Olympiad problems

2015 Federal Competition For Advanced Students, P2, 4

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)

2020 Federal Competition For Advanced Students, P2, 2

Tags: combinatorics , Coloring , points , Austria

In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: [i]There are exactly two green points that represent the distance $2020$ from that black point. [/i] Find the smallest possible number of green dots. (Walther Janous)

2017 Australian MO, 1

Tags: polynomial , algebra , Austria , AUT

Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions: (a) $P(2017)=2016$ and (b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.

2015 Federal Competition For Advanced Students, 3

Tags: Austria , game strategy

Alice and Bob play a game with a string of $2015$ pearls. In each move, one player cuts the string between two pearls and the other player chooses one of the resulting parts of the string while the other part is discarded. In the first move, Alice cuts the string, thereafter, the players take turns. A player loses if he or she obtains a string with a single pearl such that no more cut is possible. Who of the two players does have a winning strategy? (Theresia Eisenkölbl)

2019 Austrian Junior Regional Competition, 3

Tags: combinatorics , game , game strategy , Austria

Alice and Bob are playing a year number game. There will be two game numbers $19$ and $20$ and one starting number from the set $\{9, 10\}$ used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number $2019$ is reached or exceeded. Whoever reaches the number $2019$ wins. If $2019$ is exceeded, the game ends in a draw. $\bullet$ Show that Bob cannot win. $\bullet$ What starting number does Bob have to choose to prevent Alice from winning? (Richard Henner)

2015 Federal Competition For Advanced Students, P2, 6

Tags: Austria , number theory

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 [i]except to the jar $n$[/i]. 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)

2015 Federal Competition For Advanced Students, P2, 2

Tags: Austria , geometry , cyclic quadrilateral

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)

2016 Regional Competition For Advanced Students, 1

Tags: Austria , number theory

Determine all positive integers $k$ and $n$ satisfying the equation $$k^2 - 2016 = 3^n$$ (Stephan Wagner)

2016 Regional Competition For Advanced Students, 3

Tags: Austria , Game Theory

On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move. Prove that Bob wins, no matter how they play. (Richard Henner)

2015 Federal Competition For Advanced Students, P2, 1

Tags: Austria , algebra , functional equation

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 Regional Competition For Advanced Students, 4

Tags: Austria , geometry

Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$. We denote the incenter and circumcenter by $I$ and $O$, respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$. (a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular. (Walther Janous)

2005 Federal Competition For Advanced Students, Part 2, 3

Tags: geometry , Austria , AUT

Triangle $DEF$ is acute. Circle $c_1$ is drawn with $DF$ as its diameter and circle $c_2$ is drawn with $DE$ as its diameter. Points $Y$ and $Z$ are on $DF$ and $DE$ respectively so that $EY$ and $FZ$ are altitudes of triangle $DEF$ . $EY$ intersects $c_1$ at $P$, and $FZ$ intersects $c_2$ at $Q$. $EY$ extended intersects $c_1$ at $R$, and $FZ$ extended intersects $c_2$ at $S$. Prove that $P$, $Q$, $R$, and $S$ are concyclic points.

2016 Austria Beginners' Competition, 2

Tags: Austria , inequalities , algebra

Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality: $$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$ (Karl Czakler)

2018 Regional Competition For Advanced Students, 2

Tags: geometry , Equilateral Triangle , perpendicular bisector , Chords , Austria , national olympiad , AUT

Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. [i]Proposed by Stefan Leopoldseder[/i]

2015 Regional Competition For Advanced Students, 2

Tags: Austria , number theory

Let $x$, $y$, and $z$ be positive real numbers with $x+y+z = 3$. Prove that at least one of the three numbers $$x(x+y-z)$$ $$y(y+z-x)$$ $$z(z+x-y)$$ is less or equal $1$. (Karl Czakler)

2018 Federal Competition For Advanced Students, P1, 1

Tags: inequalities , Austria , AUT

Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur? [i](Proposed by Walther Janous)[/i]

2016 Regional Competition For Advanced Students, 2

Tags: inequalities , Austria , algebra

Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$. Prove that the inequality $$(a+2)(b+2) \ge cd$$ holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds. (Walther Janous)

2016 Austria Beginners' Competition, 4

Tags: Austria , geometry , pentagon

Let $ABCDE$ be a convex pentagon with ﬁve equal sides and right angles at $C$ and $D$. Let $P$ denote the intersection point of the diagonals $AC$ and $BD$. Prove that the segments $PA$ and $PD$ have the same length. (Gottfried Perz)

2017 Australian MO, 3

Tags: combinatorics , Combinatorial games , game strategy , game , Austria , AUT

Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table. Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.

2016 Federal Competition For Advanced Students, P1, 3

Tags: Austria , number theory

Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point? (Gerd Baron)

2004 Regional Competition For Advanced Students, 1

Tags: number theory , AUT , Austria , Diophantine equation

Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$

2016 Austria Beginners' Competition, 1

Tags: Austria , number theory

Determine all nonnegative integers $n$ having two distinct positive divisors with the same distance from $\tfrac{n}{3}$. (Richard Henner)

2015 Regional Competition For Advanced Students, 3

Tags: Austria , number theory

Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board. Determine the least integer that can be reached at the end by an appropriate sequence of moves. (Theresia Eisenkölbl)

2019 Federal Competition For Advanced Students, P2, 4

Tags: inequalities , BPSQ , High school olympiad , China , Austria

Let $a, b, c$ be the positive real numbers such that $a+b+c+2=abc .$ Prove that $$(a+1)(b+1)(c+1)\geq 27.$$

2008 Federal Competition For Advanced Students, P1, 3

Tags: inequalities , Sequence , recurrence relation , algebra , Austria

Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$. Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.