India National Olympiad 2015 Problems
Problem 1
Let $ABC$ be a right-angled triangle with $\angle{B}=90^{\circ}$. Let $BD$ is the altitude from $B$ on $AC$. Let $P,Q$ and $I $be the incenters of triangles $ABD,CBD$ and $ABC$ respectively.Show that circumcenter of triangle $PIQ$ lie on the hypotenuse $AC$.
Problem 2
For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.
Problem 3
Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.
Problem 4
There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person.
In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.
Problem 5
Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$
Problem 6
Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.
