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
\((m(4m^2+m+12)=3(p^n-1))\).

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.

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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.

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