Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 Federal Competition For Advanced Students, Part 1
Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1 $ or $\sqrt 2 +1$. Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$? Which integral values can this sum take?
2005 Federal Competition For Advanced Students, Part 1
Prove that there are infinitely many multiples of 2005 that contain all the digits 0, 1, 2,...,9 an equal number of times.
2005 Federal Competition For Advanced Students, Part 1
For how many integers $a$ with $|a| \leq 2005$, does the system

$x^2=y+a$

$y^2=x+a$

have integer solutions?
2005 Federal Competition For Advanced Students, Part 1
For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$.

It is known that $s_1=2$, $s_2=6$ and $s_3=14$.

Prove that for all natural numbers $n>1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$
2005 Federal Competition For Advanced Students, Part 1
We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.
2006 Federal Competition For Advanced Students, Part 1
Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$.

For a $ n$ is only $ k=k(n)\geq2$ known. In how many different ways (as a function of $ k=k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?
2006 Federal Competition For Advanced Students, Part 1
Show that the sequence $ a_n = \frac {(n + 1)^nn^{2 - n}}{7n^2 + 1}$ is strictly monotonically increasing, where $ n = 0,1,2, \dots$.
2006 Federal Competition For Advanced Students, Part 1
In the triangle $ ABC$ let $ D$ and $ E$ be the boundary points of the incircle with the sides $ BC$ and $ AC$. Show that if  $ AD=BE$ holds, then the triangle is isoceles.
2006 Federal Competition For Advanced Students, Part 1
Given is the function $ f= \lfloor x^2 \rfloor + \{ x \}$ for all positive reals $ x$. ( $ \lfloor x \rfloor$ denotes the largest integer less than or equal $ x$ and $ \{ x \} = x - \lfloor x \rfloor$.)

Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $ 3$, if they are a reduced fraction, and don’t lie in the range of the function $ f$.
2007 Federal Competition For Advanced Students, Part 1
In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field.

For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column.

$ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$.

Show that $ A+B\neq0$
2007 Federal Competition For Advanced Students, Part 1
For every positive integer $ n$ determine the highest value $ C(n)$, such that for every $ n$-tuple $ (a_1,a_2,\ldots,a_n)$ of  pairwise distinct integers



$ (n + 1)\sum_{j = 1}^n a_j^2 - \left(\sum_{j = 1}^n a_j\right)^2\geq C(n)$
2007 Federal Competition For Advanced Students, Part 1
Let $ M(n )=\{-1,-2,\ldots,-n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products?
2007 Federal Competition For Advanced Students, Part 1
Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n - 1}A_n$ $ (A_n = A_0)$ where the side $ A_{i - 1}A_i = i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i + 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum

$ \Phi = \sum_{i = 0}^{n - 1} \phi_i$

of these $ n$ angles.
2008 Federal Competition For Advanced Students, P1
What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$?
2008 Federal Competition For Advanced Students, P1
Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\
x_2x_3(3a-2x_4) = a^3\\
...\\
x_{n-2}x_{n-1}(3a-2x_n) = a^3\\
x_{n-1}x_n(3a-2x_1) = a^3 \\
x_nx_1(3a-2x_2) = a^3 \end {cases}$

.
2008 Federal Competition For Advanced Students, P1
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 $n$.
2008 Federal Competition For Advanced Students, P1
In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.
2009 Federal Competition For Advanced Students, P1
Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$

.
2009 Federal Competition For Advanced Students, P1
For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$

$(n-2k)$...$(r)$, where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$

Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square.
2009 Federal Competition For Advanced Students, P1
There are $n$ bus stops placed around the circular lake. Each bus stop is connected by a road to the two adjacent stops (we call a segment  the entire road between two stops). Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly $n+1$ segments.
2009 Federal Competition For Advanced Students, P1
Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.
2010 Federal Competition For Advanced Students, Part 1
Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$.



(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)
2010 Federal Competition For Advanced Students, Part 1
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.



(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)
2010 Federal Competition For Advanced Students, Part 1
Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$. A subset $S$ of $M_n$ is called outstanding if it is non-empty and for every natural number $k\in S$, there exists a $k$-element subset $T_k$ of $S$.

Determine the number $a(n)$ of outstanding subsets of $M_n$.



(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)
2010 Federal Competition For Advanced Students, Part 1
The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$.

(a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side.

(b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side.

(c) For which inner point does the sum of the areas of the three small triangles attain a minimum?



(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)
2011 Federal Competition For Advanced Students, Part 1
Determine all integer triplets $(x,y,z)$ such that

$$x^4+x^2=7^zy^2\mbox{.}$$
2011 Federal Competition For Advanced Students, Part 1
For a positive integer $k$ and real numbers $x$ and $y$, let

$$f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}$$

If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$.
2011 Federal Competition For Advanced Students, Part 1
A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.



How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic?



(Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as

$$A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}$$

respectively, where $H(a,b)$ is not defined for some $a$, $b$.)
2011 Federal Competition For Advanced Students, Part 1
Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$, there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.
2012 Federal Competition For Advanced Students, Part 1
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$.



Note: If $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$.
2012 Federal Competition For Advanced Students, Part 1
Determine all solutions $(n, k)$ of the equation $n!+An = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.
2012 Federal Competition For Advanced Students, Part 1
Consider a stripe of $n$ fieds, numbered from left to right with the integers $1$ to $n$ in ascending order. Each of the fields is colored with one of the colors $1$, $2$ or $3$. Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors $1$ and $3$.

How many such colorings are there such that any two neighboring fields have different colors?
2012 Federal Competition For Advanced Students, Part 1
Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$.

