Olympiad problems

2016 Federal Competition For Advanced Students, P1, 4

Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then $$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$ (Walther Janous)

2016 Austria Beginners' Competition, 3

Tags: Austria , number theory , combinatorics

We consider the following ﬁgure: [See attachment] We are looking for labellings of the nine ﬁelds with the numbers 1, 2, ..., 9. Each of these numbers has to be used exactly once. Moreover, the six sums of three resp. four numbers along the drawn lines have to be be equal. Give one such labelling. Show that all such labellings have the same number in the top ﬁeld. How many such labellings do there exist? (Two labellings are considered diﬀerent, if they disagree in at least one ﬁeld.) (Walther Janous)

2019 Federal Competition For Advanced Students, P1, 1

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

We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as $a_n = 14a_{n-1} + a_{n-2}$ , $b_n = 6b_{n-1}-b_{n-2}$. Determine whether there are infinite numbers that occur in both sequences

2015 Federal Competition For Advanced Students, 1

Tags: Austria , algebra , inequalities

Let $a$, $b$, $c$, $d$ be positive numbers. Prove that $$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$ When does equality hold? (Georg Anegg)

2016 Federal Competition For Advanced Students, P1, 1

Tags: inequalities , Austria , algebra

Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$. For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds. (Walther Janous)

2015 Federal Competition For Advanced Students, 2

Tags: Austria , geometry

Let $ABC$ be an acute-angled triangle with $AC < AB$ and circumradius $R$. Furthermore, let $D$ be the foot ofthe altitude from $A$ on $BC$ and let $T$ denote the point on the line $AD$ such that $AT = 2R$ holds with $D$ lying between $A$ and $T$. Finally, let $S$ denote the mid-point of the arc $BC$ on the circumcircle that does not include $A$. Prove: $\angle AST = 90^\circ$. (Karl Czakler)

2020 Federal Competition For Advanced Students, P1, 3

Tags: combinatorics , greatest common divisor , Austria

On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as $a, b, c$ such that $a >gcd(b, c)$, then $a$ gets replaced by $ a-gcd(b, c)$. The game ends if there is no way to denote the numbers such that $a >gcd(b, c)$. Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers. (Theresia Eisenkölbl)

2015 Federal Competition For Advanced Students, P2, 5

Tags: geometry , incenter , Austria , AUT

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)

2020 Federal Competition For Advanced Students, P1, 4

Tags: number theory , Perfect Squares , Austria

Determine all positive integers $N$ such that $$2^N-2N$$ is a perfect square. (Walther Janous)

2015 Federal Competition For Advanced Students, P2, 3

Tags: Austria , number theory , number base

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)

2019 Federal Competition For Advanced Students, P1, 3

Tags: number theory , game , winning strategy , combinatorics , game strategy , Austria

Let $n\ge 2$ be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo $n$. At the beginning there is the residue class $1$ on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class $x$ with either $x + 1$ or by $2x$. The two players take turns, with Ariane starting. Ariane wins if the residue class $0$ is reached during the game. Bérénice wins if she can prevent that permanently. Depending on $n$, determine which of the two has a winning strategy.

2017 Regional Competition For Advanced Students, 3

Tags: combinatorics , Austria , AUT

The nonnegative integers $2000$, $17$ and $n$ are written on the blackboard. Alice and Bob play the following game: Alice begins, then they play in turns. A move consists in replacing one of the three numbers by the absolute difference of the other two. No moves are allowed, where all three numbers remain unchanged. The player who is in turn and cannot make an allowed move loses the game. [list] [*] Prove that the game will end for every number $n$. [*] Who wins the game in the case $n = 2017$? [/list] [i]Proposed by Richard Henner[/i]

2020 Federal Competition For Advanced Students, P2, 3

Tags: number theory , prime numbers , Sequence , Austria

Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and $$ e_n=a + \prod_{k=0}^{n-1} e_k$$ for $n \geq 1$. Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence. (Theresia Eisenkölbl)

2016 Regional Competition For Advanced Students, 4

Tags: geometry , circumcircle , Austria

Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$. (a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle. (b) Prove that the line $ST$ is parallel to the side $BC$. (Karl Czakler)

2015 Federal Competition For Advanced Students, 4

Tags: Austria , number theory

A [i]police emergency number[/i] is a positive integer that ends with the digits $133$ in decimal representation. Prove that every police emergency number has a prime factor larger than $7$. (In Austria, $133$ is the emergency number of the police.) (Robert Geretschläger)

2013 Federal Competition For Advanced Students, Part 1, 2

Tags: algebra , polynomial , function , Rational Root Theorem , Austria , 2013 , AUT

Solve the following system of equations in rational numbers: \[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]

2016 Federal Competition For Advanced Students, P1, 2

Tags: geometry , Austria , orthocenter , Circumcenter , circumcircle

We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$. (Karl Czakler)

1991 Federal Competition For Advanced Students, P2, 5

Tags: inequalities , inequalities proposed , Austria

For all positive integers $ n$ prove the inequality: $ \left( \frac{1\plus{}(n\plus{}1)^{n\plus{}1}}{n\plus{}2} \right)^{n\minus{}1}>\left( \frac{1\plus{}n^n}{n\plus{}1} \right)^n.$

2020 Regional Competition For Advanced Students, 1

Tags: algebra , Austria

Let $a$ be a positive integer. Determine all $a$ such that the equation $$ \biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x$$ has at least one integer solution for $x$. For every such $a$ state the respective solutions. (Richard Henner)