# 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 $latex (\Gamma_1)$ 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.

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

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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.

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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.

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5     In an acute triangle ABC, let O,G,H be its circumcentre, centroid and orthocenter. Let $latex (D\in BC, E\in CA)$ 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.