\( \bf 1. \rm\) Let \(ABC\) be a non-equilateral triangle with integer sides. Let \(D\) and \(E\) be respectively the mid-points \(BC\) and \(CA\); let \(G\) be the centroid of triangle \(ABC\). Suppose \(D,C,E,G\) are concylic. Find the least possible perimeter of triangle \(ABC\).

\( \bf 2. \rm\) For any natural number \(n\), consider a \(1\times n\) rectangular board made up of \(n\) unit squares. This is covered by three types of tiles: \(1\times 1\) \(\color{red} \text{red} \color{black}\) tile, \(1\times 1\) \(\color{green} \text{green} \color{black}\) tile and \(1\times 2\) \(\color{blue} \text{blue} \color{black}\) domino. (For example, we can have \(5\) types of tiling when \(n=2\): red-red,red-green, green-red, green-green and blue.) Let \(t_n\) denote the number of ways of covering \(1\times n\) rectangular board by these three types of tiles. Prove that \(t_n\) divides \(t_{2n+1}\).

\(\bf 3. \rm \) Let \(\Gamma_1\) and \(\Gamma_2\) be two circles with respective centres \(O_1\) and \(O_2\) intersecting in two distinct points \(A\) and \(B\) such that \(\angle{O_1AO_2}\) is an obtuse angle. Let the circumcircle of \(\Delta{O_1AO_2}\) intersect \(\Gamma_1\) and \(\Gamma_2\) respectively in points \(C (\neq A)\) and \(D (\neq A)\). Let the line \(CB\) intersect \(\Gamma_2\) in \(E\) ; let the line \(DB\) intersect \(\Gamma_1\) in \(F\). Prove that, the points \(C, D, E, F\) are concyclic.

\( \bf 4. \rm \) Find all polynomials with real coefficients \(P(x)\) such that \(P(x^2+x+1)\) divides \(P(x^3-1)\).

\(\bf 5.\rm\) There are \(n\ge 3\) girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbours combined, the teacher takes away one apple from that girl and gives one apple each to her neighbours. Prove that, this process stops after a finite number of steps. (Assume that, the teacher has an abundant supply of apples.)

\( \bf 6. \rm \) Let \(\mathbb N\) denote set of all natural numbers and let \(f:\mathbb{N}\to\mathbb{N}\) be a function such that

\(\text{(a)} f(mn)=f(m)f(n)\) for all \(m,n \in\mathbb{N}\);

\(\text{(b)} m+n\) divides \(f(m)+f(n)\) for all \(m,n\in \mathbb N\).

Prove that, there exists an odd natural number \(k\) such that \(f(n)= n^k\) for all \(n\) in \(\mathbb{N}\).

\(\boxed{\text{Note that problem 6 appeared in Turkey TST!!!}}\)

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## Comments

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TopNewest\(Sol^{n}1:\) \(G = C-\) vertexed HM- point of \(\Delta ABC\)

\(\Longrightarrow AD^{2} = DG \times CD \Longrightarrow 4AB^{2} = 3CD^{2}\)

Now, applying Apollonius' theorem for median, we get:

\(2 AB^{2} = AC^{2}+ BC^{2}.\)

Now, it suffices to length chase the problem. The sides are \(7,13,17.\) \(\boxed{Ans= 37}\)

\(Sol^{n}3:\) Note that: \(BE, BF\) are the diameters \(\Longrightarrow \angle BCF = \angle BDE = 90^{\circ}\)

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Can you please specify in which year of the Turkey TEST did the problem no.6 come?

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