Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Hong Kong TST
Let $p$ be a prime number greater than 10. Prove that there exist positive integers $m$ and $n$ such that $m+n < p$ and $5^m 7^n-1$ is divisible by $p$.
2019 Hong Kong TST
Let $\Gamma_1$ and $\Gamma_2$ be two circles with different radii, with $\Gamma_1$ the smaller one. The two circles meet at distinct points $A$ and $B$. $C$ and $D$ are two points on the circles $\Gamma_1$ and $\Gamma_2$, respectively, and such that $A$ is the midpoint of $CD$. $CB$ is extended to meet $\Gamma_2$ at $F$, while $DB$ is extended to meet $\Gamma_1$ at $E$. The perpendicular bisector of $CD$ and the perpendicular bisector of $EF$ meet at $P$.



(a) Prove that $\angle{EPF} = 2\angle{CAE}$.

(b) Prove that $AP^2 = CA^2 + PE^2$.
2019 Hong Kong TST
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.
2019 Hong Kong TST
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$for all $x,y\in\mathbb{Q}_{>0}$
2019 Hong Kong TST
Find the maximal value of

$$S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},$$where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.



Proposed by Evan Chen, Taiwan
2018 Hong Kong TST
Find all positive integer(s) $n$ such that $n^2+32n+8$ is a perfect square.
2018 Hong Kong TST
Find all polynomials $f$ such that $f$ has non-negative integer coefficients, $f(1)=7$ and $f(2)=2017$.
2018 Hong Kong TST
In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.
2018 Hong Kong TST
Find infinitely many positive integers $m$ such that for each $m$, the number $\dfrac{2^{m-1}-1}{8191m}$ is an integer.
2018 Hong Kong TST
In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.
2018 Hong Kong TST
A triangle $ABC$ has its orthocentre $H$ distinct from its vertices and from the circumcenter $O$ of $\triangle ABC$. Denote by $M, N$ and $P$ respectively the circumcenters of triangles $HBC, HCA$ and $HAB$. Show that the lines $AM, BN, CP$ and $OH$ are concurrent.
2018 Hong Kong TST
Let $ABC$ be a triangle with $AB=AC$. A circle $\Gamma$ lies outside triangle $ABC$ and is tangent to line $AC$ at $C$. Point $D$ lies on $\Gamma$ such that the circumcircle of triangle $ABD$ is internally tangent to $\Gamma$. Segment $AD$ meets $\Gamma$ secondly at $E$. Prove that $BE$ is tangent to $\Gamma$
2018 Hong Kong TST
There are three piles of coins, with $a,b$ and $c$ coins respectively, where $a,b,c\geq2015$ are positive integers. The following operations are allowed:



(1) Choose a pile with an even number of coins and remove all coins from this pile. Add coins to each of the remaining two piles with amount equal to half of that removed; or



(2) Choose a pile with an odd number of coins and at least 2017 coins. Remove 2017 coins from this pile. Add 1009 coins to each of the remaining two piles.



Suppose there are sufficiently many spare coins. Find all ordered triples $(a,b,c)$ such that after some finite sequence of allowed operations. There exists a pile with at least $2017^{2017}$ coins.
2018 Hong Kong TST
Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that



