INMO Practice Test-I

Time: 4 Hours [100 marks]

The real numbers a, b, c, d satisfy simultaneously the equations

\(abc − d = 1, bcd − a = 2, cda − b = 3, dab − c = −6.\)

Prove that \(a + b + c + d \neq 0.\) [16]Consider a triangle ABC and let M be the midpoint of the side BC.

Suppose \(∠MAC = ∠ABC\) and \(∠BAM = 105◦\) . Find the measure of \(∠ABC.\) [16]There are two piles of coins, each containing 2010 pieces. Two players A and B play a game taking turns (A plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?[17]

Find all positive integers n such that \(36^n − 6\) is a product of two or more consecutive positive integers. [17]

Consider a triangle \(ABC\) with \(∠ACB = 90◦\) . Let F be the foot of the altitude from C. Circle \(w\) touches the line segment \(FB\) at point \(P\), the altitude \(CF\) at point Q and the circumcircle of ABC at point R. Prove that points \(P, Q, R\) are collinear and \(AP = AC\). [17]

The real positive numbers \(x, y, z\) satisfy the relations \(x ≤ 2, y ≤ 3, x + y + z = 11.\) Prove that \(\sqrt { xyz } ≤ 6.\)

Please post your answers!

## Comments

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TopNewesthas anyone solved 5th – Saran .P.S · 1 year, 6 months ago

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Sorry for being too late to post the solution Q.2) Use sine rules for triangles AMC and AMB. Equate the value of AM from both these expressions. Cancel out CM and MB as they both are equal. Now you are left with a simple trigonometric equation in sine(angle ABC). Solve that you will get angle ABC as 30 degrees. Please tell me if the answer is right or wrong – Shrihari B · 1 year, 7 months ago

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done! no 6 easy! – Sayantan Dutta · 1 year, 8 months ago

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A hint for 6th question

Use AM-GM on x/2,y/3 and z/6.

A hint for second question:

Prove that triangles MAC and ABC are similar . – Satvik Mashkaria · 2 years, 6 months ago

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– Saran .P.S · 1 year, 6 months ago

tnks for the hintsLog in to reply

– Siddharth G · 2 years, 6 months ago

Thanks for the 6th!Log in to reply

Can you put answers to them – Kushal Patankar · 2 years, 6 months ago

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– Siddharth G · 2 years, 6 months ago

Sketch of proof for 3: Prove that all losing positions for a person \(p\) are consecutive/same integers whose sum is divisible by \(3\). Thus, since \(A\) starts with \(2010,2010\) he would lose. We can prove this by providing a suitable algorithm for \(B\) and proving that \((2,1)\) is a losing position.Log in to reply

PLZ give the diagram of question 5! – Sayantan Dutta · 1 year, 5 months ago

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Where did you get this from? – Krishna Ar · 2 years, 6 months ago

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– Krishna Ar · 2 years, 6 months ago

Woot...Got it...you found it on BPRIM website right? :PLog in to reply

– Siddharth G · 2 years, 6 months ago

Yeah, could you post the solutions to some of these questions? I am trying, but this is terribly difficult.Log in to reply

– Krishna Ar · 2 years, 6 months ago

Oh! I haven't even tried any one of them yet. Sorry!!Log in to reply