# [Solutions] India Regional Mathematical Olympiad 2004

1. Consider in the plane a circle $\Gamma$ with centre $O$ and a line l not intersecting the circle. Prove that there is a unique point $Q$ on the perpendicular drawn from $O$ to line $l$, such that for any point $P$ on the line $l$, $PQ$ represents the length of the tangent from $P$ to the given circle.
2. Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If $T$ denotes the sum of the numbers on all the faces, find the possible values of $T$.
3. Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $$\lambda _n = \alpha ^n + \beta ^n ,\, n \geq 0.$$ Prove that
a) $\lambda _n$ is an integer,
b) $\gcd (\lambda _n , \lambda_{n+1}) = 1$.
4. Prove that the number of triples $(A,B,C)$ where $A, B, C$ are subsets of $\{ 1,2, \ldots , n \}$ such that $A \cap B \cap C = \phi$, $A \cap B \not= \phi$, $C \cap B \not= \phi$ is $7^n - 2 . 6^n + 5^n$.
5. Let $ABCD$ be a quadrilateral; $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively and lines through $X$ and $Y$ respectively parallel to $BD$, $AC$ meet in $O$. Let $P,Q,R,S$ be the midpoints of $AB, BC, CD, DA$ respectively. Prove that
a) $APOS$ and $APXS$ have the same area,
b) $APOS, BQOP, CROQ, DSOR$ have the same area.
6. Let $p_1, p_2, \ldots$ be a sequence of primes such that $p_1 =2$ and for $n\geq 1$, $p_{n+1}$ is the largest prime factor of $p_1 p_2 \ldots p_n +1$ . Prove that $p_n \not= 5$ for any $n$.
7. Let $x$ and $y$ be positive real numbers such that $y^3 + y \leq x - x^3$. Prove that
a) $y < x < 1$,
b) $x^2 + y^2 < 1$.
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