# [Solutions] India National Mathematical Olympiad 2014

1. In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.
2. Let $n$ be a natural number. Prove that, $\left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor$ is even.
3. Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$
4. Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
5. In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.
6. Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.
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