$$f(f(xy-x))+f(x+y)=yf(x)+f(y)$$

for all real numbers $x$ and $y$.
2018 Hong Kong TST
In triangle $ABC$ with incentre $I$, let $M_A,M_B$ and $M_C$ by the midpoints of $BC, CA$ and $AB$ respectively, and $H_A,H_B$ and $H_C$ be the feet of the altitudes from $A,B$ and $C$ to the respective sides. Denote by $\ell_b$ the line being tangent tot he circumcircle of triangle $ABC$ and passing through $B$, and denote by $\ell_b'$ the reflection of $\ell_b$ in $BI$. Let $P_B$ by the intersection of $M_AM_C$ and $\ell_b$, and let $Q_B$ be the intersection of $H_AH_C$ and $\ell_b'$. Defined $\ell_c,\ell_c',P_C,Q_C$ analogously. If $R$ is the intersection of $P_BQ_B$ and $P_CQ_C$, prove that $RB=RC$.
2018 Hong Kong TST
Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?
2018 Hong Kong TST
Given triangle $ABC$, let $D$ be an inner point of segment $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC, L=CP\cap AB, E=BQ\cap AC, F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.
2018 Hong Kong TST
On a rectangular board with $m$ rows and $n$ columns, where $m\leq n$, some squares are coloured black in a way that no two rows are alike. Find the biggest integer $k$ such that for every possible colouring to start with one can always color $k$ columns entirely red in such a way that no two rows are still alike.
2018 Hong Kong TST
The altitudes $AD$ and $BE$ of acute triangle $ABC$ intersect at $H$. Let $F$ be the intersection of $AB$ and a line that is parallel to the side $BC$ and goes through the circumcentre of $ABC$. Let $M$ be the midpoint of $AH$. Prove that $\angle CMF=90^\circ$
2018 Hong Kong TST
For which natural number $n$ is it possible to place natural number from 1 to $3n$ on the edges of a right $n$-angled prism (on each edge there is exactly one number placed and each one is used exactly 1 time) in such a way, that the sum of all the numbers, that surround each face is the same?
2018 Hong Kong TST
Find all primes $p$ and all positive integers $a$ and $m$ such that $a\leq 5p^2$ and $(p-1)!+a=p^m$
2017 Hong Kong TST
In $\triangle ABC$, let $AD$ be the angle bisector of $\angle BAC$, with $D$ on $BC$. The perpendicular from $B$ to $AD$ intersects the circumcircle of $\triangle ABD$ at $B$ and $E$. Prove that $E$, $A$ and the circumcenter $O$ of $\triangle ABC$ are collinear.
2017 Hong Kong TST
In a committee there are $n$ members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of $n$
2017 Hong Kong TST
Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.)
2017 Hong Kong TST
Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.
2017 Hong Kong TST
Find the first digit after the decimal point of the number $\displaystyle \frac1{1009}+\frac1{1010}+\cdots + \frac1{2016}$
2017 Hong Kong TST
Given infinite sequences $a_1,a_2,a_3,\cdots$ and $b_1,b_2,b_3,\cdots$ of real numbers satisfying $\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$ and $\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}$ for all $n\geq1$. Suppose $b_{2016}=1$ and $a_1>0$. Find all possible values of $a_1$
2017 Hong Kong TST
Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$
2017 Hong Kong TST
Let $ABCDEF$ be a convex hexagon such that $\angle ACE = \angle BDF$ and $\angle BCA = \angle EDF$. Let $A_1=AC\cap FB$, $B_1=BD\cap AC$, $C_1=CE\cap BD$, $D_1=DF\cap CE$, $E_1=EA\cap DF$, and $F_1=FB\cap EA$. Suppose $B_1, C_1, D_1, F_1$ lie on the same circle $\Gamma$. The circumcircles of $\triangle BB_1F_1$ and $ED_1F_1$ meet at $F_1$ and $P$. The line $F_1P$ meets $\Gamma$ again at $Q$. Prove that $B_1D_1$ and $QC_1$ are parrallel. (Here, we use $l_1\cap l_2$ to denote the intersection point of lines $l_1$ and $l_2$)
2017 Hong Kong TST
Let $G$ be a simple graph with $n$ vertices and $m$ edges. Two vertices are called neighbours if there is an edge between them. It turns out the $G$ does not contain any cycles of length from 3 to $2k$ (inclusive), where $k\geq2$ is a given positive integer.



a) Prove that it is possible to pick a non-empty set $S$ of vertices of $G$ such that every vertex in $S$ has at least $\left\lceil \frac mn \right\rceil$ neighbours that are in $S$. ($\lceil x\rceil$ denotes the smallest integer larger than or equal to $x$.)



