Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Federal Competition For Advanced Students, Part 2
Every brick has $5$ holes in a line. The holes can be filled with bolts (fitting in one hole) and braces (fitting into two neighboring holes). No hole may remain free.

One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent.

How many different such patterns can be produced with $n$ bricks?
2011 Federal Competition For Advanced Students, Part 2
We consider permutations $f$ of the set $\mathbb{N}$ of non-negative integers, i.e. bijective maps $f$ from $\mathbb{N}$ to $\mathbb{N}$, with the following additional properties: $$f(f(x)) = x \quad \mbox{and}\quad \left| f(x)-x\right| \leqslant 3\quad\mbox{for all } x \in\mathbb{N}\mbox{.}$$

Further, for all integers $n > 42$, $$\left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\mbox{.}\right.$$

Show that there are infinitely many natural numbers $K$ such that $f$ maps the set $$\left\{ n\mid 0\leqslant n\leqslant K\right\}$$ onto itself.
2011 Federal Competition For Advanced Students, Part 2
We are given a non-isosceles triangle $ABC$ with incenter $I$. Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$.

Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$.

Show that the four points $A$, $B$, $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.
2011 Federal Competition For Advanced Students, Part 2
Determine all pairs $(a,b)$ of non-negative integers, such that $a^b+b$ divides $a^{2b}+2b$.

(Remark: $0^0=1$.)
2011 Federal Competition For Advanced Students, Part 2
Let $k$ and $n$ be positive integers.

Show that if $x_j$ ($1\leqslant j\leqslant n$) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$, then

$$\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}$$
2011 Federal Competition For Advanced Students, Part 2
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$.

We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.)

Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$.

The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ .

Show that there is a point $Z$ that lies on all of these lines $UV$.
2012 Federal Competition For Advanced Students, Part 2
Determine the maximum value of $m$, such that the inequality



$$ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m $$



holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.

When does equality occur?
2012 Federal Competition For Advanced Students, Part 2
Solve over $\mathbb{Z}$:

$$ x^4y^3(y-x)=x^3y^4-216 $$
2012 Federal Competition For Advanced Students, Part 2
We call an isosceles trapezoid $PQRS$ interesting, if it is inscribed in the unit square $ABCD$ in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square.



Find all interesting isosceles trapezoids and their areas.
2012 Federal Competition For Advanced Students, Part 2
Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$.

If there is a real number $C$, such that

$$ (i-j)m_k+(j-k)m_i+(k-i)m_j=C$$

for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.
2012 Federal Competition For Advanced Students, Part 2
We define $N$ as the set of natural numbers $n<10^6$ with the following property:



There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$.



Find $|N|$.
2012 Federal Competition For Advanced Students, Part 2
Given an equilateral triangle $ABC$ with sidelength 2, we consider all equilateral triangles $PQR$ with sidelength 1 such that



$P$ lies on the side $AB$,

$Q$ lies on the side $AC$, and

$R$ lies in the inside or on the perimeter of $ABC$.





Find the locus of the centroids of all such triangles $PQR$.
2013 Federal Competition For Advanced Students, Part 2
For each pair $(a,b)$ of positive integers, determine all non-negative integers $n$ such that $$b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.$$
2013 Federal Competition For Advanced Students, Part 2
Let $k$ be an integer. Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=0$ and $$f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.$$
2013 Federal Competition For Advanced Students, Part 2
A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon $S$ inscribed in the circle ($S$ might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively?
2013 Federal Competition For Advanced Students, Part 2
For a positive integer $n$, let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$, the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that $$na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.$$
2013 Federal Competition For Advanced Students, Part 2
Let $n\geqslant3$ be an integer. Let $A_1A_2\ldots A_n$ be a convex $n$-gon. Consider a line $g$ through $A_1$ that does not contain a further vertice of the $n$-gon. Let $h$ be the perpendicular to $g$ through $A_1$. Project the $n$-gon orthogonally on $h$.

For $j=1,\ldots,n$, let $B_j$ be the image of $A_j$ under this projection. The line $g$ is called admissible if the points $B_j$ are pairwise distinct.

Consider all convex $n$-gons and all admissible lines $g$. How many different orders of the points $B_1,\ldots,B_n$ are possible?
2013 Federal Competition For Advanced Students, Part 2
Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$.

Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.
2014 Federal Competition For Advanced Students, P2
For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$.

For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?
2014 Federal Competition For Advanced Students, P2
Let $S$ be the set of all real numbers greater than or equal to $1$.

Determine all functions$ f: S \to S$, so that for all real numbers $x ,y  \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.
2014 Federal Competition For Advanced Students, P2
(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ?



(ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also  for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?
2014 Federal Competition For Advanced Students, P2
For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?
2014 Federal Competition For Advanced Students, P2
Show that the inequality  $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?
2014 Federal Competition For Advanced Students, P2
Let $U$ be the center of the circumcircle of the acute-angled triangle $ABC$. Let  $M_A, M_B$ and $M_C$ be the circumcenters of triangles $UBC,  UAC$ and  $UAB$ respecrively. For which triangles $ABC$ is the triangle $M_AM_BM_C$ similar to the starting triangle (with a suitable order of the vertices)?
2015 Federal Competition For Advanced Students, P2
Let $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}$ be a function with the following properties:



(i) $f(1) = 0$

(ii) $f(p) = 1$ for all prime numbers $p$

(iii) $f(xy) = y \cdot f(x) + x \cdot f(y)$ for all $x,y$ in $\mathbb{Z}_{>0}$



Determine the smallest integer $n \ge 2015$ that satisfies $f(n) = n$.



(Gerhard J. Woeginger)
2015 Federal Competition For Advanced Students, P2
We are given a triangle $ABC$.  Let $M$ be the mid-point of its side $AB$.



Let $P$ be an interior point of the triangle.  We let $Q$ denote the point symmetric to $P$ with respect to $M$.



Furthermore, let $D$ and $E$ be the common points of $AP$ and $BP$ with sides $BC$ and $AC$, respectively.



Prove that points $A$, $B$, $D$, and $E$ lie on a common circle if and only if $\angle ACP = \angle QCB$ holds.



(Karl Czakler)
2015 Federal Competition For Advanced Students, P2
We consider the following operation applied to a positive integer: The integer is represented in an arbitrary base $b \ge 2$, in which it has exactly two digits and in which both digits are different from $0$.  Then the two digits are swapped and the result in base $b$ is the new number.



Is it possible to transform every number $> 10$ to a number $\le 10$ with a series of such operations?



(Theresia Eisenkölbl)
2015 Federal Competition For Advanced Students, P2
Let $x,y,z$ be positive real numbers with $x+y+z \ge 3$.  Prove that



$\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1$



When does equality hold?



(Karl Czakler)
2015 Federal Competition For Advanced Students, P2
Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$.  The circle intersects



* the line $AI$ in points $A$ and $P$

* the line $BI$ in points $B$ and $Q$

* the line $AC$ in points $A$ and $R$

* the line $BC$ in points $B$ and $S$



with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively.



Prove that the lines $PS$, $QR$, and $CI$ meet in a single point.



(Stephan Wagner)
2015 Federal Competition For Advanced Students, P2
Max has $2015$ jars labeled with the numbers $1$ to $2015$ and an unlimited supply of coins.



Consider the following starting configurations:



(a) All jars are empty.

(b) Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins.

(c) Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin.



Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ to each jar except to the jar $n$.



Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar.



(Birgit Vera Schmidt)
2016 Federal Competition For Advanced Students, P2
Let $\alpha\in\mathbb{Q}^+$. Determine all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ that for all $x,y\in\mathbb{Q}^+$ satisfy the equation

$$ f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x.$$Here $\mathbb{Q}^+$ denote the set of positive rational numbers.



(Proposed by Walther Janous)
2016 Federal Competition For Advanced Students, P2
Let $ABC$ be a triangle. Its incircle meets the sides $BC, CA$ and $AB$ in the points $D, E$ and $F$, respectively. Let $P$ denote the intersection point of $ED$ and the line perpendicular to $EF$ and passing through $F$, and similarly let $Q$ denote the intersection point of $EF$ and the line perpendicular to $ED$ and passing through $D$.

Prove that $B$ is the mid-point of the segment $PQ$.



Proposed by Karl Czakler
2016 Federal Competition For Advanced Students, P2
Consider arrangements of the numbers $1$ through $64$ on the squares of an $8\times 8$ chess board, where each square contains exactly one number and each number appears exactly once.

A number in such an arrangement is called super-plus-good, if it is the largest number in its row and at the same time the smallest number in its column. Prove or disprove each of the following statements:

(a) Each such arrangement contains at least one super-plus-good number.

(b) Each such arrangement contains at most one super-plus-good number.



Proposed by Gerhard J. Woeginger
2016 Federal Competition For Advanced Students, P2
Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$.

Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality.



(Proposed by Karl Czakler)
2016 Federal Competition For Advanced Students, P2
Consider a board consisting of $n\times n$ unit squares where $n \ge 2$. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain $k$ tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists.

(a) Find the minimal $k$ such that the game does not end for any starting configuration and choice of cells during the game.

(b) Find the maximal $k$ such that the game ends for any starting configuration and choice of cells during the game.



Proposed by Theresia Eisenkölbl
2016 Federal Competition For Advanced Students, P2
Let $a,b,c$ be three integers for which the sum

$$ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$$is integer.

Prove that each of the three numbers

$$ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a}$$is integer.



(Proposed by Gerhard J. Woeginger)
2017 Federal Competition For Advanced Students, P2
Let $\alpha$ be a fixed real number. Find all functions $f:\mathbb R \to \mathbb R$ such that $$f(f(x + y)f(x - y)) = x^2 + \alpha yf(y)$$for all $x,y \in \mathbb R$.



Proposed by Walther Janous
2017 Federal Competition For Advanced Students, P2
A necklace contains $2016$ pearls, each of which has one of the colours black, green or blue.

In each step we replace simultaneously each pearl with a new pearl, where the colour of the new pearl is determined as follows: If the two original neighbours were of the same colour, the new pearl has their colour. If the neighbours had two different colours, the new pearl has the third colour.

