Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Hong kong National Olympiad
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) +1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.
2008 Hong kong National Olympiad
$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$.



Remark: the original question has missed the condition $ AB \neq AC$
2008 Hong kong National Olympiad
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.
2007 Hong kong National Olympiad
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:

1) $EY$ is perpendicular to $AD$;

2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
2007 Hong kong National Olympiad
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?
2007 Hong kong National Olympiad
There are $2007$ boys and $2007$ girls in a middle school,every student can attend no more than $100$ academic meetings,if we know any pair of students with different sex attend at least one common meeting.prove that there must be a meeting with at least $11$ boys and $11$ girls attend.
2007 Hong kong National Olympiad
find all positive integer pairs $(m,n)$,satisfies:

(1)$gcd(m,n)=1$,and $m\le\ 2007$

(2)for any $k=1,2,...2007$,we have $[\frac{nk}{m}]=[\sqrt{2}k]$
2006 Hong kong National Olympiad
A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$, $x+ y$ does not divide $z$. Determine the largest possible size of $M$.
2006 Hong kong National Olympiad
For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.
2006 Hong kong National Olympiad
A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.
2006 Hong kong National Olympiad
Let $(a_n)_{n\ge 1}$ be a sequence of positive numbers. If there is a constant $M > 0$ such that $a_2^2 + a_2^2 +\ldots  + a_n^2 < Ma_{n+1}^2$ for all $n$, then prove that there is a constant $M ' > 0$ such that $a_1 + a_2 +\ldots + a_n < M ' a_{n+1}$ .
2005 Hong kong National Olympiad
On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.
2005 Hong kong National Olympiad
Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.
2005 Hong kong National Olympiad
Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.
2005 Hong kong National Olympiad
Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that$$ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8}  $$
2004 Hong kong National Olympiad
Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$
2004 Hong kong National Olympiad
In a school there $b$ teachers and $c$ students. Suppose that

a) each teacher teaches exactly $k$ students, and

b)for any two (distinct) students , exactly $h$ teachers teach both of them.

Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.
2004 Hong kong National Olympiad
Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.
2004 Hong kong National Olympiad
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions

a)$f(1)=1$

b)$f$ is bijective

c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
2003 Hong kong National Olympiad
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
2003 Hong kong National Olympiad
Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$.
2003 Hong kong National Olympiad
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.
2003 Hong kong National Olympiad
Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.
2002 Hong kong National Olympiad
Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$.



$(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$.

$(b)$ Prove that these two tangents meet on $KM$.
2002 Hong kong National Olympiad
In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.
2002 Hong kong National Olympiad
Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.
2002 Hong kong National Olympiad
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.
2001 Hong kong National Olympiad
A triangle $ABC$ is given. A circle $\Gamma$, passing through $A$, is tangent to side $BC$ at point $P$ and intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Prove that the smaller arcs $MP$ and $NP$ of $\Gamma$ are equal iff $\Gamma$ is tangent to the circumcircle of $\Delta ABC$ at $A$.
2001 Hong kong National Olympiad
Find, with proof, all positive integers $n$ such that the equation $x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}$ has a solution in positive integers.
2001 Hong kong National Olympiad
Let $k\geq 4$ be an integer number. $P(x)\in\mathbb{Z}[x]$ such that $0\leq P(c)\leq k$ for all $c=0,1,...,k+1$. Prove that $P(0)=P(1)=...=P(k+1)$.
2001 Hong kong National Olympiad
There are $212$ points inside or on a given unit circle. Prove that there are at least $2001$ pairs of points having distances at most $1$.
2000 Hong kong National Olympiad
Let $O$ be the circumcentre of a triangle $ABC$ with $AB > AC > BC$. Let $D$ be a point on the minor arc $BC$ of the circumcircle and let $E$ and $F$ be points on $AD$ such that $AB \perp OE$ and $AC \perp OF$ . The lines $BE$ and $CF$ meet at $P$. Prove that if $PB=PC+PO$, then $\angle BAC = 30^{\circ}$.
2000 Hong kong National Olympiad
Define $a_1=1$ and $a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$ for $n\in\mathbb{N}$. Find the greatest integer not exceeding $a_{2000}$ and prove your claim.
2000 Hong kong National Olympiad
Find all prime numbers $p$ and $q$ such that $\frac{(7^{p}-2^{p})(7^{q}-2^{q})}{pq}$ is an integer.
2000 Hong kong National Olympiad
Find all positive integers $n \ge 3$ such that there exists an $n$-gon with vertices on lattice points of the coordinate plane and all sides of equal length.
1999 Hong kong National Olympiad
Find all positive rational numbers $r\not=1$ such that $r^{\frac{1}{r-1}}$ is rational.
1999 Hong kong National Olympiad
Let $I$ be the incentre and $O$ the circumcentre of a non-equilateral triangle $ABC$. Prove that $\angle AIO \le 90^{\circ}$ if and only if $2BC\le AB+AC$.
1999 Hong kong National Olympiad
Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.
1999 Hong kong National Olympiad
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$f(x+yf(x))=f(x)+xf(y) \quad \text{for all}\ x,y \in\mathbb{R}$$
1998 Hong kong National Olympiad
In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.
1998 Hong kong National Olympiad
The underside of a pyramid is a convex nonagon ,  paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )
1998 Hong kong National Olympiad
Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair ,  A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" .

(1) Is $(s,t)$ a good pair ?

(2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".
1998 Hong kong National Olympiad
Define a function $f$ on positive real numbers to satisfy

$$f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},$$

where $g(x) $ is a function defined on real numbers and for all real numbers $y,z$ and $0\leq t \leq 1$, it satisfies

$$g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).$$

(1) Prove: for any integer $n$ and $0 \leq t \leq 1$, we have

$$t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].$$

(2) Prove that $$\frac{4}{3} \leq  f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.$$
2021 India National Olympiad
Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$.



Proposed by B Sury
2021 India National Olympiad
Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$has all the roots to be integers.



Proposed by Prithwijit De and Sutanay Bhattacharya
2021 India National Olympiad
Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge?



Note. A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.



Proposed by Prithwijit De and Sutanay Bhattacharya
2021 India National Olympiad
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.



Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.



Proposed by Anant Mudgal
2021 India National Olympiad
In a convex quadrilateral $ABCD$, $\angle ABD=30^\circ$, $\angle BCA=75^\circ$, $\angle ACD=25^\circ$ and $CD=CB$. Extend $CB$ to meet the circumcircle of triangle $DAC$ at $E$. Prove that $CE=BD$.



Proposed by BJ Venkatachala
2021 India National Olympiad
Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:



$f$ maps the zero polynomial to itself,

for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and

for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.





Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha
2020 India National Olympiad
Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. The tangent to $\Gamma_1$ at $B$ intersects $\Gamma_2$ again in $C$, different from $B$; the tangent to $\Gamma_2$ at $B$ intersects $\Gamma_1$ again at $D$, different from $B$. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in $X$ and $Y$, respectively. Let $P$ and $Q$ be the circumcentres of triangles $ACD$ and $XAY$, respectively. Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$.



Proposed by Prithwijit De
2020 India National Olympiad
Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$.



Proposed by C.R. Pranesacher
2020 India National Olympiad
Let $S$ be a subset of $\{0,1,2,\cdots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.



Proposed by Sutanay Bhattacharya



Original WordingAs pointed out by Wizard_32, the original wording is:



Let $X=\{0,1,2,\dots,9\}.$ Let $S \subset X$ be such that any positive integer $n$ can be written as $p+q$ where the non-negative integers $p, q$ have all their digits in $S.$ Find the smallest possible number of elements in $S.$
2020 India National Olympiad
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$

Proposed by Kapil Pause
2020 India National Olympiad
Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.



(a) Show that $3, 4, 6$ are frameable.

(b) Show that any integer $n \geqslant 7$ is not frameable.

(c) Determine whether $5$ is frameable.



Proposed by Muralidharan
2020 India National Olympiad
A stromino is a $3 \times 1$ rectangle. Show that a $5 \times 5$ board divided into twenty-five $1 \times 1$ squares cannot be covered by $16$ strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.)



Proposed by Navilarekallu Tejaswi
2019 India National OIympiad
Let $ABC$ be a triangle with $\angle{BAC} > 90$. Let $D$ be a point on the segment $BC$ and $E$ be a point on line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ is perpendicular to $AD$. Given that $CA=CD$ and $AE=CE$. Determine $\angle{BCA}$ in degrees.
2019 India National OIympiad
Let $A_1B_1C_1D_1E_1$ be a regular pentagon.For $ 2 \le n \le 11$, let $A_nB_nC_nD_nE_n$ be the pentagon whose vertices are the midpoint of the sides $A_{n-1}B_{n-1}C_{n-1}D_{n-1}E_{n-1}$. All the $5$ vertices of each of the $11$ pentagons are arbitrarily coloured red or blue. Prove that four points among these $55$ points have the same colour and form the vertices of a cyclic quadrilateral.
2019 India National OIympiad
Let $m,n$ be distinct positive integers. Prove that

$$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$Further, determine when equality holds.
2019 India National OIympiad
Let $n$ and $M$ be positive integers such that $M>n^{n-1}$. Prove that there are $n$ distinct primes $p_1,p_2,p_3 \cdots ,p_n$ such that $p_j$ divides $M + j$ for all $1 \le j \le n$.
2019 India National OIympiad
Let $AB$ be the diameter of a circle $\Gamma$ and let $C$ be a point on $\Gamma$ different from $A$ and $B$. Let $D$ be the foot of perpendicular from $C$ on to $AB$.Let $K$ be a point on the segment $CD$ such that $AC$ is equal to the semi perimeter of $ADK$.Show that the excircle of $ADK$ opposite $A$ is tangent to $\Gamma$.
2019 India National OIympiad
Let $f$ be a function defined from $((x,y) : x,y$ real, $xy\ne 0)$ to the set of all positive real numbers such that

$ (i) f(xy,z)= f(x,z)\cdot f(y,z)$ for all $x,y \ne 0$

$ (ii) f(x,yz)= f(x,y)\cdot f(x,z)$ for all $x,y \ne 0$

$ (iii) f(x,1-x) = 1 $ for all $x \ne 0,1$

Prove that

$ (a) f(x,x) = f(x,-x) = 1$ for all $x \ne 0$

$(b) f(x,y)\cdot f(y,x) = 1 $ for all $x,y \ne 0$



The condition (ii) was left out in the paper leading to an incomplete problem during contest.
2018 India National Olympiad
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta{ABC}$.
2018 India National Olympiad
For any natural number $n$, consider a $1\times n$ rectangular board made up of $n$ unit squares. This is covered by $3$ types of tiles : $1\times 1$ red tile, $1\times 1$ green tile and $1\times 2$ domino. (For example, we can have $5$ types of tiling when $n=2$ : red-red ; red-green ; green-red ; green-green ; and blue.) Let $t_n$ denote the number of ways of covering $1\times n$ rectangular board by these $3$ types of tiles. Prove that, $t_n$ divides $t_{2n+1}$.
2018 India National Olympiad
Let $\Gamma_1$ and $\Gamma_2$ be two circles with respective centres $O_1$ and $O_2$ intersecting in two distinct points $A$ and $B$ such that $\angle{O_1AO_2}$ is an obtuse angle. Let the circumcircle of $\Delta{O_1AO_2}$ intersect $\Gamma_1$ and $\Gamma_2$ respectively in points $C (\neq A)$ and $D (\neq A)$. Let the line $CB$ intersect $\Gamma_2$ in $E$ ; let the line $DB$ intersect $\Gamma_1$ in $F$. Prove that, the points $C, D, E, F$ are concyclic.
2018 India National Olympiad
Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.
2018 India National Olympiad
There are $n\ge 3$ girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbours combined, the teacher takes away one apple from that girl and gives one apple each to her neighbours. Prove that, this process stops after a finite number of steps.

(Assume that, the teacher has an abundant supply of apples.)
2018 India National Olympiad
Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that



$\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$;



$\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$.



Prove that, there exists an odd natural number  $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.
2017 India National Olympiad
In the given figure, $ABCD$ is a square sheet of paper. It is folded along $EF$ such that $A$ goes to a point $A'$ different from $B$ and $C$, on the side $BC$ and $D$ goes to $D'$. The line $A'D'$ cuts $CD$ in $G$. Show that the inradius of the triangle $GCA'$ is the sum of the inradii of the triangles $GD'F$ and $A'BE$.



[asy]

size(5cm);

pair A=(0,0),B=(1,0),C=(1,1),D=(0,1),Ap=(1,0.333),Dp,Ee,F,G;

Ee=extension(A,B,(A+Ap)/2,bisectorpoint(A,Ap));

F=extension(C,D,(A+Ap)/2,bisectorpoint(A,Ap));

Dp=reflect(Ee,F)*D;

G=extension(C,D,Ap,Dp);

D(MP("A",A,W)--MP("E",Ee,S)--MP("B",B,E)--MP("A^{\prime}",Ap,E)--MP("C",C,E)--MP("G",G,NE)--MP("D^{\prime}",Dp,N)--MP("F",F,NNW)--MP("D",D,W)--cycle,black);

draw(Ee--Ap--G--F);

dot(A);dot(B);dot(C);dot(D);dot(Ap);dot(Dp);dot(Ee);dot(F);dot(G);

draw(Ee--F,dashed);

[/asy]
2017 India National Olympiad
Suppose $n \ge 0$ is an integer and all the roots of $x^3 + 

\alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$.
2017 India National Olympiad
Find the number of triples $(x, a, b)$ where $x$ is a real numbe and $a$, $b$ belong to the set $\{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$ such that $$x^2 - a\{x\} + b = 0,$$where $\{x\}$ denotes the fractional part of the real number $x$. (For example $\{1.1\} = .1 = \{-0.9\}$.)
2017 India National Olympiad
Let $ABCDE$ be a convex pentagon in which $\angle{A}=\angle{B}=\angle{C}=\angle{D}=120^{\circ}$ and the side lengths are five consecutive integers in some order. Find all possible values of $AB+BC+CD$.
2017 India National Olympiad
Let $ABC$ be a triangle with $\angle{A}=90^{\circ}$ and $AB<AC$. Let $AD$ be the altitude from $A$ on to $BC$, Let $P,Q$ and $I$ denote respectively the incentres of triangle $ABD,ACD$ and $ABC$. Prove that $AI$ is perpendicular to $PQ$ and $AI=PQ$.
2017 India National Olympiad
Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.
2016 India National Olympiad
Let $ABC$ be a triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$.
2016 India National Olympiad
For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.
2016 India National Olympiad
Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\rightarrow\mathbb{N}$ by $T(2k)=k$ and $T(2k+1)=2k+2$. We write $T^2(n)=T(T(n))$ and in general $T^k(n)=T^{k-1}(T(n))$ for any $k>1$.



(i) Show that for each $n\in\mathbb{N}$, there exists $k$ such that $T^k(n)=1$.



(ii) For $k\in\mathbb{N}$, let $c_k$ denote the number of elements in the set $\{n: T^k(n)=1\}$. Prove that $c_{k+2}=c_{k+1}+c_k$, for $k\ge 1$.
2016 India National Olympiad
Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.