Indian National Math Olympiad 2013

This post contains problems from Indian National Mathematics Olympiad, INMO 2013. Try them and share your solution in the comments.

Problem 1

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles touching each other externally at $R$ . Let $l_{1}$ be a line which is tangent to $\Gamma_{2}$ at $P$ and passing through the center $O_{1}$ of $\Gamma_{1}$ . Similarly, let $l_{2}$ be a line which is tangent to $\Gamma_{2}$ at $Q$ and passing through the center $O_{2}$ of $\Gamma_{2}$ . Suppose $l_{1}$ and $l_{2}$ are not parallel and interesct at $K .$ If $K P=K Q,$ prove that the triangle $P Q R$ is equilateral.

Solution

Problem 2

Find all positive integers $m, n,$ and primes $p \geq 5$ such that
m(4m2+m+12)=3(pn−1)

Problem 3

Let $a, b, c, d$ be positive integers such that $a \geq b \geq c \geq d$ . Prove that the equation $x^{4}-a x^{3}-b x^{2}-c x-d=0$ has no integer solution.

Solution

Problem 4

Let $n$ be a positive integer. Call a nonempty subset $S$ of ${1,2, \ldots, n}$ good if the arithmetic mean of the elements of $S$ is also an integer. Further let $t_{n}$ denote the number of good subsets of ${1,2, \ldots, n} .$ Prove that $t_{n}$ and $n$ are both odd or both even.

Solution

Problem 5

In an acute triangle $A B C, O$ is the circumcenter, $H$ is the orthocenter and $G$ is the centroid. Let $O D$ be perpendicular to $B C$ and $H E$ be perpendicular to $C A,$ with $D$ on $B C$ and $E$ on CA. Let $F$ be the midpoint of $A B$ . Suppose the areas of triangles ODC,HEA and GFB are equal. Find all the possible values of $\widehat{C}$ .

Problem 6

Let $a, b, c, x, y, z$ be positive real numbers such that $a+b+c=x+y+z$ and $a b c=x y z .$ Further, suppose that $a \leq x<y<z \leq c$ and $a<b<c .$ Prove that $a=x, b=y$ and $c=z .$

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles touching each other externally at $R$ . Let $l_{1}$ be a line which is tangent to $\Gamma_{2}$ at $P$ and passing through the center $O_{1}$ of $\Gamma_{1}$ . Similarly, let $l_{2}$ be a line which is tangent to $\Gamma_{2}$ at $Q$ and passing through the center $O_{2}$ of $\Gamma_{2}$ . Suppose $l_{1}$ and $l_{2}$ are not parallel and interesct at $K .$ If $K P=K Q,$ prove that the triangle $P Q R$ is equilateral.

Solution

Problem 2

Find all positive integers $m, n,$ and primes $p \geq 5$ such that
m(4m2+m+12)=3(pn−1)

Problem 3

Let $a, b, c, d$ be positive integers such that $a \geq b \geq c \geq d$ . Prove that the equation $x^{4}-a x^{3}-b x^{2}-c x-d=0$ has no integer solution.

Solution

Problem 4

Let $n$ be a positive integer. Call a nonempty subset $S$ of ${1,2, \ldots, n}$ good if the arithmetic mean of the elements of $S$ is also an integer. Further let $t_{n}$ denote the number of good subsets of ${1,2, \ldots, n} .$ Prove that $t_{n}$ and $n$ are both odd or both even.

Solution

Problem 5

In an acute triangle $A B C, O$ is the circumcenter, $H$ is the orthocenter and $G$ is the centroid. Let $O D$ be perpendicular to $B C$ and $H E$ be perpendicular to $C A,$ with $D$ on $B C$ and $E$ on CA. Let $F$ be the midpoint of $A B$ . Suppose the areas of triangles ODC,HEA and GFB are equal. Find all the possible values of $\widehat{C}$ .

Problem 6

Let $a, b, c, x, y, z$ be positive real numbers such that $a+b+c=x+y+z$ and $a b c=x y z .$ Further, suppose that $a \leq x<y<z \leq c$ and $a<b<c .$ Prove that $a=x, b=y$ and $c=z .$