Let AOB be a given angle less than \( 180^o \) and let P be an interior point of the angular region determined by \( \angle AOB \) . Show, with proof, how to construct, using only ruler and compass, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB and CP:PD = 1:2.

RMO 2017, Problem 2:

Show that the equation $$ a^3 + (a+1)^3 + (a+2)^3 + (a+3)^3 + (a+4)^3 + (a+5)^3 + (a+6)^3 = b^4 + (b+1)^4 $$ has no solutions in integer a, b.

RMO 2017, Problem 3:

Let P(x)=x2+12x+b and Q(x)=x2+cx+d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all real roots of P(Q(x)) = 0

RMO 2017, Problem 4:

Consider n2 unit squares in the xy-plane centered at the point (i, j) with integer coordinates, 1≤i≤n , 1≤j≤n . It is required to color each unit square in such a way that whenever 1≤ i < j and 1 ≤ k < l ≤ n the three squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?

RMO 2017, Problem 5:

Let Ω be a circle with a chord AB which is not a diameter. Let Γ1 be a circle on one side of AB such that it is tangent to AB at C and internally tangent to Ω at D. Likewise let Γ2 be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to Ω at F. Suppose the line DC intersects Ω at X≠D and the line FE intersects Ω at Y≠F. Prove that XY is a diameter of Ω

RMO 2017, Problem 6:

Let x, y, z be real numbers, each greater than 1. Prove that