(a) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if half of the pearls were black and half of the pearls were green at the start?

(b) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if thousand of the pearls were black at the start and the rest green?

(c) Is it possible to transform a necklace that contains exactly two adjacent black pearls and $2014$ blue pearls to a necklace that contains one green pearl and $2015$ blue pearls?



Proposed byTheresia Eisenkölbl
2017 Federal Competition For Advanced Students, P2
Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge  0$.

Show that the sequence does not contain a square of a rational number.



Proposed by Theresia Eisenkölbl
2017 Federal Competition For Advanced Students, P2
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$.

(b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$  and $x + y + z = M$.



Proposed by Karl Czakler
2017 Federal Competition For Advanced Students, P2
Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$.



Proposed by Karl Czakler
2017 Federal Competition For Advanced Students, P2
Let $S = \{1,2,..., 2017\}$.

Find the maximal $n$ with the property that there exist $n$ distinct subsets of $S$ such that for no two subsets their union equals $S$.



Proposed by Gerhard Woeginger
2018 Federal Competition For Advanced Students, P2
Let $a \ne 0$ be a real number.

Find all functions $f : R_{>0}\to  R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$for all $x, y \in  R_{>0}$.



(Proposed by Walther Janous)
2018 Federal Competition For Advanced Students, P2
Let $A, B, C$ and $D$ be four different points lying on a common circle in this order. Assume that the line segment $AB$ is the (only) longest side of the inscribed quadrilateral $ABCD$. Prove that the inequality $AB + BD > AC + CD$ holds.



(Proposed by Karl Czakler)
2018 Federal Competition For Advanced Students, P2
There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule:

In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece.

Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy.



(Proposed by Theresia Eisenkölbl)
2018 Federal Competition For Advanced Students, P2
Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and  $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$.



(Proposed by Walther Janous)
2018 Federal Competition For Advanced Students, P2
On a circle $2018$ points are marked. Each of these points is labeled with an integer.

Let each number be larger than the sum of the preceding two numbers in clockwise order.

Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers.



(Proposed by Walther Janous)
2018 Federal Competition For Advanced Students, P2
Determine all digits $z$ such that for each integer $k \ge 1$  there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$.



(Proposed by Walther Janous)
2019 Federal Competition For Advanced Students, P2
Determine all functions $f: R\to R$, such that  $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$.



(Gerhard Kirchner)
2019 Federal Competition For Advanced Students, P2
A (convex) trapezoid $ABCD$ is good, if it is inscribed in a circle, sides $AB$ and $CD$ are the bases and $CD$ is shorter than $AB$. For a good trapezoid $ABCD$ the following terms are defined:

$\bullet$ The parallel to $AD$ passing through $B$ intersects the extension of side $CD$ at point $S$.

$\bullet$ The two tangents passing through $S$ on the circumircle of the trapezoid touch the circle at $E$ and $F$, where $E$ lies on the same side of the straight line $CD$ as $A$.

Give the simplest possible equivalent condition (expressed in side lengths and / or angles of the trapezoid) so that with a good trapezoid $ABCD$ the two angles $\angle BSE$ and  $\angle FSC$ have the same measure.



(Walther Janous)
2019 Federal Competition For Advanced Students, P2
In Oddland there are stamps with values of $1$ cent, $3$ cents, $5$ cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value.

In Squareland, on the other hand, there are stamps with values of $1$ cent, $4$ cents, $9$ cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations.

Prove for every positive integer $n$: there are the same number in the two countries possibilities to send a letter with stamps worth a total of $n$ cents. It makes no difference if you have the same stamps on arrange a letter differently.



(Stephan Wagner)
2019 Federal Competition For Advanced Students, P2
Let $a, b, c$ be the positive real numbers such that $a+b+c+2=abc .$ Prove that $$(a+1)(b+1)(c+1)\geq 27.$$
2019 Federal Competition For Advanced Students, P2
Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the feet of the altitudes on the sides $BC$ or $AC$. Points $F$ and $G$ are located on the lines $AD$ and $BE$ in such a way that$ \frac{AF}{FD}=\frac{BG}{GE}$. The line passing through $C$ and $F$ intersects $BE$ at point $H$, and the line passing through $C$ and $G$ intersects $AD$ at point $I$. Prove that points $F, G, H$ and $I$ lie on a circle.



(Walther Janous)
2019 Federal Competition For Advanced Students, P2
Find the smallest possible positive integer n with the following property:

For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$.



(Gerhard J. Woeginger)
2021 Federal Competition For Advanced Students, P2
Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$.

When does equality holds?



(Karl Czakler)
2021 Federal Competition For Advanced Students, P2
Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game:

Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front.

At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds.

For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front?



(Birgit Vera Schmidt)



original wording, in case it doesn't make much senseHerr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen.

Ganz genau gesagt spielen die beiden das folgende Spiel:

Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne.

Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen.

Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können?



(Birgit Vera Schmidt)
2021 Federal Competition For Advanced Students, P2
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied.



(Walther Janous)