Let ABC be an acute angled scalene triangle with circumcenter O and orthocenter H. If M is the midpoint of BC, then show that AO and HM intersect at the circumcircle of ABC.

Let n be a positive integer such that 2n + 1 and 3n + 1 are both perfect squares. Show that 5n + 3 is a composite numbers.

Let a, b, c > 0. If 1/a , 1/b and 1/c are in arithmetic progression, and if \(a^2+b^2 , b^2+c^2 , c^2+a^2 \) are in geometric progression, prove that a=b=c.

Find the number of 4 digit numbers with distinct digits chosen from the set {0, 1, 2, 3, 4, 5} in which no two adjacent digits are even.

Let ABCD be a convex quadrilateral. Let E, F, G, H be midpoints of AB, BC, CD, DA respectively. If AC, BD, EG, FH concur at a point O, prove that ABCD is a parallelogram.

Find the largest real constant \(\gamma \) such that \(\frac{{\gamma}abc}{a+b+c} \le (a+b)^2 +(a+b+4c)^2 \) for all positive real numbers a, b and c.

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