Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Middle European Mathematical Olympiad
The numbers $ 0$, $ 1$, $ \dots$, $ n$ ($ n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $ g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $ g(n)$ for every $ n$.
2009 Middle European Mathematical Olympiad
We colour every square of the $ 2009$ x $ 2009$ board with one of $ n$ colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares having another colour (a chess quen moves horizontally, vertically or diagonally). Find the maximum $ n$, such that for every colouring of the board at least on colour present at the board is connected.
2009 Middle European Mathematical Olympiad
Let $ ABCD$ be a parallelogram with $ \angle BAD = 60$ and denote by $ E$ the intersection of its diagonals. The circumcircle of triangle $ ACD$ meets the line $ BA$ at $ K \ne A$, the line $ BD$ at $ P \ne D$ and the line $ BC$ at $ L\ne C$. The line $ EP$ intersects the circumcircle of triangle $ CEL$ at points $ E$ and $ M$. Prove that triangles $ KLM$ and $ CAP$ are congruent.
2009 Middle European Mathematical Olympiad
Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD=DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC=2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.
2009 Middle European Mathematical Olympiad
Find all pairs $ (m$, $ n)$ of integers which satisfy the equation

$$ (m + n)^4 = m^2n^2 + m^2 + n^2 + 6mn.$$
2009 Middle European Mathematical Olympiad
Find all non-negative integer solutions of the equation

$$ 2^x+2009=3^y5^z.$$
2008 Middle European Mathematical Olympiad
Let $ (a_n)^{\infty}_{n=1}$ be a sequence of integers with $ a_{n} < a_{n+1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i + l = j + k$ we have the inequality $ a_{i} + a_{l} > a_{j} + a_{k}.$ Determine the least possible value of $ a_{2008}.$
2008 Middle European Mathematical Olympiad
Consider a $ n \times n$ checkerboard with $ n > 1, n \in \mathbb{N}.$ How many possibilities are there to put $ 2n - 2$ identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the $ n \times n$ square.
2008 Middle European Mathematical Olympiad
Determine that all $ k \in \mathbb{Z}$ such that $ \forall n$ the numbers $ 4n+1$ and $ kn+1$ have no common divisor.
2008 Middle European Mathematical Olympiad
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that

$$ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad  \forall  x,y \in \mathbb{R}.$$
2008 Middle European Mathematical Olympiad
On a blackboard there are $ n \geq 2, n \in \mathbb{Z}^{+}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $ n$ for which it is possible to yield $ n$ identical number after a finite number of steps.
2008 Middle European Mathematical Olympiad
Let $ ABC$ be an acute-angled triangle. Let $ E$ be a point such $ E$ and $ B$ are on distinct sides of the line $ AC,$ and $ D$ is an interior point of segment $ AE.$ We have $ \angle ADB = \angle CDE,$ $ \angle BAD = \angle ECD,$ and $ \angle ACB = \angle EBA.$ Prove that $ B, C$ and $ E$ lie on the same line.
2008 Middle European Mathematical Olympiad
Prove: If the sum of all positive divisors of $ n \in \mathbb{Z}^{+}$ is a power of two, then the number/amount of the divisors is a power of two.
2007 Middle European Mathematical Olympiad
Let $ a,b,c,d$ be positive real numbers with $ a+b+c+d = 4$.

Prove that

$$ a^{2}bc+b^{2}cd+c^{2}da+d^{2}ab\leq 4.$$
2007 Middle European Mathematical Olympiad
A set of balls contains $ n$ balls which are labeled with numbers $ 1,2,3,\ldots,n.$ We are given $ k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that



(a) the balls labeled with the same number are of the same colour,



(b) any subset of $ k+1$ balls with (not necessarily different) labels $ a_{1},a_{2},\ldots,a_{k+1}$ satisfying the condition $ a_{1}+a_{2}+\ldots+a_{k}= a_{k+1}$, contains at least one ball of each colour.



