Navvye/Olympiad-Problems-CSV huggingface dataset:
1991 Vietnam Team Selection Test
For every natural number $n$  we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) =  f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.
1991 Vietnam Team Selection Test
Let a set $X$ be given which consists of $2 \cdot n$ distinct real numbers ($n \geq 3$). Consider a set $K$ consisting of some pairs $(x, y)$ of distinct numbers $x, y \in X$, satisfying the two conditions:



I. If $(x, y) \in K$ then $(y, x) \not \in K$.

II. Every number $x \in X$ belongs to at most 19 pairs of $K$.



Show that we can divide the set $X$ into 5 non-empty disjoint sets $X_1, X_2, X_3, X_4, X_5$ in such a way that for each $i = 1, 2, 3, 4, 5$ the number of pairs $(x, y) \in K$ where $x, y$ both belong to $X_i$ is not greater than $3 \cdot n$.
1990 Vietnam Team Selection Test
Let be given a convex polygon $ M_0M_1\ldots M_{2n}$ ($ n\ge 1$), where $ 2n + 1$ points $ M_0$, $ M_1$, $ \ldots$, $ M_{2n}$ lie on a circle $ (C)$ with diameter $ R$ in an anticlockwise direction. Suppose that there is a point $ A$ inside this convex polygon such that $ \angle M_0AM_1$, $ \angle M_1AM_2$, $ \ldots$, $ \angle M_{2n - 1}AM_{2n}$, $ \angle M_{2n}AM_0$ are equal. Assume that $ A$ is not coincide with the center of the circle $ (C)$ and $ B$ be a point lies on $ (C)$ such that $ AB$ is perpendicular to the diameter of $ (C)$ passes through $ A$. Prove that

$$ \frac {2n + 1}{\frac {1}{AM_0} + \frac {1}{AM_1} + \cdots + \frac {1}{AM_{2n}}} < AB < \frac {AM_0 + AM_1 + \cdots + AM_{2n}}{2n + 1} < R

$$
1990 Vietnam Team Selection Test
Let be given four positive real numbers $ a$, $ b$, $ A$, $ B$. Consider a sequence of real numbers $ x_1$, $ x_2$, $ x_3$, $ \ldots$ is given by $ x_1 = a$, $ x_2 = b$ and $ x_{n + 1} = A\sqrt [3]{x_n^2} + B\sqrt [3]{x_{n - 1}^2}$ ($ n = 2, 3, 4, \ldots$). Prove that there exist limit $ \lim_{n\to + \propto}x_n$ and find this limit.
1990 Vietnam Team Selection Test
Prove that there is no real function $ f(x)$ satisfying $ f\left(f(x)\right) = x^2 - 2$ for all real number $ x$.
1990 Vietnam Team Selection Test
Let $ T$ be a finite set of positive integers, satisfying the following conditions:

1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$.

2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$.



For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.
1990 Vietnam Team Selection Test
Given a tetrahedron such that product of the opposite edges is $ 1$. Let the angle between the opposite edges be $ \alpha$, $ \beta$, $ \gamma$, and circumradii of four faces be $ R_1$, $ R_2$, $ R_3$, $ R_4$. Prove that

$$ \sin^2\alpha + \sin^2\beta + \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}}

$$
1990 Vietnam Team Selection Test
There are $n\geq 3$ pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number $M$ such that after $M$ times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils $ A$ and $ B$, and if $ A$ initially stands on the left side of $ B$ then $ A$ will finally stand on the right side of $ B$.
1985 Vietnam Team Selection Test
The sequence $ (x_n)$ of real numbers is defined by $ x_1=\frac{29}{10}$ and $ x_{n+1}=\frac{x_n}{\sqrt{x_n^2-1}}+\sqrt{3}$ for all  $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k-1}>a>x_{2k}$.
1985 Vietnam Team Selection Test
Let $ ABC$ be a triangle with $ AB = AC$. A ray $ Ax$ is constructed in space such that the three planar angles of the trihedral angle $ ABCx$ at its vertex $ A$ are equal. If a point $ S$ moves on $ Ax$, find the locus of the incenter of triangle $ SBC$.
1985 Vietnam Team Selection Test
Does there exist a triangle $ ABC$ satisfying the following two conditions:

