Navvye/Olympiad-Problems-CSV huggingface dataset:
1997 Flanders Math Olympiad
$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$.

Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$,  going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there)

Show $A_k < S$ for every positive integer $k$.
1997 Flanders Math Olympiad
Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)
1996 Flanders Math Olympiad
In triangle $\Delta ADC$ we got $AD=DC$ and $D=100^\circ$.

In triangle $\Delta CAB$ we got $CA=AB$ and $A=20^\circ$.



Prove that $AB=BC+CD$.
1996 Flanders Math Olympiad
Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.
1996 Flanders Math Olympiad
Consider the points $1,\frac12,\frac13,...$ on the real axis. Find the smallest value $k \in \mathbb{N}_0$ for which all points above can be covered with 5 closed intervals of length $\frac1k$.
1996 Flanders Math Olympiad
Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$.

(a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$.

(b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have $$ p(x+1)+p(x-1)-rp(x)-s=0 $$Prove $\deg(p(x))\le 2$.

(c) Show, in (b) that $s=0$ implies $a_2=0$.
1995 Flanders Math Olympiad
Four couples play chess together. For the match, they're paired as follows:  ("man Clara" indicates the husband of Clara, etc.)

$$Bea \Longleftrightarrow Eddy$$

$$An \Longleftrightarrow man\ Clara$$

$$Freddy \Longleftrightarrow woman\ Guy$$

$$Debby \Longleftrightarrow man\ An$$

$$Guy \Longleftrightarrow woman\ Eddy$$

Who is $Hubert$ married to?
1995 Flanders Math Olympiad
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?
1995 Flanders Math Olympiad
Points $A,B,C,D$ are on a circle with radius $R$.

$|AC|=|AB|=500$, while the ratio between $|DC|, |DA|, |DB|$ is $1,5,7$. Find $R$.
1995 Flanders Math Olympiad
Given a regular $n$-gon inscribed in a circle of radius 1, where $n > 3$.

Define $G(n)$ as the average length of the diagonals of this $n$-gon.



Prove that if $n \rightarrow \infty,  G(n) \rightarrow \frac{4}{\pi}$.
1994 Flanders Math Olympiad
Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.
1994 Flanders Math Olympiad
Determine all integer solutions (a,b,c) with $c\leq 94$ for which:

$(a+\sqrt c)^2+(b+\sqrt c)^2 = 60 + 20\sqrt c$
1994 Flanders Math Olympiad
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)



What's the volume of $A\cup B$?
1994 Flanders Math Olympiad
Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$)

(a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined.

(b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.
1993 Flanders Math Olympiad
The 20 pupils in a class each send 10 cards to 10 (different) class members. [note: you cannot send a card to yourself.]

(a) Show at least 2 pupils sent each other a card.

(b) Now suppose we had $n$ pupils sending $m$ cards each. For which $(m,n)$ is the above true? (That is, find minimal $m(n)$ or maximal $n(m)$)
1993 Flanders Math Olympiad
A jeweler covers the diagonal of a unit square with small golden squares in the following way:

- the sides of all squares are parallel to the sides of the unit square

- for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex)

- each midpoint of a square has distance to the vertex of the unit square equal to $\dfrac12, \dfrac14, \dfrac18, ...$ of the diagonal. (so real length: $\times \sqrt2$)

- all midpoints are on the diagonal



(a) What is the side length of the middle square?

(b) What is the total gold-plated area?
1993 Flanders Math Olympiad
For $a,b,c>0$ we have:

$$ -1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1 $$
1993 Flanders Math Olympiad
Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$.



Find $\lim\limits_{n\rightarrow+\infty} oa_n$.
1992 Flanders Math Olympiad
For every positive integer $n$, determine the biggest positive integer $k$ so that $2^k |\ 3^n+1$
1992 Flanders Math Olympiad
It has come to a policeman's ears that 5 gangsters (all of different height) are meeting, one of them is the clan leader, he's the tallest of the 5. He knows the members will leave the building one by one, with a 10-minute break between them, and too bad for him Belgium has not enough policemen to follow all gangsters, so he's on his own to spot the clanleader, and he can only follow one member.



