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.