b) Suppose a set $S$ as described in (a) is chosen. Let $H$ be the graph consisting of the vertices in $S$ and the edges between those vertices only. Let $v$ be a vertex of $H$. Prove that at least $\left\lceil \left(\frac mn -1\right)^k \right\rceil$ vertices of $H$ can be reached by starting at $v$ and travelling across the edges of $H$ for at most $k$ steps. (Note that $v$ itself satisfies this condition, since it can be reached by starting at $v$ and travelling along the edges of $H$ for 0 steps.)
2017 Hong Kong TST
Let $n$ be a positive integer with the following property: $2^n-1$ divides a number of the form $m^2+81$, where $m$ is a positive integer. Find all possible $n$.
2017 Hong Kong TST
a) Do there exist 5 circles in the plane such that each circle passes through exactly 3 centers of other circles?



b) Do there exist 6 circles in the plane such that each circle passes through exactly 3 centers of other circles?
2017 Hong Kong TST
Suppose all of the 200 integers lying in between (and including) 1 and 200 are written on a blackboard. Suppose we choose exactly 100 of these numbers and circle each one of them. By the score of such a choice, we mean the square of the difference between the sum of the circled numbers and the sum of the non-circled numbers. What is the average scores over all possible choices for 100 numbers?
2017 Hong Kong TST
At a mathematical competition $n$ students work on 6 problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is 0 or 2. Find the maximum possible value of $n$.
2017 Hong Kong TST
Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$
2017 Hong Kong TST
Two circles $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ intersects the circle $\omega$ through $AO_1O_2$ at $F$. Prove that the length of segment $EF$ is equal to the diameter of $\omega$.
2017 Hong Kong TST
Let a sequence of real numbers $a_0, a_1,a_2, \cdots$ satisfies the condition:



$$\sum_{n=0}^ma_n\cdot(-1)^n\cdot{m\choose n}=0$$

for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\geq 0$
2016 Hong Kong TST
Find all natural numbers $n$ such that $n$, $n^2+10$, $n^2-2$, $n^3+6$, and $n^5+36$ are all prime numbers.
2016 Hong Kong TST
Find the largest possible positive integer $n$ , so that there exist$n$ distinct positive real numbers $x_1,x_2,...,x_n$ satisfying the following inequality : for any  $1\le i,j \le n,$ $(3x_i-x_j) (x_i-3x_j)\geq (1-x_ix_j)^2$
2016 Hong Kong TST
Let $ABC$ be a triangle such that $AB \neq AC$. The incircle with centre $I$ touches $BC$ at $D$. Line $AI$ intersects the circumcircle $\Gamma$ of $ABC$ at $M$, and $DM$ again meets $\Gamma$ at $P$. Find $\angle API$
2016 Hong Kong TST
Find all triples $(m,p,q)$ such that

\begin{align*}

2^mp^2 +1=q^7,

\end{align*}where $p$ and $q$ are ptimes and $m$ is a positive integer.
2016 Hong Kong TST
Let $ABCD$ be inscribed in a circle with center $O$. Let $E$ be the intersection of $AC$ and $BD$. $M$ and $N$ are the midpoints of the arcs $AB$ and $CD$ respectively (the arcs not containing any other vertices). Let $P$ be the intersection point of $EO$ and $MN$. Suppose $BC=5$, $AC=11$, $BD=12$, and $AD=10$. Find $\frac{MN}{NP}$
2016 Hong Kong TST
4031 lines are drawn on a plane, no two parallel or perpendicular, and no three lines meet at a point. Determine the maximum number of acute-angled triangles that may be formed.
2016 Hong Kong TST
During a school year 44 competitions were held. Exactly 7 students won in each of the competition. For any two competitions, there exists exactly 1 student who won both competitions. Is it true that there exists a student who won all the competitions?
2016 Hong Kong TST
Let $\Gamma$ be a circle and $AB$ be a diameter. Let $l$ be a line outside the circle, and is perpendicular to $AB$. Let $X$, $Y$ be two points on $l$. If $X'$, $Y'$ are two points on $l$ such that $AX$, $BX'$ intersect on $\Gamma$ and such that $AY$, $BY'$ intersect on $\Gamma$. Prove that the circumcircles of triangles $AXY$ and $AX'Y'$ intersect at a point on $\Gamma$ other than $A$, or the three circles are tangent at $A$.
2016 Hong Kong TST
Let $a,b,c$ be positive real numbers satisfying $abc=1$. Determine the smallest possible value of



$$\frac{a^3+8}{a^3(b+c)}+\frac{b^3+8}{b^3(a+c)}+\frac{c^3+8}{c^3(b+a)}$$
2016 Hong Kong TST
Mable and Nora play a game according to the following steps in order:



1. Mable writes down any 2015 distinct prime numbers in ascending order in a row. The product of these primes is Marble's score.