So he decides to let go the first 2 people, and then follow the first one that is taller than those two. What's the chance he actually catches the clan leader like this?
1992 Flanders Math Olympiad
a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?
1992 Flanders Math Olympiad
Let $A,B,P$ positive reals with $P\le A+B$.

(a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that $$ A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)} $$

(b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$.

(c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$, $z,w>0$ and $z^2+w^2<x^2+y^2$.

Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$, the cyclical one has the greatest area.
1991 Flanders Math Olympiad
Show that the number $111...111$ with 1991 times the number 1, is not prime.
1991 Flanders Math Olympiad
(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$. Call this $x_n$.

(b) Find $\lim\limits_{n\rightarrow+\infty}x_n$.
1991 Flanders Math Olympiad
Given $\Delta ABC$ equilateral, with $X\in[A,B]$. Then we define unique points Y,Z so that $Y\in[B,C]$, $Z\in[A,C]$, $\Delta XYZ$ equilateral.



If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.
1991 Flanders Math Olympiad
A word of length $n$ that consists only of the digits $0$ and $1$, is called a bit-string of length $n$. (For example, $000$ and $01101$ are bit-strings of length 3 and 5.) Consider the sequence $s(1), s(2), ...$ of bit-strings of length $n > 1$ which is obtained as follows :

(1) $s(1)$ is the bit-string $00...01$, consisting of $n - 1$ zeros and a $1$ ;

(2) $s(k+1)$ is obtained as follows :

(a) Remove the digit on the left of $s(k)$. This gives a bit-string $t$ of length $n - 1$.

(b) Examine whether the bit-string $t1$ (length $n$, adding a $1$ after $t$) is already in $\{s(1), s(2), ..., s(k)\}$. If this is the not case, then $s(k+1) = t1$. If this is the case then $s(k+1) = t0$.



For example, if $n = 3$ we get :

$s(1) = 001 \rightarrow s(2) = 011 \rightarrow s(3) = 111 \rightarrow s(4) = 110 \rightarrow s(5) = 101$

$\rightarrow s(6) = 010 \rightarrow s(7) = 100 \rightarrow s(8) = 000 \rightarrow s(9) = 001 \rightarrow ...$



Suppose $N = 2^n$.

Prove that the bit-strings $s(1), s(2), ..., s(N)$ of length $n$ are all different.
1990 Flanders Math Olympiad
On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0].

You get a figure as below, find the area of the colored part.



Invalid image file
1990 Flanders Math Olympiad
Let $a$ and $b$ be two primes having at least two digits, such that $a > b$.



Show that $$240|\left(a^4-b^4\right)$$ and show that 240 is the greatest positive integer having this property.
1990 Flanders Math Olympiad
We form a decimal code of $21$ digits. the code may start with $0$. Determine the probability that the fragment $0123456789$ appears in the code.
1990 Flanders Math Olympiad
Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function.



(a)   Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$

Prove that $a_n$ is ascending, that $\displaystyle\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty$



(b)   Prove that there exist such a sequence ($a_n$) in $\mathbb{R}^+_0$ if you know $\displaystyle\lim_{x\rightarrow +\infty} f(x)=0$.
1989 Flanders Math Olympiad
Show that every subset of {1,2,...,99,100} with 55 elements contains at least 2 numbers with a difference of 9.
1989 Flanders Math Olympiad
When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?
1989 Flanders Math Olympiad
Show that:$$\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2

(k\in \mathbb{Z}) \Longleftrightarrow\  |{\tan \alpha}| + |{\cot

\alpha}|  = 4$$
1989 Flanders Math Olympiad
Let $D$ be the set of positive reals different from $1$ and let $n$ be a positive integer.

If for $f: D\rightarrow \mathbb{R}$ we have $x^n f(x)=f(x^2)$,

and if $f(x)=x^n$ for $0<x<\frac{1}{1989}$ and for $x>1989$,

then prove that $f(x)=x^n$ for all $x \in D$.
1988 Flanders Math Olympiad
show that the polynomial $x^4+3x^3+6x^2+9x+12$ cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.
1988 Flanders Math Olympiad
A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it.

