How Cheenta works to ensure student success?
Explore the Back-Story

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.
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.

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

One comment on “Indian National Math Olympiad 2014 (INMO 2014)”

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
Math Olympiad Program
magic-wandrockethighlight