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## INMO 2007

Try to solve these interesting INMO 2007 Questions.

1. In a triangle ”ABC” right-angled at ”C”, the median through ”B” bisects the angle between ”BA” and the bisector of ”$\angle B$”. Prove that $\frac{5}{2} < \frac{AB}{BC} < 3$.
2. Let ”n” be a natural number such that $n = a^2 +b^2 +c^2$ , for some natural numbers $a, b, c$. Prove that $9n = (p_{1a} + q_{1b} + r_{1c})^2 + (p_{2a} + q_{2b} + r_{2c})^2 + (p_{3a}+ q_{3b} + r_{3c})^2$,where $p_{j} ‘s$, $q_{j}$’s, $r_{j}$’s are all nonzero integers. Further, of 3 does not divide at least one of ”a, b, c,” prove that 9n can be expressed in the form $x^2 + y^2 + z^2$, where ”x, y, z” are natural numbers none of which is divisible by 3.
3. Let ”m” and ”n” be positive integers such that the equation $x^2−mx+n = 0$ has real roots $\alpha$ and  $\beta$. Prove that $\alpha$ and $\beta$ are integers if and only if  $[m\alpha]+[m\beta]$ is the square of an integer. (Here [x] denotes the largest integer not exceeding x.)
4. Let $\sigma = (a_{1}, a_{2}, a_{3}, . . . , a_{n}$  be a permutation of  (1, 2, 3, . . . , n) . A pair     $a_{i}, a_{j}$ is said to correspond to an inversion of $\sigma$, if  < j  but $a_{i} > a_{j}$ . (Example: In the permutation  (2, 4, 5, 3, 1) , there are 6 inversions corresponding to the pairs  (2, 1), (4, 3), (4, 1), (5, 3), (5, 1), (3, 1) How many permutations of (1, 2, 3, . . . , n), (n>3) , have exactly two inversions.
5. Let ABC be a triangle in which AB = AC. Let D be the midpoint of BC and P be a point on AD. Suppose E is the foot of the perpendicular from P on AC. If $\frac{AP}{PD}=\frac{BP}{PE}= \lambda$,$\frac{BD}{AD}= m$ and $z = m^2(1 + \lambda)$, prove that $z^2− (\sigma^3− \sigma^2− 2)z + 1 = 0$. Hence show that $\sigma ≥ 2$ and $\lambda = 2$ if and only if ABC is equilateral.
6. If x, y, z are positive real numbers, prove that $(x+y+z)^2(yz+zx+xy)^2≤ 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$.
7. Let  f :Z mapsto Z be a function satisfying $f(0) \neq 0$ , $f(1)=0$ and 1.f(xy) + f(x)f(y) = f(x) + f(y)\), 2. $(f(x-y) – f(0))f(x)f(y) = 0$ for all x , y in Z  simultaneously. 1. Find the set of all possible values of the function f. 2. If $f(10) \neq 0$ and  f(2) = 0 , find the set of all integers n such that $f(n)\neq 0$ .

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## A trigonometric polynomial ( INMO 2020 Problem 2)

Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$.

Using a very standard trigometric identity , we can easily convert the following ,
\begin{align*} P(\cos\theta + \sin\theta) &= P(\cos\theta - \sin\theta) \\ \implies P\left(\sqrt{2}\sin\left(\frac{\pi}{4} + \theta\right)\right) &= P\left(\sqrt{2}\cos\left(\frac{\pi}{4} + \theta\right)\right) \\ \implies P(\sqrt{2}\sin x) &= P(\sqrt{2}\cos x) \\ \end{align*} Assuming ,  $(\frac{\pi}{4}+\theta) = x$ for all reals $$x$$. So,
$$P(-\sqrt{2}\sin(x)) = P(\sqrt{2}\sin(-x)) = P(\sqrt{2}\cos(-x)) = P(\sqrt{2}\cos(x)) = P(\sqrt{2}\sin(x))$$for all $$x\in\mathbb{R}$$. Since $$P(x) = P(-x)$$ holds for infinitely many $$x$$, it must hold for all $$x$$ (since $$P(x)$$ is a polynomial). so we get that ,  $P(x)$ is a even polynomial .

Also
$$P(\sqrt{2}\cos(x)) = P(\sqrt{2}\sin(x))$$ implies that
$$P(t) = P(\sqrt{2}\sin(\cos^{-1}(t/\sqrt{2})))$$putting , $x=\cos^{-1}(t/\sqrt{2})$
for infinitely many $$t$$ $\in [-\sqrt2 ,\sqrt2]$.
$$\sqrt{2}\sin(\cos^{-1}(t/\sqrt{2})) = \sqrt{2 - t^2}$$so we get , $$P(x) = P(\sqrt{2-t^2})$$
Again as it is a polynomial function we can extend it all $\mathbb{R}$. And we get , $$P(x) = P(\sqrt{2-x^2})$$ for all reals $$x$$
Since $$P(x)$$ is even , we can choose a even polynomial $Q(x)$ such that ,$$Q(x) = P(\sqrt{x+1})$$. $$P(\sqrt{1+x}) = Q(x) = a_0 + a_1x^2 + a_2x^4 + \cdots + a_nx^{2n}$$now take , $\sqrt{1+x} = y$ and you get the polynomial of required form .

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#### Pre RMO 2018

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# Understand the problem

Let $\Gamma_1$ and $\Gamma_2$ be two circles with unequal radii, with centers $O_1$ and $O_2$ respectively, in the plane intersecting in two distinct points A and B. Assume that the center of each of the circles $\Gamma_1$ and $\Gamma_2$ are outside each other. The tangent to $\Gamma_ 1$ at B intersects $\Gamma_2$ again at C, different from B; the tangent to $\Gamma_2$ at B intersects $\Gamma_1$ again in D different from B. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in X and Y, respectively. different from A. Let P and Q be the circumcenters of the triangles ACD and XAY, respectively. Prove that PQ is perpendicular bisector of the line segment $O_1 O_2$.

# Tutorial Problems… try these before watching the video.

## Pigeonhole Principle

“The Pigeonhole principle” ~ Students who have never heard may think that it is a joke. Pigeonhole Principle is one of the simplest but most useful ideas in mathematics. Let’s learn the Pigeonhole Principle with some applications. Pigeonhole Principle Definition: In...

## Triangle Problem | PRMO-2018 | Problem No-24

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

## Even Parity and Odd Parity

Parity in Mathematics is a term which we use to express if a given integer is even or odd. It is basically depend on the remainder when we divide a number by 2. Parity can be divided into two categories – 1. Even Parity 2. Odd Parity Even Parity : If we...

## Value of Sum | PRMO – 2018 | Question 16

Try this Integer Problem from Number theory from PRMO 2018, Question 16 You may use sequential hints to solve the problem.

## Chessboard Problem | PRMO-2018 | Problem No-26

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

## Measure of Angle | PRMO-2018 | Problem No-29

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

## Good numbers Problem | PRMO-2018 | Question 22

Try this good numbers Problem from Number theory from PRMO 2018, Question 22 You may use sequential hints to solve the problem.

## Polynomial Problem | PRMO-2018 | Question 30

Try this Integer Problem from Number theory from PRMO 2018, Question 30 You may use sequential hints to solve the problem.

## Digits Problem | PRMO – 2018 | Question 19

Try this Integer Problem from Number theory from PRMO 2018, Question 19 You may use sequential hints to solve the problem.