The cross is inscribed in a circle with radius 1. What's its volume?
1988 Flanders Math Olympiad
Work base 3. (so each digit is 0,1,2)



A good number of size $n$ is a number in which there are no consecutive $1$'s and no consecutive $2$'s. How many good 10-digit numbers are there?
1988 Flanders Math Olympiad
Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad).



Prove $2R\theta$ is between $1$ and $\pi$.
1987 Flanders Math Olympiad
Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.$$
1987 Flanders Math Olympiad
Show that for $p>1$ we have $$\lim_{n\rightarrow+\infty}\frac{1^p+2^p+...+(n-1)^p+n^p+(n-1)^p+...+2^p+1^p}{n^2} = +\infty$$

Find the limit if $p=1$.
1986 Flanders Math Olympiad
Prove that for integer $n$ we have:

$$n! \le \left( \frac{n+1}{2} \right)^n$$

(please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)
1986 Flanders Math Olympiad
Let $\{a_k\}_{k\geq 0}$ be a sequence given by $a_0 = 0$, $a_{k+1}=3\cdot a_k+1$ for $k\in \mathbb{N}$.



Prove that $11 \mid a_{155}$
1986 Flanders Math Olympiad
Given a cube in which you can put two massive spheres of radius 1.

What's the smallest possible value of the side - length of the cube?

Prove that your answer is the best possible.
2021 German National Olympiad
Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.
2021 German National Olympiad
Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent:



(1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter.



(2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.
2021 German National Olympiad
For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once.



Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
2021 German National Olympiad
Let $OFT$ and $NOT$ be two similar triangles (with the same orientation) and let $FANO$ be a parallelogram. Show that

$$\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.$$
2021 German National Olympiad
a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$, one has

$$\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.$$b) For this real number $A$, find all triples $(x,y,z)$ of non-negative real numbers for which equality holds in the above inequality.
2021 German National Olympiad
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers  such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.
2020 German National Olympiad
Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.
2020 German National Olympiad
In ancient times there was a Celtic tribe consisting of several families. Many of these families were at odds with each other, so that their chiefs would not shake hands.

At some point at the annual meeting of the chiefs they found it even impossible to assemble four or more of them in a circle with each of them being willing to shake his neighbour's hand.

To emphasize the gravity of the situation, the Druid collected three pieces of gold from each family. The Druid then let all those chiefs shake hands who were willing to. For each handshake of two chiefs he paid each of them a piece of gold as a reward.



Show that the number of pieces of gold collected by the Druid exceeds the number of pieces paid out by at least three.
2020 German National Olympiad
Show that the equation

$$x(x+1)(x+2)\dots (x+2020)-1=0$$has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies

$$\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.$$
2020 German National Olympiad
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.
2020 German National Olympiad
Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$.

Prove the inequality

$$\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.$$
2020 German National Olympiad
The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.
2019 German National Olympiad
Determine all real solutions $(x,y)$ of the following system of equations:

\begin{align*}

x&=3x^2y-y^3,\\
y &= x^3-3xy^2

\end{align*}
2019 German National Olympiad
Let $a$ and $b$ be two circles, intersecting in two distinct points $Y$ and $Z$. A circle $k$ touches the circles $a$ and $b$ externally in the points $A$ and $B$.



Show that the angular bisectors of the angles $\angle ZAY$ and $\angle YBZ$ intersect on the line $YZ$.
2019 German National Olympiad
In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is below another rectangle if there is a line $g$ parallel to the $x$-axis such that the first rectangle is below $g$, the second one above $g$ and both rectangles do not touch $g$.



Similarly, we say that one rectangle is to the right of another rectangle if there is a line $h$ parallel to the $y$-axis such that the first rectangle is to the right of $h$, the second one to the left of $h$ and both rectangles do not touch $h$.



Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$
2019 German National Olympiad
Show that for each non-negative integer $n$ there are unique non-negative integers $x$ and $y$ such that we have

$$n=\frac{(x+y)^2+3x+y}{2}.$$
2019 German National Olympiad
We are given two positive integers $p$ and $q$.



