Indian National Math Olympiad 2013

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.  

Sketch of the solution

2     Find all \(m,n\in\mathbb N\) and primes \(p\geq 5\) satisfying
\(m(4m^2+m+12)=3(p^n-1)\).    

Sketch of the solution

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.   

Sketch of the solution

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.    

Sketch of the solution

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 \(\angle C\).    

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.