Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Lusophon Mathematical Olympiad
In the center of a square is a rabbit and at each vertex of this even square, a wolf. The wolves only move along the sides of the square and the rabbit moves freely in the plane. Knowing that the rabbit move at a speed of $10$ km / h and that the wolves move to a maximum speed of $14$ km / h, determine if there is a strategy for the rabbit

leave the square without being caught by the wolves.
2015 Lusophon Mathematical Olympiad
Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers .

Prove that if  $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$  , then $m \cdot  n = p \cdot q$.
2015 Lusophon Mathematical Olympiad
Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.
2015 Lusophon Mathematical Olympiad
Let {An} be defined by

A1=2

A(n+1)=(An)^3 - An + 1

Prove that if  p | An => p>n
2014 Lusophon Mathematical Olympiad
Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers would have the same quantity of money. How much money does each brother have?
2014 Lusophon Mathematical Olympiad
Each white point in the figure below has to be completed with one of the integers $1, 2, ..., 9$, without repetitions, such that the sum of the three numbers in the external circle is equal to the sum of the four numbers in each internal circle that don't belong to the external circle.



$(a)$ Show a solution.



$(b)$ Prove that, in any solution, the number $9$ must belong to the external circle.
2014 Lusophon Mathematical Olympiad
In a convex quadrilateral $ABCD$, $P$ and $Q$ are points on sides $BC$ and $DC$ such that $B\hat{A}P = D\hat{A}Q$. If the line that passes through the orthocenters of $\triangle ABP$ and $\triangle ADQ$ is perpendicular to $AC$, prove that the area of these triangles are equals.
2014 Lusophon Mathematical Olympiad
From a point $K$ of a circle, a chord $KA$ (arc $AK$ is greather than $90^{o}$) and a tangent $l$ are drawn. The line that passes through the center of the circle and that is perpendicular to the radius $OA$, intersects $KA$ at $B$ and $l$ at $C$. Show that $KC = BC$.
2014 Lusophon Mathematical Olympiad
Find all quadruples of positive integers $(k,a,b,c)$ such that $2^k=a!+b!+c!$ and $a\geq b\geq c$.
2014 Lusophon Mathematical Olympiad
Kilua and Ndoti play the following game in a square $ABCD$: Kilua chooses one of the sides of the square and draws a point $X$ at this side. Ndoti chooses one of the other three sides and draws a point Y. Kilua chooses another side that hasn't been chosen and draws a point Z. Finally, Ndoti chooses the last side that hasn't been chosen yet and draws a point W. Each one of the players can draw his point at a vertex of $ABCD$, but they have to choose the side of the square that is going to be used to do that. For example, if Kilua chooses $AB$, he can draws $X$ at the point $B$ and it doesn't impede Ndoti of choosing $BC$. A vertex cannot de chosen twice. Kilua wins if the area of the convex quadrilateral formed by $X$, $Y$, $Z$, and $W$ is greater or equal than a half of the area of $ABCD$. Otherwise, Ndoti wins. Which player has a winning strategy? How can he play?
2013 Lusophon Mathematical Olympiad
If Xiluva puts two oranges in each basket, four oranges are in excess. If she puts five oranges in each basket, one basket is in excess. How many oranges and baskets has Xiluva?
2013 Lusophon Mathematical Olympiad
Let $ABC$ be an acute triangle. The circumference with diameter $AB$ intersects sides $AC$ and $BC$ at $E$ and $F$ respectively. The tangent lines to the circumference at the points $E$ and $F$ meet at $P$. Show that $P$ belongs to the altitude from $C$ of triangle $ABC$.
2013 Lusophon Mathematical Olympiad
An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$, $1986$, $1996$, $2008$ and it didn't occur in $1976$, $1993$, $2006$, $2013$. What is the first year in that the event will occur again?
2013 Lusophon Mathematical Olympiad
Find all the pairs $(x,y)$ of positive integers that satisfy the equation $x^2-xy+2x-3y=2013$.
2013 Lusophon Mathematical Olympiad
Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.
2013 Lusophon Mathematical Olympiad
Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.
2012 Lusophon Mathematical Olympiad
Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?
2012 Lusophon Mathematical Olympiad
Maria has a board of size $n \times  n$, initially with all the houses painted white. Maria decides to paint black some houses on the board, forming a mosaic, as shown in the figure below, as follows: she paints black all the houses from the edge of the board, and then leaves white the houses that have not yet been painted. Then she paints the houses on the edge of the next remaining board again black, and so on.

a) Determine a value of $n$ so that the number of black houses equals $200$.

b) Determine the smallest value of $n$ so that the number of black houses is greater than $2012$.
2012 Lusophon Mathematical Olympiad
Let $n$ be a positive integer, the players A and B play the following game: we have $n$ balls with the numbers of $1, 2, 3, 4,...., n$ this balls will be in two boxes with the symbols $\prod$ and $\sum$.

In your turn, the player can choose one ball and the player will put this ball in some box, in the final all the balls of the box $\prod$ are multiplied and we will get a number $P$, after this all the balls of the box $\sum$ are added up and we will get a number $Q$(if the box $\prod$ is empty $P = 1$, if the box $\sum$ is empty $Q = 0$).

The player(s) play alternately, player A starts, if $P + Q$ is even player A wins, otherwise player B wins.



a)If $n= 6$, which player has the winning strategy???

b)If $n = 2012$, which player has the winning strategy???
2012 Lusophon Mathematical Olympiad
An ant decides to walk on the perimeter of an $ABC$ triangle. The ant can start at any vertex. Whenever the ant is in a vertex, it chooses one of the adjacent vertices and walks directly (in a straight line) to the chosen vertex.

a) In how many ways can the ant walk around each vertex exactly twice?

b) In how many ways can the ant walk around each vertex exactly three times?



Note: For each item, consider that the starting vertex is visited.
2012 Lusophon Mathematical Olympiad
5)Players $A$ and $B$ play the following game: a player writes, in a board, a positive integer $n$, after this they delete a number in the board and write a new number where can be:

i)The last number $p$, where the new number will be $p - 2^k$ where $k$ is greatest number such that $p\ge 2^k$

ii) The last number $p$, where the new number will be $\frac{p}{2}$ if $p$ is even.

The players play alternately, a player win(s) if the new number is equal to $0$ and player $A$ starts.



a)Which player has the winning strategy with $n = 40$??

b)Which player has the winning strategy with $n = 2012$??
2012 Lusophon Mathematical Olympiad
A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure.

a) Show that the triangles $AOB$ and $COD$ have the equal areas.

b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.
2011 Lusophon Mathematical Olympiad
Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.
2011 Lusophon Mathematical Olympiad
A non-negative integer $n$ is said to be squaredigital if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.
2011 Lusophon Mathematical Olympiad
Consider a sequence of equilateral triangles $T_{n}$ as represented below:

[asy]

defaultpen(linewidth(0.8));size(350);

real r=sqrt(3);

path p=origin--(2,0)--(1,sqrt(3))--cycle;

int i,j,k;

for(i=1; i<5; i=i+1) {

for(j=0; j<i; j=j+1) {

for(k=0; k<j; k=k+1) {

draw(shift(5*i-5+(i-2)*(i-1)*1,0)*shift(2(j-k)+k, k*r)*p);

}}}[/asy]

The length of the side of the smallest triangles is $1$. A triangle is called a delta if its vertex is at the top; for example, there are $10$ deltas in $T_{3}$. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in $T_{20}$?
2011 Lusophon Mathematical Olympiad
Each one of three friends, Mário, João and Filipe, does one, and only one, of the following sports: football, basketball and swimming. None of these sports is done by more than one of the friends. Each one of the friends likes a certain kind of fruit: one likes oranges, another likes bananas and the other likes papayas. Find, for each one, which sport he plays and which fruit he prefers, given that:



* Mário doesn't like oranges;



* João doesn't play football;



* The swimmer hates bananas;



* The swimmer and the one who likes oranges do different sports;



* The one who likes papayas and the footballer visit Filipe every Saturday.
2011 Lusophon Mathematical Olympiad
Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.
2011 Lusophon Mathematical Olympiad
Let $d$ be a positive real number. The scorpion tries to catch the flea on a  $10\times 10$ chessboard. The length of the side of each small square of the chessboard is $1$. In this game, the flea and the scorpion move alternately. The flea is always on one of the $121$ vertexes of the chessboard and, in each turn, can jump from the vertex where it is to one of the adjacent vertexes. The scorpion  moves on the boundary line of the chessboard, and, in each turn, it can walk along any path of length less than $d$.

At the beginning, the flea is at the center of the chessboard and the scorpion is at a point that he chooses on the boundary line. The flea is the first one to play. The flea is said to escape if it reaches a point of the boundary line, which the scorpion can't reach in the next turn. Obviously, for big values of $d$, the scorpion has a strategy to prevent the flea's escape. For what values of $d$ can the flea escape? Justify your answer.
2021 Mediterranean Mathematics Olympiad
Determine the smallest positive integer $M$ with the following property:

For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients  so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$.



Proposed by Gerhard Woeginger, Austria
2021 Mediterranean Mathematics Olympiad
For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$:

$$p_1\, p_2\, \cdots\, p_8 

\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)

~~=~~ N $$

Proposed by Gerhard Woeginger, Austria
2021 Mediterranean Mathematics Olympiad
Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$.



(The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)
2021 Mediterranean Mathematics Olympiad
Let $x_1,x_2,x_3,x_4,x_5$ ve non-negative real numbers, so that

$x_1\le4$ and

$x_1+x_2\le13$ and

$x_1+x_2+x_3\le29$ and

$x_1+x_2+x_3+x_4\le54$ and

$x_1+x_2+x_3+x_4+x_5\le90$.

Prove that $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20$.
2020 Mediterranean Mathematics Olympiad
Determine all integers $m\ge2$ for which there exists an integer $n\ge1$ with

$\gcd(m,n)=d$ and $\gcd(m,4n+1)=1$.



Proposed by Gerhard Woeginger, Austria
2020 Mediterranean Mathematics Olympiad
Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$.



Proposed by Gerhard Woeginger, Austria
2020 Mediterranean Mathematics Olympiad
Prove that all postive real numbers $a,b,c$ with $a+b+c=4$ satisfy the inequality

$$\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}}

\le\frac43 \sqrt[4]{12}$$
2020 Mediterranean Mathematics Olympiad
Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$.   Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.
2019 Mediterranean Mathematics Olympiad
Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that

$$ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)$$
2019 Mediterranean Mathematics Olympiad
Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that

$$ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). $$

(Proposed by Gerhard Woeginger, Austria)
2019 Mediterranean Mathematics Olympiad
Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$.



(Proposed by Gerhard Woeginger, Austria)
2019 Mediterranean Mathematics Olympiad
Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that

$$ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz $$
2018 Mediterranean Mathematics OIympiad
Let $a_1, a_2, ..., a_n$ be more than one real numbers, such that $0\leq a_i\leq \frac{\pi}{2}$. Prove that

$$\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.$$
2018 Mediterranean Mathematics OIympiad
Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that

$$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$
2018 Mediterranean Mathematics OIympiad
An integer $a\ge1$ is called Aegean, if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime.

Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$.



(Proposed by Gerhard Woeginger, Austria)
2018 Mediterranean Mathematics OIympiad
Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties:

$*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering.

$*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$.

$*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$.



(Proposed by Gerhard Woeginger, Austria)
2017 Mediterranean Mathematics Olympiad
Let $ABC$ be an equilateral triangle, and let $P$ be some point in its circumcircle.

Determine all positive integers $n$, for which the value of the sum

$S_n (P) = |PA|^n +  |PB|^n +  |PC|^n$ is independent of the choice of point $P$.
2017 Mediterranean Mathematics Olympiad
Determine the smallest integer $n$ for which there exist integers $x_1,\ldots,x_n$ and positive integers $a_1,\ldots,a_n$ so that

\begin{align*}

x_1+\cdots+x_n &=0,\\
a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\
a_1^2x_1+\cdots+a_n^2x_n &<0.

\end{align*}
2017 Mediterranean Mathematics Olympiad
A set $S$ of integers is Balearic, if there are two (not necessarily distinct) elements

$s,s'\in S$ whose sum $s+s'$ is a power of two; otherwise it is called a non-Balearic set.

Find an integer $n$ such that $\{1,2,\ldots,n\}$ contains a 99-element non-Balearic set,

whereas all the 100-element subsets are Balearic.
2017 Mediterranean Mathematics Olympiad
Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that

$$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} +  \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$
2016 Mediterranean Mathematics Olympiad
Let $ABC$ be a triangle. Let $D$ be the intersection point of the angle bisector at $A$ with $BC$.

Let $T$ be the intersection point of the tangent line to the circumcircle of triangle $ABC$ at point $A$ with the line through $B$ and $C$.

Let $I$ be the intersection point of the orthogonal line to $AT$ through point $D$ with the altitude $h_a$ of the triangle at point $A$.

Let $P$ be the midpoint of $AB$, and let $O$ be the circumcenter of triangle $ABC$.

Let $M$ be the intersection point of $AB$ and $TI$, and let $F$ be the intersection point of $PT$ and $AD$.

Prove: $MF$ and $AO$ are orthogonal to each other.
2016 Mediterranean Mathematics Olympiad
Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that

$$ \sqrt{\frac{b}{a^2+3}}+ 

   \sqrt{\frac{c}{b^2+3}}+

   \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}$$
2016 Mediterranean Mathematics Olympiad
Consider a $25\times25$ chessboard with cells $C(i,j)$ for $1\le i,j\le25$. Find the smallest possible number $n$ of colors with which these cells can be colored subject to the following condition: For $1\le i<j\le25$ and for $1\le s<t\le25$, the three cells $C(i,s)$, $C(j,s)$, $C(j,t)$ carry at least two different colors.



(Proposed by Gerhard Woeginger, Austria)
2016 Mediterranean Mathematics Olympiad
Determine all integers $n\ge1$ for which the number $n^8+n^6+n^4+4$ is prime.



(Proposed by Gerhard Woeginger, Austria)
2015 Mediterranean Mathematical Olympiad
Let $P(x)=x^4-x^3-3x^2-x+1.$ Prove that there are infinitely many positive integers $n$ such that $P(3^n)$ is not a prime.
2015 Mediterranean Mathematical Olympiad
Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and

each cevian starting from that vertex, is possible to construct a triangle.
2015 Mediterranean Mathematical Olympiad
In the Cartesian plane $\mathbb{R}^2,$ each triangle contains a Mediterranean point on its sides or in its interior, even if the triangle is degenerated into a segment or a point. The Mediterranean points have the following properties:

(i) If a triangle is symmetric with respect to a line which passes through the origin $(0,0)$, then the Mediterranean point lies on this line.

(ii) If the triangle $DEF$ contains the triangle $ABC$ and if the triangle $ABC$ contains the Mediterranean points $M$ of $DEF,$ then $M$ is the Mediterranean point of the triangle $ABC.$



Find all possible positions for the Mediterranean point of the triangle with vertices $(-3,5),\ (12,5),\ (3,11).$
2015 Mediterranean Mathematical Olympiad
In a mathematical contest, some of the competitors are friends and friendship is mutual. Prove that there is a subset $M$ of the competitors such that each element of $M$ has at most three friends in $M$ and such that each competitor who is not in $M,$ has at least four friends in $M.$
2014 Mediterranean Mathematics Olympiad
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that

$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$

(Proposed by Gerhard Woeginger, Austria)
2014 Mediterranean Mathematics Olympiad
Consider increasing integer sequences with elements from $1,\ldots,10^6$. Such a sequence is Adriatic if its first element equals 1 and if every element is at least twice the preceding element. A sequence is Tyrrhenian if its final element equals $10^6$ and if every element is strictly greater than the sum of all preceding elements.

Decide whether the number of Adriatic sequences is smaller than, equal to, or greater than the number of Tyrrhenian sequences.

(Proposed by Gerhard Woeginger, Austria)
2014 Mediterranean Mathematics Olympiad
Prove that for every integer $S\ge100$ there exists an integer $P$ for which the following story could hold true:

The mathematician asks the shop owner: ``How much are the table, the cabinet and the bookshelf?''  The shop owner replies: ``Each item costs a positive integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and their product is $P$.''

The mathematician thinks and complains: ``This is not enough information to determine the three prices!''

(Proposed by Gerhard Woeginger, Austria)
2014 Mediterranean Mathematics Olympiad
In triangle $ABC$ let $A'$, $B'$, $C'$ respectively be the midpoints of the sides $BC$, $CA$, $AB$. Furthermore let $L$, $M$, $N$ be the projections of the orthocenter on the three sides $BC$, $CA$, $AB$, and let $k$ denote the nine-point circle. The lines $AA'$, $BB'$, $CC'$ intersect $k$ in the points $D$, $E$, $F$. The tangent lines on $k$ in $D$, $E$, $F$ intersect the lines $MN$, $LN$ and $LM$ in the points $P$, $Q$, $R$.

Prove that $P$, $Q$ and $R$ are collinear.
2013 Mediterranean Mathematics Olympiad
Do there exist two real monic polynomials $P(x)$ and $Q(x)$ of degree 3,such that the roots of $P(Q(X))$ are nine pairwise distinct nonnegative integers that add up to $72$?

(In a monic polynomial of degree 3, the coefficient of $x^{3}$ is $1$.)