2. Nora writes down a positive integer



3. Mable draws a vertical line between two adjacent primes she has written in step 1, and compute the product of the prime(s) on the left of the vertical line



4. Nora must add the product obtained by Marble in step 3 to the number she has written in step 2, and the sum becomes Nora's score.



If Marble and Nora's scores have a common factor greater than 1, Marble wins, otherwise Nora wins.



Who has a winning strategy?
2016 Hong Kong TST
Find all prime numbers $p$ and $q$ such that $p^2|q^3+1$ and $q^2|p^6-1$
2016 Hong Kong TST
Suppose that $I$ is the incenter of triangle $ABC$. The perpendicular to line $AI$ from point $I$ intersects sides $AC$ and $AB$ at points $B'$ and $C'$ respectively. Points $B_1$ and $C_1$ are placed on half lines $BC$ and $CB$ respectively, in such a way that $AB=BB_1$ and $AC=CC_1$. If $T$ is the second intersection point of the circumcircles of triangles $AB_1C'$ and $AC_1B'$, prove that the circumcenter of triangle $ATI$ lies on the line $BC$
2016 Hong Kong TST
2016 circles with radius 1 are lying on the plane. Among these 2016 circles, show that one can select a collection $C$ of 27 circles satisfying the following: either every pair of two circles in $C$ intersect or every pair of two circles in $C$ does not intersect.
2016 Hong Kong TST
Let $O$ be the circumcenter of a triangle $ABC$, and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$. Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$.
2016 Hong Kong TST
Determine all positive integers $n$ for which there exist pairwise distinct positive real numbers $a_1, a_2, \cdots, a_n$ satisfying $\displaystyle \left\{a_i+\frac{(-1)^i}{a_i}\mid 1\leq i \leq n\right\}=\{a_i\mid 1\leq i \leq n\}$
2016 Hong Kong TST
Let $p$ be a prime number greater than 5. Suppose there is an integer $k$ satisfying that $k^2+5$ is divisible by $p$. Prove that there are positive integers $m$ and $n$ such that $p^2=m^2+5n^2$
2008 Hong Kong TST
Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of $$ \sin\alpha_1\cos\alpha_2 + \sin\alpha_2\cos\alpha_3 + \cdots + \sin\alpha_{2007}\cos\alpha_{2008} + \sin\alpha_{2008}\cos\alpha_1$$
2008 Hong Kong TST
Find the total number of solutions to the following system of equations: $$ \begin{cases} a^2+bc\equiv a\pmod {37}\\ b(a+d)\equiv b\pmod {37}\\ c(a+d)\equiv c\pmod{37}\\ bc+d^2\equiv d\pmod{37}\\ ad-bc\equiv 1\pmod{37}\end{cases}$$
2008 Hong Kong TST
Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE = A'$, $ CE \cap DA = B'$, $ DA\cap EB = C'$, $ EB\cap AC = D'$ and $ AC \cap BD = E'$. Suppose also that $ (ABD')\cap (AC'E) = A''$, $ (BCE')\cap (BD'A) = B''$, $ (CDA')\cap (CE'B) = C''$, $ (DEB')\cap DA'C = D''$ and $ (EAC')\cap (EB'D) = E''$. Prove that $ AA''$, $ BB''$, $ CC''$, $ DD''$ and $ EE''$ are concurrent.
2008 Hong Kong TST
In a school there are $ 2008$ students. Students are members of certain committees. A committee has at most $ 1004$ members and every two students join a common committee.

(i) Determine the smallest possible number of committees in the school.

