India National Olympiad 2013 Problems
Problem 1
Let $\Gamma_1$ and $\Gamma_2$ be two circles touching each other externally at $R.$ Let $O_1$ and $O_2$ be the centres of $\Gamma_1$ and $\Gamma_2,$ respectively. Let $\ell_1$ be a line which is tangent to $\Gamma_2$ at $P$ and passing through $O_1,$ and let $\ell_2$ be the line tangent to $\Gamma_1$ at $Q$ and passing through $O_2.$ Let $K=\ell_1\cap \ell_2.$ If $KP=KQ$ then prove that the triangle $PQR$ is equilateral.
Problem 2
Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying
\[m(4m^2+m+12)=3(p^n-1).\]
Problem 3
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 – ax^3 – bx^2 – cx -d = 0$ has no integer solution.
Problem 4
Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,…..,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n – n$ is always even.
Problem 5
In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$
Problem 6
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$
