India National Olympiad 2019 Problems
Problem 1
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.
Problem 2
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.
Problem 3
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.
Problem 4
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 $1 \le j \le n$.
Problem 5
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 $\Gamma$.
Problem 6
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.