(ii) If it is further required that the union of any two committees consists of at most $ 1800$ students, will your answer in (i) still hold?
2008 Hong Kong TST
Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of $$ \frac{a^2+b^2+c^2}{ab+bc+ca}$$
2008 Hong Kong TST
Show that the equation $ y^{37} = x^3 +11\pmod p$ is solvable for every prime $ p$, where $ p\le 100$.
2008 Hong Kong TST
Let $ f: Z \to Z$ be such that $ f(1) = 1, f(2) = 20, f(-4) = -4$ and $ f(x+y) = f(x) +f(y)+axy(x+y)+bxy+c(x+y)+4 \forall x,y \in Z$, where $ a,b,c$ are constants.



(a) Find a formula for $ f(x)$, where $ x$ is any integer.

(b) If $ f(x) \geq mx^2+(5m+1)x+4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.
2008 Hong Kong TST
Define a $ k$-clique to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.
2008 Hong Kong TST
Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}+4y^{4}+5z^{3}-y^{4}z \equiv 0$ is $ p^2$
2008 Hong Kong TST
Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB = 90^{\circ}$.



(a) Show that all such lines $ AB$ are concurrent.

(b) Find the locus of midpoints of all such segments $ AB$.
1994 Hong Kong TST
In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.
1994 Hong Kong TST
In a table-tennis tournament of $10$ contestants, any $2$ contestants meet only once.

We say that there is a winning triangle if the following situation occurs: $i$-th contestant defeated the $j$-th contestant, $j$-th contestant defeated the $k$-th contestant, and, $k$-th contestant defeated the $i$-th contestant.

Let, $W_i$ and $L_i $ be respectively the number of games won and lost by the $i$-th contestant.

Suppose, $L_i+W_j\geq 8$ whenever the $j$-th contestant defeats the $i$-th contestant.

Prove that, there are exactly $40$ winning triangles in this tournament.
1994 Hong Kong TST
Find all non-negative integers $x, y$ and $z$ satisfying the equation: $$7^{x}+1=3^{y}+5^z$$
1994 Hong Kong TST
Suppose, $x, y, z \in \mathbb{R}_+$ such that $xy+yz+zx=1$. Prove that, $$x(1-y^2)(1-z^2)+y(1-z^2)(1-x^2)+z(1-x^2)(1-y^2)\leq \frac{4\sqrt{3}}{9}$$
1994 Hong Kong TST
Given that, a function $f(n)$, defined on the natural numbers, satisfies the following conditions: (i)$f(n)=n-12$ if $n>2000$; (ii)$f(n)=f(f(n+16))$ if $n \leq 2000$.

(a) Find $f(n)$.

(b) Find all solutions to $f(n)=n$.
1994 Hong Kong TST
Let $m$ and $n$ be positive integers where $m$ has $d$ digits in base ten and $d\leq n$. Find the sum of all the digits (in  base ten) of the product $(10^n-1)m$.
2019 India IMO Training Camp
In an acute angled triangle $ABC$ with $AB < AC$, let $I$ denote the incenter and $M$ the midpoint of side $BC$. The line through $A$ perpendicular to $AI$ intersects the tangent from $M$ to the incircle (different from line $BC$) at a point $P$> Show that $AI$ is tangent to the circumcircle of triangle $MIP$.



Proposed by Tejaswi Navilarekallu
2019 India IMO Training Camp
Show that there do not exist natural numbes $a_1, a_2, \cdots, a_{2018}$ such that the numbers $$ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \cdots, (a_{2018})^{2018}+a_1 $$are all powers of $5$



Proposed by Tejaswi Navilarekallu
2019 India IMO Training Camp
Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two sepearate tilings of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours.



Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$



 Proposed by Tejaswi Navilarekallu
2019 India IMO Training Camp
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
2019 India IMO Training Camp
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:



(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;



(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2019 India IMO Training Camp
Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$.



Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$.  Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$.



Proposed by Anant Mudgal
2019 India IMO Training Camp
Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.
2019 India IMO Training Camp
Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$



 Proposed by Anant Mudgal
2019 India IMO Training Camp
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$for all $x,y>0$.
2019 India IMO Training Camp
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.