Indian National Math Olympiad, INMO 2019 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2019. Try them and share your solution in the comments.

INMO 2019, Problem 1

Let $ABC$ be a triangle with $\angle BAC > 90^\circ$ . Let $D$ be a point on the line segment $BC$ and $E$ be a point on the line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ perpendicular to $AD$ . Given that $CA=CD$ and $AE=CE$ , determine $\angle BCA$ in degrees

INMO 2019, Problem 2

Let ${A_1}{B_1}{C_1}{D_1}{E_1}$ be a regular pentagon. For $2\leq n \leq 11$ , let ${A_n}{B_n}{C_n}{D_n}{E_n}$ whose vertices are the midpoints of the sides of ${A_{n-1}}{B_{n-1} }{C_{n-1} }{D_{n-1}}{E_{n-1}}$ . All the $5$ vertices of each of the $11$ pentagons are arbitrarily coloured red or blue. Prove that four points among these $55$ have the same colour and form the vertices of a cyclic quadrilateral.

INMO 2019, Problem 3

Let $m$ , $n$ be distinct positive integers. Prove that $\gcd (m,n) + \gcd (m+1,n+1) + \gcd (m+2,n+2) \leq 2|m-n|+1$ . Further, determine when equality holds.

INMO 2019, Problem 4

Let $n$ and $M$ be positive integers such that $M>n^{n-1}$ . Prove that there are $n$ distinct primes $p_1$ , $,<script type="math/tex" id="MathJax-Element-31">,....,pn<script type="math/tex" id="MathJax-Element-32">p_n$ such that $p_j$ divides $M+j$ for $1 \leq j \leq n$ .

INMO 2019, Problem 5

Let $AB$ be a 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 of the segment $CD$ such that $AC$ is equal to the semiperimeter of the triangle $ADK$ . Show that the excircle of triangle $ADK$ opposite $A$ is tangent to $\Gamma$ .

INMO 2019, Problem 6

Let $f$ be a function defined from the set $\{(x,y) : x,y$ are real, $xy \neq 0\}$ to the set of all positive real number such that

(i)$f(x y, z)=f(x, z) f(y, z),$ for all $x, y \neq 0$
(ii)$\quad f(x, y z)=f(x, y) f(x, z),$ for all $x, y \neq 0$
(iii)$f(x, 1-x)=1,$ for all $x \neq 0,1$
Prove that
(a) $\quad f(x, x)=f(x,-x)=1,$ for all $x \neq 0$
(b) $f(x, y) f(y, x)=1,$ for all $x, y \neq 0$