Find, depending on $ k$ the greatest possible number $ n$ which admits such a colouring.
2007 Middle European Mathematical Olympiad
Let $ k$ be a circle and $ k_{1},k_{2},k_{3},k_{4}$ four smaller circles with their centres $ O_{1},O_{2},O_{3},O_{4}$ respectively, on $ k$. For $ i = 1,2,3,4$ and $ k_{5}= k_{1}$ the circles $ k_{i}$ and $ k_{i+1}$ meet at $ A_{i}$ and $ B_{i}$ such that $ A_{i}$ lies on $ k$. The points $ O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$ lie in that order on $ k$ and are pairwise different.



Prove that $ B_{1}B_{2}B_{3}B_{4}$ is a rectangle.
2007 Middle European Mathematical Olympiad
Determine all pairs $ (x,y)$ of positive integers satisfying the equation

$$ x!+y!=x^{y}.$$
2007 Middle European Mathematical Olympiad
Let $ a,b,c,d$ be real numbers which satisfy $ \frac{1}{2}\leq a,b,c,d\leq 2$ and $ abcd=1$. Find the maximum value of

$$ \left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{d}\right)\left(d+\frac{1}{a}\right).$$
2007 Middle European Mathematical Olympiad
For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.

Find the maximum value of $ s(P)$ over all such sets $ P$.
2007 Middle European Mathematical Olympiad
A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$.

(a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)=n$.

(b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)=2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
2007 Middle European Mathematical Olympiad
Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a+k)^{3}-a^{3}$ is a multiple of $ 2007$.
2021 Nordic
On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.
2021 Nordic
Find all functions $f:R->R$ satisfying that for every $x$ (real number):

$f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$
2021 Nordic
Let $n$ be a positive integer. Alice and Bob play the following game. First, Alice picks $n + 1$ subsets $A_1,...,A_{n+1}$ of $\{1,... ,2^n\}$ each of size $2^{n-1}$. Second, Bob picks $n + 1$ arbitrary integers $a_1,...,a_{n+1}$. Finally, Alice picks an integer $t$. Bob wins if there exists an integer $1 \le i \le n + 1$ and $s \in A_i$ such that $s + a_i \equiv t$ (mod $2^n$). Otherwise, Alice wins.

Find all values of $n$ where Alice has a winning strategy.
2021 Nordic
Let $A, B, C$ and $D$ be points on the circle $\omega$ such that $ABCD$ is a convex quadrilateral. Suppose that $AB$ and $CD$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $BD$ and $AC$ intersect at a point $F$. Let $X \ne D$ be the point on $\omega$ such that $DX$ and $EF$ are parallel. Let $Y$ be the reflection of $D$ through $EF$ and suppose that $Y$ is inside the circle $\omega$.

Show that $A, X$, and $Y$ are collinear.
2020 Nordic
For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds: The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.
2020 Nordic
Georg has $2n + 1$ cards with one number written on each card. On one card the integer $0$ is written, and among the rest of the cards, the integers $k = 1, ... , n$ appear, each twice. Georg wants to place the cards in a row in such a way that the $0$-card is in the middle, and for each $k = 1, ... , n$, the two cards with the number $k$ have the distance $k$ (meaning that there are exactly $k - 1$ cards between them).

For which $1 \le n \le 10$ is this possible?
2020 Nordic
Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle  EPM$ and $\vartriangle FNQ$.
2020 Nordic
Find all functions $f : R- \{-1\} \to  R$ such that

$$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right) $$for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$.
2019 Nordic
A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers.

$a)$ Does there exist a meaningful set which consists of $2019$ numbers?

$b)$ Does there exist an infinite meaningful set?

Note: The geometric mean of the non-negative numbers $a_1, a_2,\cdots, $ $a_n$ is defined as $\sqrt[n]{a_1a_2\cdots a_n} .$
2019 Nordic
Let $a, b, c $ be the side lengths of a right angled triangle with c > a, b. Show that