Show that $\gamma$ is the second-largest angle in the triangle $ABC$.
2013 Federal Competition For Advanced Students, Part 1
Show that if for non-negative integers $m$, $n$, $N$, $k$ the equation $$(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m$$ holds, then $m = 1$.
2013 Federal Competition For Advanced Students, Part 1
Solve the following system of equations in rational numbers:

$$ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.$$
2013 Federal Competition For Advanced Students, Part 1
Arrange the positive integers into two lines as follows:

\begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\
\mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\ \ldots\end{align*}We start with writing $1$ in the upper line, $2$ in the lower line and $3$ again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block.

Let $a_1$, $a_2$, $a_3$, $\ldots$ be the numbers in the upper line. Give an explicit formula for $a_n$.
2013 Federal Competition For Advanced Students, Part 1
Let $A$, $B$ and $C$ be three points on a line (in this order).

For each circle $k$ through the points $B$ and $C$, let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$. Further, let $E$ be the second point of intersection of the line $AD$ with $k$.

Show that for each circle $k$, the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.
2014 Federal Competition For Advanced Students
Determine all real numbers $x$ and $y$ such that

$x^2 + x = y^3 - y$,

$y^2 + y = x^3 - x$
2014 Federal Competition For Advanced Students
We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length $n$. Determine the largest possible number of squares in such a friendly set.
2014 Federal Competition For Advanced Students
Let $a_n$ be a sequence defined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot  3^n$ for $n \ge 0$.

Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.
2014 Federal Competition For Advanced Students
We are given a right-angled triangle $MNP$ with right angle in $P$. Let $k_M$ be the circle with center $M$ and radius $MP$, and let $k_N$ be the circle with center $N$ and radius $NP$. Let $A$ and $B$ be the common points of $k_M$ and the line $MN$, and let $C$ and $D$ be the common points of $k_N$ and the line $MN$ with with $C$ between $A$ and $B$. Prove that the line $PC$ bisects the angle $\angle APB$.
2015 Federal Competition For Advanced Students
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)
2015 Federal Competition For Advanced Students
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)
2015 Federal Competition For Advanced Students
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)
2015 Federal Competition For Advanced Students
A police emergency number 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)
2016 Federal Competition For Advanced Students, P1
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)
2016 Federal Competition For Advanced Students, P1
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)
2016 Federal Competition For Advanced Students, P1
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)
2016 Federal Competition For Advanced Students, P1
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)
2017 Federal Competition For Advanced Students
Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions :

(a) $P(2017) = 2016$ and

(b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$.



proposed by Walther Janous
2017 Federal Competition For Advanced Students
Let $ABCDE$ be a regular pentagon with center $M$. A point $P$ (different from $M$) is chosen on the line

segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and

the line through $P$ perpendicular to $CD$ in $P$ and $R$.

Prove that $AR$ and $QR$ have same length.



proposed by Stephan Wagner
2017 Federal Competition For Advanced Students
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table.

Determine the smallest number $N\ge100000$ which Berta has wining strategy.



proposed by Gerhard Woeginger
2017 Federal Competition For Advanced Students
Find all pairs $(a,b)$ of non-negative integers such that:

$$2017^a=b^6-32b+1$$

proposed by Walther Janous
2018 Federal Competition For Advanced Students, P1
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?

(Proposed by Walther Janous)
2018 Federal Competition For Advanced Students, P1
Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear.



(Proposed by Karl Czakler)
2018 Federal Competition For Advanced Students, P1
Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$.  Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$.

Can Alice avoid that?



(Proposed by Richard Henner)
2018 Federal Competition For Advanced Students, P1
Let $M$ be a set containing positive integers with the following three properties:

(1) $2018 \in M$.

(2) If $m \in M$, then all positive divisors of m are also elements of $M$.

(3) For all elements $k, m \in M$ with $1 < k < m$, the number $km + 1$ is also an element of $M$.

Prove that $M = Z_{\ge 1}$.



(Proposed by Walther Janous)
2019 Federal Competition For Advanced Students, P1
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
2019 Federal Competition For Advanced Students, P1
Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$.  The circle passing through $B, C$ and $I$ intersects the line $AC$ for  second time at point $Y$. Show that the segments $AY$ and $BX$ have  equal length.
2019 Federal Competition For Advanced Students, P1
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.
2019 Federal Competition For Advanced Students, P1
Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor  = b   \cdot \lfloor  a  \cdot n  \rfloor$ applies to all positive integers$ n$.

(For a real number $x, \lfloor x\rfloor$  denotes the largest integer that is less than or equal to $x$.)
2021 Federal Competition For Advanced Students, P1
Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$(Marian Dinca)
2021 Federal Competition For Advanced Students, P1
Let $ABC$ denote a triangle. The point $X$ lies on the extension of $AC$ beyond $A$, such that $AX = AB$. Similarly, the point $Y$ lies on the extension of $BC$ beyond $B$ such that $BY = AB$. Prove that the circumcircles of $ACY$ and $BCX$ intersect a second time in a point different from $C$ that lies on the bisector of the angle $\angle BCA$.



(Theresia Eisenkölbl)
2021 Federal Competition For Advanced Students, P1
Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$.



(Walther Janous)
2021 Federal Competition For Advanced Students, P1
On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game.

Alice starts and they alternately play the following moves:

$\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$

$\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$.

Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps.

Prove that Alice has a winning strategy.



(Daniel Holmes)
2019 Regional Competition For Advanced Students
Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$
2019 Regional Competition For Advanced Students
The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point  $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.
2019 Regional Competition For Advanced Students
Let $n\ge 2$ be a natural number.

An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers  $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined.

(a) Show that for no odd number $n$ there is a label with $S_n = 0$.

(b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$.
2019 Regional Competition For Advanced Students
Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors.