Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 IMO Shortlist
For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define

$$\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.$$That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.



Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that

$$\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.$$

Proposed by Rodrigo Sanches Angelo, Brazil
2014 IMO Shortlist
Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that

$$ n^2+4f(n)=f(f(n))^2 $$

for all $n\in \mathbb{Z}$.



Proposed by Sahl Khan, UK
2014 IMO Shortlist
We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:

1. The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.

2. If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$.

3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$  then $A$ also beats $C$.

How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other.



Proposed by Ilya Bogdanov, Russia
2014 IMO Shortlist
Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves.



Proposed by Vladislav Volkov, Russia
2014 IMO Shortlist
There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$.

Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd.



Proposed by Tejaswi Navilarekallu, India
2014 IMO Shortlist
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$ . Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$ , respectively. We call the pair $(E, F )$ $\textit{interesting}$, if the quadrilateral $KSAT$ is cyclic.

Suppose that the pairs $(E_1 , F_1 )$ and $(E_2 , F_2 )$ are interesting. Prove that $\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}$

Proposed by Ali Zamani, Iran
2014 IMO Shortlist
Let $a_1 < a_2 <  \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2  \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 ,   \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.



Proposed by Serbia
2014 IMO Shortlist
For every real number $x$, let $||x||$ denote the distance between $x$ and the nearest integer.

Prove that for every pair $(a, b)$ of positive integers there exist an odd prime $p$ and a positive integer $k$ satisfying $$\displaystyle\left|\left|\frac{a}{p^k}\right|\right|+\left|\left|\frac{b}{p^k}\right|\right|+\left|\left|\frac{a+b}{p^k}\right|\right|=1.$$



Proposed by Geza Kos, Hungary
2013 IMO Shortlist
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation

$$ f(f(f(n))) = f(n+1 ) +1 $$

for all $ n\in \mathbb{Z}_{\ge 0}$.
2013 IMO Shortlist
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that

$$ (x^3 - mx^2 +1 ) P(x+1)  + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) $$

for all real number $x$.
2013 IMO Shortlist
A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.

(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.

(ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment.



Prove that the physicist may apply a sequence of much operations resulting in a family of imons, no two of which are entangled.
2013 IMO Shortlist
In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that  there is no city such that more than $2550$ other cities have distance exactly four from it.
2013 IMO Shortlist
Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened.

Decide whether there exists a strategy for player $A$ to win in a finite number of moves.
2013 IMO Shortlist
Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses.



An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise.



Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.
2013 IMO Shortlist
Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying

$$ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) $$

for all  $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)
2013 IMO Shortlist
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called good if

$$a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.$$ A good pair $(a,b)$ is called excellent if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good.



Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.
2012 IMO Shortlist
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions

$$f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},$$

and $f(-1) \neq 0$.
2012 IMO Shortlist
We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form

$$f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),$$

where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.
2012 IMO Shortlist
In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain.



Proposed by Merlijn Staps, The Netherlands
2012 IMO Shortlist
The columns and the row of a $3n \times 3n$ square board are numbered $1,2,\ldots ,3n$. Every square $(x,y)$ with $1 \leq x,y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0,1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^2$ tokens of each color.

Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.
2012 IMO Shortlist
There are given $2^{500}$ points on a circle labeled $1,2,\ldots ,2^{500}$ in some order. Prove that one can choose $100$ pairwise disjoint chords joining some of theses points so that the $100$ sums of the pairs of numbers at the endpoints of the chosen chord are equal.
2012 IMO Shortlist
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
2012 IMO Shortlist
In an acute triangle $ABC$ the points $D,E$ and $F$ are the feet of the altitudes through $A,B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1$ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.
2012 IMO Shortlist
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$.
2012 IMO Shortlist
Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.
2012 IMO Shortlist
Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.



Proposed by Cosmin Pohoata, Romania
2012 IMO Shortlist
Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.
2012 IMO Shortlist
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.
2011 IMO Shortlist
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $$f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1$$ for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$.



Proposed by Bojan Bašić, Serbia
2011 IMO Shortlist
Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.



Proposed by Canada
2011 IMO Shortlist
Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that



$$\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.$$



Proposed by Titu Andreescu, Saudi Arabia
2011 IMO Shortlist
Let $n$ be a positive integer, and let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word, consisting of just letters $a$ and/or $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^n$.



A finite nonempty word $U$ is said to appear in $W$ if there exist indices $k \leq \ell$ such that $U=x_k x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called ubiquitous if the four words $Ua$, $Ub$, $aU$, and $bU$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words.



Proposed by Grigory Chelnokov, Russia
2011 IMO Shortlist
On a square table of $2011$ by $2011$ cells we place a finite number of napkins that each cover a square of $52$ by $52$ cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of $k$?



Proposed by Ilya Bogdanov and Rustem Zhenodarov, Russia
2011 IMO Shortlist
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that

$$\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.$$



Proposed by Alexey Gladkich, Israel
2011 IMO Shortlist
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.



Proposed by Irena Majcen and Kris Stopar, Slovenia
2011 IMO Shortlist
Let $ABCDEF$ be a convex hexagon all of whose sides are tangent to a circle $\omega$ with centre $O$. Suppose that the circumcircle of triangle $ACE$ is concentric with $\omega$. Let $J$ be the foot of the perpendicular from $B$ to $CD$. Suppose that the perpendicular from $B$ to $DF$ intersects the line $EO$ at a point $K$. Let $L$ be the foot of the perpendicular from $K$ to $DE$. Prove that $DJ=DL$.



Proposed by Japan
2011 IMO Shortlist
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$



Proposed by Suhaimi Ramly, Malaysia
2011 IMO Shortlist
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$



Proposed by Mihai Baluna, Romania
2011 IMO Shortlist
For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences



$$t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)$$ are divisible by 4.



Proposed by Gerhard Wöginger, Austria
2011 IMO Shortlist
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.



Proposed by Oleksiy Klurman, Ukraine
2011 IMO Shortlist
Let $k \in \mathbb{Z}^+$ and set $n=2^k+1.$ Prove that $n$ is a prime number if and only if the following holds: there is a permutation $a_{1},\ldots,a_{n-1}$ of the numbers $1,2, \ldots, n-1$ and a sequence of integers $g_{1},\ldots,g_{n-1},$ such that $n$ divides $g^{a_i}_i - a_{i+1}$ for every $i \in \{1,2,\ldots,n-1\},$ where we set $a_n = a_1.$



Proposed by Vasily Astakhov, Russia
2010 IMO Shortlist
Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$



Proposed by Alex Schreiber, Germany
2010 IMO Shortlist
Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that

$$2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.$$





Proposed by Sung Yun Kim, South Korea
2010 IMO Shortlist
$n \geq 4$ players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players $bad$ if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let $w_i$ and $l_i$ be respectively the number of wins and losses of the $i$-th player. Prove that $$\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.$$



Proposed by Sung Yun Kim, South Korea
2010 IMO Shortlist
Given a positive integer $k$ and other two integers $b > w > 1.$ There are two strings of pearls, a string of $b$ black pearls and a string of $w$ white pearls. The length of a string is the number of pearls on it. One cuts these strings in some steps by the following rules. In each step:



(i) The strings are ordered by their lengths in a non-increasing order. If there are some strings of equal lengths, then the white ones precede the black ones. Then $k$ first ones (if they consist of more than one pearl) are chosen; if there are less than $k$ strings longer than 1, then one chooses all of them.

(ii) Next, one cuts each chosen string into two parts differing in length by at most one. (For instance, if there are strings of $5, 4, 4, 2$ black pearls, strings of $8, 4, 3$ white pearls and $k = 4,$ then the strings of 8 white, 5 black, 4 white and 4 black pearls are cut into the parts $(4,4), (3,2), (2,2)$ and $(2,2)$ respectively.) The process stops immediately after the step when a first isolated white pearl appears.



Prove that at this stage, there will still exist a string of at least two black pearls.



Proposed by Bill Sands, Thao Do, Canada
2010 IMO Shortlist
Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied:



(i) each integer belongs to at least one of them;

(ii) each progression contains a number which does not belong to other progressions.



Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization.



Prove that $$s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).$$



Proposed by Dierk Schleicher, Germany
2010 IMO Shortlist
Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too.



Proposed by Géza Kós, Hungary



[asy]

pathpen=black;

size(400);

pair A=(0,0), B=(4,0), C=(10,0);

draw(L(A,C,0.3));

MP("A",A); MP("B",B); MP("C",C);

pair X=(5,-7);

path G1=D(arc(X,C,A));

pair Y=(5,7), Z=(9,6);

draw(Z--B--Y);

struct T {pair C;real r;};

T f(pair X, pair B, pair Y, pair Z)

{

pair S=unit(Y-B)+unit(Z-B);

real s=abs(sin(angle((Y-B)/(Z-B))/2));

real t=10, r=abs(X-A);

pair Q;

for(int k=0;k<30;++k)

{

Q=B+t*S;

t-=(abs(X-Q)-r)/abs(S)-s*t;

}



T T=new T;

T.C=Q; T.r=s*t*abs(S);

return T;

}

void g(pair Q, real r)

{

real t=0;

for(int k=0;k<30;++k)

{

X=(5,t);

t+=(abs(X-Q)+r-abs(X-A));

}

}

pair Z1=(1.07,6);

draw(B--Z1);

T T=f(X,B,Y,Z1);

draw(CR(T.C,T.r));

T T=f(X,B,Y,Z);

draw(CR(T.C,T.r));

g(T.C,T.r);

path G2=D(arc(X,C,A));

T T=f(X,B,Y,Z1);

draw(CR(T.C,T.r));

T=f(X,B,Y,Z);

draw(CR(T.C,T.r));

g(T.C,T.r);

path G3=D(arc(X,C,A));

pen p=black+fontsize(8);

MC("\gamma_1",G1,0.85,p);

MC("\gamma_2",G2,0.85,NNW,p);

MC("\gamma_3",G3,0.85,WNW,p);

MC("h_1",B--Z1,0.95,E,p);

MC("h_2",B--Y,0.95,E,p);

MC("h_3",B--Z,0.95,E,p);

path[] G={G1,G2,G3};

path[] H={B--Z1,B--Y,B--Z};

pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}};

for(int i=0;i<3;++i)

for(int j=0;j<3;++j)

MP("V_{"+string(i+1)+string(j+1)+"}",IP(H[i],G[j]),al[i][j],fontsize(8));[/asy]
2010 IMO Shortlist
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying $$x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.$$



Proposed by Mariusz Skałba, Poland
2010 IMO Shortlist
The rows and columns of a $2^n \times 2^n$ table are numbered from $0$ to $2^{n}-1.$ The cells of the table have been coloured with the following property being satisfied: for each $0 \leq i,j \leq 2^n - 1,$ the $j$-th cell in the $i$-th row and the $(i+j)$-th cell in the $j$-th row have the same colour. (The indices of the cells in a row are considered modulo $2^n$.) Prove that the maximal possible number of colours is $2^n$.



Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran
2009 IMO Shortlist
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:

$$\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.$$

Proposed by Juhan Aru, Estonia