ table with the digits $1$, $2$, $3$ and $4$, such that in every row and column, and also in the diagonal going from the top left cell to the bottom right, each digit appears exactly once. What is the highest possible value of the sum of the digits in the six grey cells?
Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$? Do not disturb my circles!
One day Mnemosyne decided to colour all natural numbers in increasing order. She coloured $0$, $1$ and $2$ in brown, and her favourite number, $3$, in gold. From then on, for any number whose sum of digits (in the decimal system) was a golden number less than the number itself, she coloured it gold, but coloured the rest of the numbers brown. How many four-digit numbers were coloured gold by Mnemosyne? The set of natural numbers includes $0$.
The binary sudoku is a puzzle in which a table should be filled with digits $0$ and $1$ such that in each row and column, the number of 0s is equal to the number of $1$s. Furthermore, there cannot exist three adjacent cells in a row or in a column such that they have the same digit written in them. Solving the given binary sudoku, what is the sum of the numbers in the two diagonals?
Marvin really likes pancakes, so he asked his grandma to make pancakes for him. Every time Grandma sends pancakes, she sends a package of $32$. When Marvin is in the mood for pancakes, he eats half of the pancakes he has. Marvin ate $157$ pancakes for lunch today. At least how many times has Grandma sent pancakes to Marvin so far? Marvin does not necessarily eat an integer number of pancakes at once, and he is in the mood for pancakes at most once a day.
A country has $2023$ cities and there are flights between these cities. Each flight connects two cities in both directions. We know that you can get from any city to any other using these flights, and from each city there are flights to at most $4$ other cities. What is the maximum possible number of cities in the country from which there is a flight to only one city?
Zeus’s lightning is made of a copper rod of length $60$ by bending it $4$ times in alternating directions so that the angle between two adjacent parts is always $60^o$. What is the minimum value of the square of the distance between the two endpoints of the lightning? All five segments of the lightning lie in the same plane.
Csongi bought a $12$-sided convex polygon-shaped pizza. The pizza has no interior point with three or more distinct diagonals passing through it. Áron wants to cut the pizza along $3$ diagonals so that exactly $6$ pieces of pizza are created. In how many ways can he do this? Two ways of slicing are different if one of them has a cut line that the other does not have.
For the Dürer final results announcement, four loudspeakers are used to provide sound in the hall. However, there are only two sockets in the wall from which the power comes. To solve the problem, Ádám got two extension cords and two power strips. One plug can be plugged into an extension cord, and two plugs can be plugged into a power strip. Gábor, in his haste before the announcement of the results, quickly plugs the $8$ plugs into the $8$ holes. Every possible way of plugging has the same probability, and it is also possible for Gábor to plug something into itself. What is the probability that all $4$ speakers will have sound at the results announcement? For the solution, give the sum of the numerator and the denominator in the simplified form of the probability. A speaker sounds when it is plugged directly or indirectly into the wall.
Category E+
Day 1
Prove that for any real $r>0$, one can cover the circumference of a $1\times r$ rectangle with non-intersecting disks of unit radius.
a) Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes. b) Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.
same as E5 Day 1 - 3
For a given integer $n\geq2$, a pyramid of height $n$ if defined as a collection of $1^2+2^2+\dots+n^2$ stone cubes of equal size stacked in $n$ layers such that the cubes in the $k$-th layer form a square with sidelength $n+1-k$ and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?
For an acute triangle $ABC$, let $O$ be its circumcenter, and let $O_A,O_B,O_C$ be the circumcenter of $BCO,CAO,ABO$ respectively. Show that $AO_A,BO_B,CO_C$ are concurrent.
same as E6 Day 1 - 6
Day 2
Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?
same as Day 2 E7 - 2
Hapi, the god of the annual flooding of the Nile is preparing for this year’s flooding. The shape of the channel of the Nile can be described by the function $y = \frac{-1000}{ x^2+100}$ where the $x$ and $y$ coordinates are in metres. The depth of the river is $5$ metres now. Hapi plans to increase the water level by $3$ metres. How many metres wide will the river be after the flooding? The depth of the river is always measured at its deepest point.
same as Day 2 E6 - 4
same as Day 2 E9 - 6
King Minos divided his rectangular island of Crete between his 3 sons as follows: he built a wall along one diagonal of the island and gave one half of the island to his eldest son. Then, in the remaining triangular area, from the right-angled vertex he built a wall perpendicular to the other wall. Of the two areas thus obtained, the larger was given to the middle son and the smaller to the youngest. Each of the three sons had the largest possible square palace built on his own land. How many times is the area of the eldest son’s palace larger than the area of the youngest son’s palace if the side lengths of the island are $30$ m and $210$ m?
same as Day 2 E10 - 7
same as Day 2 E8 - 8
same as Day 2 E11 - 9
same as Day 2 E12 - 10
same as Day 2 E13 - 11
same as Day 2 E14 - 12
same as Day 2 E15 - 13
same as Day 2 E16 - 14
What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?
What is the remainder of $2025\wedge (2024\wedge (2022\wedge (2021\wedge (2020\wedge ...\wedge (2\wedge 1) . . .)))))$ when it is divided by $2023$? Here $\wedge$ is the exponential symbol, for example $2\wedge (3\wedge 2) = 2\wedge 9 = 512$. The power tower contains the integers from $2025$ to $1$ exactly once, except that the number $2023$ is missing.