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February 2, 2014

Indian National Math Olympiad 2014 (INMO 2014)

Problem 1
In a triangle $A B C,$ let $D$ be a point on the segment $B C$ such that $A B+B D=A C+C D .$ Suppose that the points $B, C$ and the centroids of triangles $A B D$ and $A C D$ lie on a circle. Prove that $A B=A C .$
Solution
 
 
Problem 2
Let $n$ be a natural number. Prove that
$$
\left[\frac{n}{1}\right]+\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\cdots\left[\frac{n}{n}\right]+[\sqrt{n}]
$$
is even. (Here $[x]$ denotes the largest integer smaller than or equal to $x$.
 
 
Problem 3
Let $a, b$ be natural numbers with $a b>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisible by $a+b$. Prove that the quotient is at most $(a+b) / 4$. When is this quotient exactly equal to $(a+b) / 4 ?$
 
 
Problem 4
Written on a blackboard is the polynomial $x^{2}+x+2014$. Calvin and Hobbes take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or decrease the coefficient of $x$ by 1 . And during his 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 strategy.
 
 
Problem 5
In an acute-angled triangle $A B C,$ a point $D$ lies on the segment $B C .$ Let $O_{1}, O_{2}$ denote the circumcentres of triangles $A B D$ and $A C D,$ respectively. Prove that the line joining the circumcentre of triangle $A B C$ and the orthocentre of triangle $O_{1} O_{2} D$ is parallel to $B C$.
 
 
Problem 6
Let $n$ be a natural number and $X={1,2, \ldots, n} .$ For subsets $A$ and $B$ of $X$ we define $A \Delta B$ to be the set of all those elements of $X$ which belong to exactly one of $A$ and $B$. Let $\mathcal{F}$ be a collection of subsets of $X$ such that for any two distinct elements $A$ and $B$ in $\mathcal{F}$ the set $A \Delta B$ has at least two elements. Show that $\mathcal{F}$ has at most $2^{n-1}$ elements. Find all such collections $\mathcal{F}$ with $2^{n-1}$ elements.

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