$$3<\frac{c^3-a^3-b^3}{c(c-a)(c-b)}\leq \sqrt{2}+2.$$
2019 Nordic
The quadrilateral $ABCD$ satisfies $\angle ACD = 2\angle CAB, \angle ACB = 2\angle CAD $ and $CB = CD.$ Show that $$\angle CAB=\angle CAD.$$
2019 Nordic
Let $n$ be an integer with $n\geq 3$ and assume that $2n$ vertices of a regular $(4n + 1)-$gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.
2018 Nordic
Let $k$ be a positive integer and $P$ a point in the plane. We wish to draw lines, none passing through $P$, in such a way that any ray starting from $P$ intersects at least $k$ of these lines. Determine the smallest number of lines needed.
2018 Nordic
A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$, and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$. Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$.
2018 Nordic
Let $ABC$ be a triangle with $AB < AC$. Let $D$ and $E$ be on the lines $CA$ and $BA$, respectively, such that $CD = AB$, $BE = AC$, and $A$, $D$ and $E$ lie on the same side of $BC$. Let $I$ be the incenter of triangle $ABC$, and let $H$ be the orthocenter of triangle $BCI$. Show that $D$, $E$, and $H$ are collinear.
2018 Nordic
Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and $$ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). $$Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?
2017 Nordic
Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that $$ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. $$
2017 Nordic
Let $a, b, \alpha, \beta$ be real numbers such that $0 \leq a, b \leq 1$, and $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. Show that if $$ ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)}, $$then $$ a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha). $$
2017 Nordic
Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$, respectively, of an acute triangle $ABC$, $AB \neq AC$. Let $\omega_B$ be the circle centered at $M$ passing through $B$, and let $\omega_C$ be the circle centered at $N$ passing through $C$. Let the point $D$ be such that $ABCD$ is an isosceles trapezoid with $AD$ parallel to $BC$. Assume that $\omega_B$ and $\omega_C$ intersect in two distinct points $P$ and $Q$. Show that $D$ lies on the line $PQ$.
2017 Nordic
Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.
2016 Nordic
Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$.
2016 Nordic
Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.
2016 Nordic
Find all $a\in\mathbb R$ for which there exists a function $f\colon\mathbb R\rightarrow\mathbb R$, such that

(i) $f(f(x))=f(x)+x$, for all $x\in\mathbb R$,

(ii) $f(f(x)-x)=f(x)+ax$, for all $x\in\mathbb R$.
2016 Nordic
King George has decided to connect the $1680$ islands in his kingdom by bridges. Unfortunately the rebel movement will destroy two bridges after all the bridges have been built, but not two bridges from the same island. What is the minimal number of bridges the King has to build in order to make sure that it is still possible to travel by bridges between any two of the $1680$ islands after the rebel movement has destroyed two bridges?
2015 Nordic
Let ${ABC}$ be a triangle and ${\Gamma}$ the circle with diameter ${AB}$. The bisectors of ${\angle BAC}$ and ${\angle ABC}$ intersect ${\Gamma}$  (also) at ${D}$ and ${E}$, respectively. The incircle of ${ABC}$ meets ${BC}$ and ${AC}$ at ${F}$ and ${G}$, respectively. Prove that ${D, E, F}$ and ${G}$ are collinear.





I shall post in order to be collected, all the problems of the Nordic Mathematical Contest, already missing  in Aops.
2015 Nordic
Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.
2015 Nordic
Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted

with multiplicity). Let the polynomial $q$ be defined by

$$q(x) = 	\prod_{j=1}^{2015} p(x + j)$$.

We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$

such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.
2015 Nordic
An encyclopedia consists of ${2000}$  numbered volumes. The volumes are stacked in order with number ${1}$  on top and ${2000}$  in the bottom. One may perform two operations with the stack:

(i) For ${n}$ even, one may take the top ${n}$ volumes and put them in the bottom of the stack without changing the order.

(ii) For ${n}$ odd, one may take the top ${n}$ volumes, turn the order around and put them on top of the stack again.