Step by step, a rope of length $1$ is cut into smaller pieces as follows: In each step all the currently longest pieces are cut into two pieces with the ratio $p:q$  at the same time.

After an unknown number of such operations, the currently longest pieces have the length $x$.



Determine in terms of $x$ the number $a(x)$ of different lengths of pieces of rope existing at that time.
2019 German National Olympiad
Suppose that real numbers $x,y$ and $z$ satisfy the following equations:



\begin{align*}

x+\frac{y}{z} &=2,\\
y+\frac{z}{x} &=2,\\
z+\frac{x}{y} &=2.

\end{align*}

Show that $s=x+y+z$ must be equal to $3$ or $7$.



Note: It is not required to show the existence of such numbers $x,y,z$.
2018 German National Olympiad
Find all real numbers $x,y,z$ satisfying the following system of equations:

\begin{align*}

xy+z&=-30\\
yz+x &= 30\\
zx+y &=-18

\end{align*}
2018 German National Olympiad
We are given a tetrahedron with two edges of length $a$ and the remaining four edges of length $b$ where $a$ and $b$ are positive real numbers. What is the range of possible values for the ratio $v=a/b$?
2018 German National Olympiad
Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes.

Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions:

1) There are no two red boxes in the same column or in the same row.

2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column.

Proposed by Christian Reiher
2018 German National Olympiad
a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that

$$a^2+b^2+c^2+abc<8.$$b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have

$$a^2+b^2+c^2+abc<d \quad$$where $a,b$ and $c$ are the side lengths of the triangle?
2018 German National Olympiad
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.
2018 German National Olympiad
Let $P$ be a point in the interior of a triangle $ABC$ and let the rays $\overrightarrow{AP}, \overrightarrow{BP}$ and $\overrightarrow{CP}$ intersect the sides $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$, respectively. Let $D$ be the foot of the perpendicular from $A_1$ to $B_1C_1$. Show that

$$\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.$$
2017 German National Olympiad
Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$:

\begin{align*} x^2+py+q&=0,\\
y^2+px+q&=0.

\end{align*}Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.
2017 German National Olympiad
Let $ABC$ be a triangle such that $\vert AB\vert \ne \vert AC\vert$. Prove that there exists a point $D \ne A$ on its circumcircle satisfying the following property:

For any points $M, N$ outside the circumcircle on the rays $AB$ and $AC$, respectively, satisfying $\vert BM\vert=\vert CN\vert$, the circumcircle of $AMN$ passes through $D$.
2017 German National Olympiad
General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!).

To this end, they arrange $N$ tin soldiers in $M$ companies and command them by turns.

Both of them must give a command and execute it in their turn.



Only two commands are possible: The command "Divide!" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier.

On the other hand, the command "Rule!" removes exactly one tin soldier from each company.



The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command.



a) Can he force Tilly to lose if they start with $7$ companies of $7$ tin soldiers each?



b) Who loses if they start with $M \ge 1$ companies consisting of $n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1$ $(n_1+n_2+\dotsc+n_M=N)$ tin soldiers?
2017 German National Olympiad
Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$. Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$.



Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$.
2017 German National Olympiad
Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has

$$1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.$$
2017 German National Olympiad
Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.
2016 German National Olympiad
Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*}(German MO 2016 - Problem 1)
2016 German National Olympiad
A very well known family of mathematicians has three children called Antonia, Bernhard and Christian. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days.



Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day.



Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.
2016 German National Olympiad
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.

Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
2016 German National Olympiad
Find all positive integers $m,n$ with $m \leq 2n$ that solve the equation $$ m \cdot \binom{2n}{n} = \binom{m^2}{2}. $$(German MO 2016 - Problem 4)
2016 German National Olympiad
Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.
2016 German National Olympiad
Let $$ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 $$be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$.

Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.
2015 German National Olympiad
Determine all pairs of real numbers $(x,y)$ satisfying

\begin{align*} x^3+9x^2y&=10,\\
y^3+xy^2 &=2.

\end{align*}
2015 German National Olympiad
A positive integer $n$ is called smooth if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying

$$a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.$$Find all smooth numbers.