# 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.