India National Olympiad 2003 Problems
Problem 1
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.
Problem 2
Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+…+1=q^2+q+1$.
Problem 3
Show that $8x^4 – 16x^3 + 16x^2 – 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.
Problem 4
Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$.
Problem 5
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.
Problem 6
Each lottery ticket has a 9-digit numbers, which uses only the digits $1$, $2$, $3$. Each ticket is colored red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket $122222222$ is red, and ticket $222222222$ is green. What color is ticket $123123123$ ?