(a) ${ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C}$

(b) $ S\ge a^2 - (b - c)^2$ where $ S$ is the area of the triangle $ ABC$.
1985 Vietnam Team Selection Test
A convex polygon $ A_1,A_2,\cdots ,A_n$ is inscribed in  a circle with center $ O$ and radius $ R$ so that $ O$ lies inside the polygon. Let the inradii of the triangles $ A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n - 1}A_n$ be denoted by $ r_1,r_2,\cdots ,r_{n - 2}$. Prove that  $ r_1 + r_2 + ... + r_{n - 2}\leq R(n\cos \frac {\pi}{n} - n + 2)$.
1985 Vietnam Team Selection Test
Find all real values of a for which the equation $ (a - 3x^2 + \cos \frac {9\pi x}{2})\sqrt {3 - ax} = 0$ has an odd number of solutions in the interval $ [ - 1,5]$
1985 Vietnam Team Selection Test
Suppose a function  $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) = - x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.
2021-IMO Problems
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
2021-IMO Problems
Show that the inequality $$\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}$$holds for all real numbers $x_1,\ldots x_n.$
2021-IMO Problems
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.
2021-IMO Problems
Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that $$A D+D T+T X+X A=C D+D Y+Y Z+Z C.$$

Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland
2021-IMO Problems
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.



Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.
2021-IMO Problems
Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.
2020-IMO
Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold:

$$\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC$$Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.



Proposed by Dominik Burek, Poland
2020-IMO
The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that

$$(a+2b+3c+4d)a^ab^bc^cd^d<1$$Proposed by Stijn Cambie, Belgium
2020-IMO
There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:



The total weights of both piles are the same.

Each pile contains two pebbles of each colour.



Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA
2020-IMO
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies.



Proposed by Tejaswi Navilarekallu, India
2020-IMO
A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.

For which $n$ does it follow that the numbers on the cards are all equal?



Proposed by Oleg Košik, Estonia
2020-IMO
Prove that there exists a positive constant $c$ such that the following statement is true:

Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$.



(A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.)



Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.



Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan
2019 IMO
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
2019 IMO
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.



Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.



Proposed by Anton Trygub, Ukraine
2019 IMO
A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time:



Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged.



Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.



Proposed by Adrian Beker, Croatia
2019 IMO
Find all pairs $(k,n)$ of positive integers such that $$ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). $$Proposed by Gabriel Chicas Reyes, El Salvador
2019 IMO
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT  \to HTT \to TTT$, which stops after three operations.



(a) Show that, for each initial configuration, Harry stops after a finite number of operations.



(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.



Proposed by David Altizio, USA
2019 IMO
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.



Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.



Proposed by Anant Mudgal, India
2018 IMO
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.



Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece
2018 IMO
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and

$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.



Proposed by Patrik Bak, Slovakia
2018 IMO
An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$.

$$\begin{array}{

c@{\hspace{4pt}}c@{\hspace{4pt}}

c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c

} \vspace{4pt}

 & & & 4 & & &  \\\vspace{4pt}

 & & 2 & & 6 & &  \\\vspace{4pt}

 & 5 & & 7 & & 1 & \\\vspace{4pt}

 8 & & 3 & & 10 & & 9 \\\vspace{4pt}

\end{array}$$Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$?



Proposed by Morteza Saghafian, Iran
2018 IMO
A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.



Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.



Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.



Proposed by Gurgen Asatryan, Armenia
2018 IMO
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number

$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.



Proposed by Bayarmagnai Gombodorj, Mongolia
2018 IMO
A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that $$\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.$$Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$.



Proposed by Tomasz Ciesla, Poland
2017 IMO
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as

$$a_{n+1} = 

\begin{cases}

\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}

\end{cases}

$$Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.



Proposed by Stephan Wagner, South Africa
2017 IMO
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, $$ f(f(x)f(y)) + f(x+y) = f(xy). $$

Proposed by Dorlir Ahmeti, Albania
2017 IMO
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order:



The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$



A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$



The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$



Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$



Proposed by Gerhard Woeginger, Austria
2017 IMO
Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$.



Proposed by Charles Leytem, Luxembourg
2017 IMO
An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold:

($1$) no one stands between the two tallest players,

($2$) no one stands between the third and fourth tallest players,

$\;\;\vdots$

($N$) no one stands between the two shortest players.



Show that this is always possible.



Proposed by Grigory Chelnokov, Russia
2017 IMO
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have:

$$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$

Proposed by John Berman, United States
2016 IMO
Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.
2016 IMO
Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:



in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and 

in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.



Note. The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$  for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.
2016 IMO
Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.
2016 IMO
A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements.  Let $P(n)=n^2+n+1$.  What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$is fragrant?
2016 IMO
The equation

$$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
2016 IMO
There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.



(a) Prove that Geoff can always fulfill his wish if $n$ is odd.



(b) Prove that Geoff can never fulfill his wish if $n$ is even.
2015 IMO
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.



(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.



(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.



Proposed by Netherlands
2015 IMO
Find all positive integers $(a,b,c)$ such that

$$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$.



Proposed by Serbia