INMO Practice paper

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 $\sqrt { xyz } ≤ 6.$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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