Indian INMO 2018 - problems

1. \bf 1. \rm Let ABCABC be a non-equilateral triangle with integer sides. Let DD and EE be respectively the mid-points BCBC and CACA; let GG be the centroid of triangle ABCABC. Suppose D,C,E,GD,C,E,G are concylic. Find the least possible perimeter of triangle ABCABC.

2. \bf 2. \rm For any natural number nn, consider a 1×n1\times n rectangular board made up of nn unit squares. This is covered by three types of tiles: 1×11\times 1 red\color{#D61F06} \text{red} \color{#333333} tile, 1×11\times 1 green\color{#20A900} \text{green} \color{#333333} tile and 1×21\times 2 blue\color{#3D99F6} \text{blue} \color{#333333} domino. (For example, we can have 55 types of tiling when n=2n=2: red-red,red-green, green-red, green-green and blue.) Let tnt_n denote the number of ways of covering 1×n1\times n rectangular board by these three types of tiles. Prove that tnt_n divides t2n+1t_{2n+1}.

3.\bf 3. \rm Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles with respective centres O1O_1 and O2O_2 intersecting in two distinct points AA and BB such that O1AO2\angle{O_1AO_2} is an obtuse angle. Let the circumcircle of ΔO1AO2\Delta{O_1AO_2} intersect Γ1\Gamma_1 and Γ2\Gamma_2 respectively in points C(A)C (\neq A) and D(A)D (\neq A). Let the line CBCB intersect Γ2\Gamma_2 in EE ; let the line DBDB intersect Γ1\Gamma_1 in FF. Prove that, the points C,D,E,FC, D, E, F are concyclic.

4. \bf 4. \rm Find all polynomials with real coefficients P(x)P(x) such that P(x2+x+1)P(x^2+x+1) divides P(x31)P(x^3-1).

5.\bf 5.\rm There are n3n\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.)

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

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

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

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

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

Note by Áron Bán-Szabó
3 years, 4 months ago

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Soln1:Sol^{n}1: G=CG = C- vertexed HM- point of ΔABC\Delta ABC

AD2=DG×CD4AB2=3CD2\Longrightarrow AD^{2} = DG \times CD \Longrightarrow 4AB^{2} = 3CD^{2}

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

2AB2=AC2+BC2.2 AB^{2} = AC^{2}+ BC^{2}.

Now, it suffices to length chase the problem. The sides are 7,13,17.7,13,17. Ans=37\boxed{Ans= 37}

Soln3:Sol^{n}3: Note that: BE,BFBE, BF are the diameters BCF=BDE=90\Longrightarrow \angle BCF = \angle BDE = 90^{\circ}

Vishwash Kumar ΓΞΩ - 3 years, 4 months ago

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@Áron Bán-Szabó Long time no see, since I left the team.!! Do you happen to know the latest Komal problems? It's been three months since they are updated officially. So, asking you :)

Vishwash Kumar ΓΞΩ - 2 years, 9 months ago

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

Alex Wat - 3 years, 3 months ago

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kvs raman - 3 years, 3 months ago

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