**Regional Mathematical Olympiad 2014 (Mumbai Region)**

Instructions

- There are six questions in this paper. Answer all questions.
- Each question carries 10 points
- Use of protractors, calculators, mobile phone is forbidden.
- Time alloted: 3 hours

Questions

**1** Three positive real numbers \(a, b, c\) are such that \(a^2 + 5b^2 + 4c^2 - 4ab - 4bc = 0\). Can \(a, b, c\) be the lengths of sides of a triangle? Justify your answer.

**2** The roots of the equation

\[x^3 -3ax^2 + bx + 18c = 0\]

form a non-constant arithmetic progression and the roots of the equation

\[x^3 + bx^2 + x - c^3 = 0\]

form a non-constant geometric progression. Given that \(a, b, c\) are real numbers, find all positive integral values \(a\) and \(b\).

**3** Let \(ABC\) be an acute-angled triangle in which \(\angle ABC\) is the largest angle. Let \(O\) be its circumcentre. The perpendicular bisectors of \(BC\) and \(AB\) meet \(AC\) at \(X\) and \(Y\) respectively. The internal bisectors of \(\angle AXB\) and \(\angle BYC\) meet \(AB\) and \(BC\) at \(D\) and \(E\) respectively. Prove that \(BO\) is perpendicular to \(AC\) if \(DE\) is parallel to \(AC\).

**4** A person moves in the \(x-y\) plane moving along points with integer co-ordinates \(x\) and \(y\) only. When she is at point \((x,y)\), she takes a step based on the following rules:

(a) if \(x+y\) is even she moves either to \((x+1,y)\) or \((x+1, y+1)\);

(b) if \(x+y\) is odd she moves either to \((x,y+1)\) or \((x+1, y+1)\).

How many distinct paths can she take to go from \((0,0)\) to \((8,8)\) given that she took exactly three steps to right \(((x,y)\) to \((x+1,y))\)?

**5** Let \(a, b, c\) be positive numbers such that

\[\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} \leq 1.\]

Prove that \((1+a^2 )(1+b^2)(1+c^2) \geq 125\). When does the equality hold?

**6** Let \(D, E, F\) be the points of contact of the incircle of an acute-angled triangle \(ABC\) with \(BC, CA, AB\) respectively. Let \(I_1, I_2, I_3\) be the incenters of the triangles \(AFE, BDF, CED,\) respectively. Prove that the lines \(I_1D, I_2E, I_3F\) are concurrent.

Post your innovative solutions below!! Enjoy!!!!!!!!

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## Comments

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TopNewestDid anyone get the proofs for question5 &6

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There are 2 ways to solve this. 1. AM-GM inequality. 2. Without using any theorems, we can prove the square of each individual term to be greater than 4(try figuring out on your own first) and then add 1 to each term. Finally multiply all the terms to get the final result. Equality holds true at a=b=c=2(which is clearly visible)

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Positive integer solution of 1/a+1/b+1/c+1/d =1

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6 is easy

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Well the Maharastra is much ahead in the race for IMO in India. I tried the geometry ones(nice). Both Mumbai and Pune are doing well. I couldn't try the non geometric one due to lack of time. Here are the geometric ones.

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Nicely done :)

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What is the meaning of a non constant arithmetic and geometric progression

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Pranshu, can you tell me which latex you used to separate the questions by a horizontal line?

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Priyanshu, this is not latex; it is markdown. Enter three hyphens (shown above) in a new line to get a horizontal line.

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I am preparing for RMO and I am in class 10 please someone help me in preparing.and pls help me in the 2nd Question i tried but i dont know where i am wrong

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i am selected for inmo

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Comment deleted Mar 15, 2015

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i didn't get selected. did you?

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in q3 triangle abc is isoceles can be proved.so the perpendicular bisector of side ac is a cevian so BO perpendicular to AC

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1 and 5 were so easy. I wonder why they asked them. I was on the right track for the 2nd question however, a shitty mistake while writing the equation led to me getting no solutions :-( I also attempted the 4 and the 6th but I am not sure of the solutions. How much marks do you expect? What is the expected cut-off?

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The cutoff must be somewhere around 40.

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Did you get the 3rd one? I tried but I could not get anywhere in that problem.

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5th is so easy. Basic CS/AM-HM.

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whats the answer for 4th question

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I don't have the official answers right now, but when I solved it I got 462 distinct paths. (I may be wrong)

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How ???

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image

According to the condition in the question, the person must step on exactly 3 of the red arrows. Can you continue now?

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Positive integer solution of 1/a+1/b+1/c+1/d =1

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what is the ans to question 2 i m getting a =2 b=9 there could be more values

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\(a=2\) and \(b=9\) are the only positive integer solutions.

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ya even i got that. and for question 1 i got that a,b and c can never form the sides of a triangle.

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