Olympiad problems

2017 German National Olympiad, 4

Tags: geometry , geometry proposed , cyclic quadrilateral , Cyclic Quadrilaterals , tangent circles , Germany

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 $AB$.

2004 Germany Team Selection Test, 4

Tags: Germany , TST , Team Selection Test , equation , integer equation

Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.

2017 Germany, Landesrunde - Grade 11/12, 4

Tags: combinatorics , number theory , number theory unsolved , Germany , divisor

Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

2015 German National Olympiad, 5

Tags: geometry , geometry proposed , tangent circles , parallel , Germany

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ touches the line $CD$. Prove that that the circle with diameter $CD$ touches the line $AB$ if and only if $BC$ and $AD$ are parallel.

2005 Germany Team Selection Test, 2

Tags: inequalities , function , 3-variable inequality , Nesbitt , Germany , Japan

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

2004 Germany Team Selection Test, 2

Tags: function , algebra proposed , algebra , Functional Equations , Germany , TST , Team Selection Test

Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties: (a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$. (b) We have $f\left(2\right) = 0$. (c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$. [b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.

2005 Germany Team Selection Test, 2

Tags: inequalities , function , 3-variable inequality , Nesbitt , Germany , Japan

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

2016 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , combinatorics unsolved , pigeonhole principle , graph theory , Germany , blackboard

There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.

2015 Bundeswettbewerb Mathematik Germany, 3

Tags: geometry , BWM , Germany , ray , concurrence , angle , Triangle

Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$. Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$. Both points, $X$ and $Y$, are on the same side as $C$ with respect to line $AB$. Show that the rays $[AX$ and $[BY$ intersect on line $CM$.

2017 German National Olympiad, 2

Tags: geometry , circumcircle , Fixed point , Germany , geometry proposed

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 Germany, Landesrunde - Grade 11/12, 2

Tags: ratio , invariant , geometry , circle , Germany

Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.

2015 German National Olympiad, 1

Tags: system of equations , algebra , algebra proposed , equation , equations , Germany

Determine all pairs of real numbers $(x,y)$ satisfying \begin{align*} x^3+9x^2y&=10,\\ y^3+xy^2 &=2. \end{align*}

2004 Germany Team Selection Test, 2

Tags: geometry proposed , geometry , Locus problems , Geometric Inequalities , Germany , TST , Team Selection Test

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

2015 German National Olympiad, 2

Tags: number theory , number theory proposed , equation , Germany

A positive integer $n$ is called [i]smooth[/i] 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.

2017 German National Olympiad, 6

Tags: number theory , number theory proposed , Germany , Diophantine equation , Diophantine Equations

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 $m>1$.

1998 German National Olympiad, 6a

Tags: algebra , system of equations , powers , symmetry , algebra unsolved , Germany , Real

Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3 \\ y^5 &= x^3+21y^3. \end{align}

2017 German National Olympiad, 1

Tags: quadratics , system of equations , equation , quadratic equation , algebra , algebra proposed , Germany

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 $q$.

2015 German National Olympiad, 3

Tags: combinatorics , graph theory , Germany , combinatorics proposed

To prepare a stay abroad, students meet at a workshop including an excursion. To promote interaction between the students, they are to be distributed to two busses such that not too many of the students in the same bus know each other. Every student knows all those who know her. The number of such pairwise acquaintances is $k$. Prove that it is possible to distribute the students such that the number of pairwise acquaintances in each bus is at most $\frac{k}{3}$.