INMO 2008 Question Paper | Math Olympiad Problems

This post contains the problems from Indian National Mathematics Olympiad, INMO 2008 Question Paper. Do try to find their solutions.

Let $ABC$ be a triangle, $I$ its in-centre; $ A_1, B_1, C_1 $ be the reﬂections of $I$ in BC, CA, AB respectively. Suppose the circum-circle of triangle $ A_1 B_1 C_1 $ passes through A. Prove that $ B_1, C_1, I, I_1 $ are concyclic, where $ I_1 $ is the in-centre of triangle $ A_1 B_1 C_1 $.
Find all triples $ (p, x, y) $ such that $ p^x = y^4 + 4 $, where $p $ is a prime and $ x, y $ are natural numbers.
Let $A$ be a set of real numbers such that $A$ has at least four elements. Suppose $A$ has the property that $a^2+b c$ is a rational number for all distinct numbers $a, b, c$ in $A$. Prove that there exists a positive integer $M$ such that $a \sqrt{M}$ is a rational number for every $a$ in $A$.
All the points with integer coordinates in the xy-plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point (0, 0) is coloured red and the point (0, 1) is coloured blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.
Let $A B C$ be a triangle; $\Gamma_A, \Gamma_B, \Gamma_C$ be three equal, disjoint circles inside $A B C$ such that $\Gamma_A$ touches $A B$ and $A C ; \Gamma_B$ touches $A B$ and $B C$; and $\Gamma_C$ touches $B C$ and $C A$. Let $\Gamma$ be a circle touching circles $\Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $O$ and the in-centre $I$ of triangle $A B C$ passes through the centre of $\Gamma$.
Let $ P(x) $ be a given polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x) $ and $ R(x) $, again with integer coefficients, such that
1. $ P(x)Q(x) $ is a polynomial in $ x^2 $; and
2. $ P(x)R(x) $is a polynomial in $ x^3 $.

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INMO 2020 Problems, Solutions and Hints

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Inequality of square root function

This post contains a problem from TIFR 2013 Math paper D based on Inequality of square root function.

The inequality $ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $ is false for all in n such that $ 101 \le n \le 2000 $

False

Discussion:

$ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $

By cross multiplying we have $ \sqrt {(n+1)n} - n < 1 $. That is $ \sqrt {n(n+1)} < (n+1) $ or $ n(n+1) < (n+1) ^2 $ or $n < n+1$

This is true for all $n$.

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Automorphism of the Additive Group of Rationals

Any automorphism of the group Q under addition is of the form x → qx for some q ∈ Q.

True

Discussion: Suppose f is an automorphism of the group Q. Let f(1) = m (of course 'm' will be different for different automorphisms). Now $f(x+y) = f(x) + f(y)$ implies $f(x) = mx$ where m is a constant and x belongs to set of integers (Cauchy's functional equation).

Now suppose x is rational. Then x = p/q where p and q are integers. Hence $f(p) = mp$. But $p = qx$ hence $f(p) = f(qx) = f(x+x+ ... + x) = qf(x)$

Therefore, $mp = qf(x)$ implies $ m \times {\frac{p}{q} }= f(x) \implies f(x) = mx $ where $m = f(1)$