How many different permutations of the volumes can be obtained by using these two operations repeatedly?
2014 Nordic
Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$  for all ${x, y \in N}$ with ${x \ge y}$.
2014 Nordic
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.
2014 Nordic
Find all nonnegative integers $a, b, c$ such that

$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$
2014 Nordic
A game is played on an ${n \times  n}$  chessboard. At the beginning there are ${99}$  stones on each square. Two players ${A}$  and ${B}$  take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$  takes the first turn. Determine all n for which player ${A}$  has a winning strategy.
2013 Nordic
Let ${(a_n)_{n\ge1}} $ be a sequence with  ${a_1 = 1} $  and  ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
2013 Nordic
In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).
2013 Nordic
Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further  ${q_k = n_k }$   / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence   ${(q_k)_{k\ge 1}}$
2013 Nordic
Let ${ABC}$ be an acute angled triangle, and ${H}$ a point in its interior. Let the reflections of ${H}$ through the sides ${AB}$ and ${AC}$ be called ${H_{c} }$ and ${H_{b} }$ , respectively, and let the reflections of H through the midpoints of these same sidesbe called ${H_{c}^{'} }$ and ${H_{b}^{'} }$, respectively. Show that the four points ${H_{b}, H_{b}^{'} , H_{c}}$, and  ${H_{c}^{'} }$  are concyclic if and only if at least two of them coincide or ${H}$ lies on the altitude from ${A}$ in triangle ${ABC}$.
2012 Nordic
The real numbers $a, b, c$ are such that $a^2 + b^2 = 2c^2$, and also such that $a \ne b, c \ne -a, c \ne -b$. Show that

$$\frac{(a+b+2c)(2a^2-b^2-c^2)}{(a-b)(a+c)(b+c)}$$

is an integer.
2012 Nordic
Given a triangle $ABC$, let $P$ lie on the circumcircle of the triangle and be the midpoint of the arc $BC$ which does not contain $A$. Draw a straight line $l$ through $P$ so that $l$ is parallel to $AB$. Denote by $k$ the circle which passes through $B$, and is tangent to $l$ at the point $P$. Let $Q$ be the second point of intersection of $k$ and the line $AB$ (if there is no second point of intersection, choose $Q = B$). Prove that $AQ = AC$.
2012 Nordic
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that

$$x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,$$

but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.
2012 Nordic
The number $1$ is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a - 1$ and $a + 1$; if the number $0$ occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show $1$ after $0$ steps; $2$ after $1$ step; $1, 3$ after $2$ steps; $2, 2, 4$ after $3$ steps, and so on. How many numbers will there be on the blackboard after $n$ steps?
2011 Nordic
When $a_0, a_1, \dots , a_{1000}$ denote digits, can the sum of the $1001$-digit numbers $a_0a_1\cdots a_{1000}$ and $a_{1000}a_{999}\cdots a_0$ have odd digits only?
2011 Nordic
In a triangle $ABC$ assume $AB = AC$, and let $D$ and $E$ be points on the extension of segment $BA$ beyond $A$ and on the segment $BC$, respectively, such that the lines $CD$ and $AE$ are parallel. Prove $CD \ge \frac{4h}{BC}CE$, where $h$ is the height from $A$ in triangle $ABC$. When does equality hold?
2011 Nordic
Find all functions $f$ such that

$$f(f(x) + y) = f(x^2-y) + 4yf(x)$$

for all real numbers $x$ and $y$.
2011 Nordic
Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$.

(Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)
2010 Nordic
A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.
2010 Nordic
Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that

$$\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.$$
2010 Nordic
Laura has $2010$ lamps connected with $2010$ buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too.

a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly?

b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?
2010 Nordic
A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n = a_1 \pm a_2 \pm a_3 \pm \cdots \pm a_k$ where $a_1, \dots , a_k$ are simple.
2009 Nordic
A point $P$ is chosen in an arbitrary triangle. Three lines are drawn through $P$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $f$ be the ratio between the total area of the three smaller triangles and the area of the given triangle. Prove that $f\ge\frac{1}{3}$ and determine those points $P$ for which $f =\frac{1}{3}$ .
2009 Nordic
On a faded piece of paper it is possible to read the following:

$$(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.$$

Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term $a$? We assume that all polynomials in the statement have only integer coefficients.
2009 Nordic
The integers $1$, $2$, $3$, $4$, and $5$ are written on a blackboard. It is allowed to wipe out two integers $a$ and $b$ and replace them with $a + b$ and $ab$. Is it possible, by repeating this procedure, to reach a situation where three of the five integers on the blackboard are $2009$?
2009 Nordic
$32$ competitors participate in a tournament. No two of them are equal and in a one against one match the better always wins. Show that the gold, silver, and bronze medal winners can be found in $39$ matches.