INMO 1995

In an acute angled triangle $ABC$ , $\angle A = 30^{\circ}$ , $H$ is the orthocenter, and $M$ is the midpoint of $BC$ . On the line $HM$ , take a point $T$ such that $HM = MT$ . Show that $AT = 2 BC$ .

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 x^2 + 2ax + b &=& 0 \end{eqnarray*} have integer roots.

Show that the number of $3-$ element subsets $\{ a , b, c \}$ of $\{ 1 , 2, 3, \ldots, 63 \}$ with $a+b +c < 95$ is less than the number of those with $a + b +c \geq 95.$

Let $ABC$ be a triangle and a circle $\Gamma'$ be drawn lying outside the triangle, touching its incircle $\Gamma$ externally, and also the two sides $AB$ and $AC$ . Show that the ratio of the radii of the circles $\Gamma'$ and $\Gamma$ is equal to $\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.$

Let $n \geq 2$ . Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$ . Show that $$\dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 .$$

Find all primes $p$ for which the quotient $$\dfrac{2^{p-1} - 1 }{p}$$ is a square.