2 Find all \((m,n\in\mathbb N)\) and primes \((p\geq 5)\) satisfying

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

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.

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.

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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