# Indian National Math Olympiad, INMO 2019 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2019. Try them and share your solution in the comments.

INMO 2019, Problem 1

Let ABC$ABC$ be a triangle with BAC>90$\angle BAC > 90^\circ$. Let D$D$ be a point on the line segment BC$BC$ and E$E$ be a point on the line AD$AD$ such that AB$AB$ is tangent to the circumcircle of triangle ACD$ACD$ at A$A$ and BE$BE$ perpendicular to AD$AD$. Given that CA=CD$CA=CD$ and AE=CE$AE=CE$, determine BCA$\angle BCA$ in degrees

INMO 2019, Problem 2

Let A1B1C1D1E1${A_1}{B_1}{C_1}{D_1}{E_1}$ be a regular pentagon. For 2n≤11$2\leq n \leq 11$, let AnBnCnDnEn${A_n}{B_n}{C_n}{D_n}{E_n}$ whose vertices are the midpoints of the sides of An−1Bn−1Cn−1Dn−1En−1${A_{n-1}}{B_{n-1} }{C_{n-1} }{D_{n-1}}{E_{n-1}}$. All the 5$5$ vertices of each of the 11$11$ pentagons are arbitrarily coloured red or blue. Prove that four points among these 55$55$ have the same colour and form the vertices of a cyclic quadrilateral.

INMO 2019, Problem 3

Let m$m$, n$n$ be distinct positive integers. Prove that gcd(m,n)+gcd(m+1,n+1)+gcd(m+2,n+2)≤2|mn|+1$\gcd (m,n) + \gcd (m+1,n+1) + \gcd (m+2,n+2) \leq 2|m-n|+1$. Further, determine when equality holds.

INMO 2019, Problem 4

Let n$n$ and M$M$ be positive integers such that M>nn−1$M>n^{n-1}$. Prove that there are n$n$ distinct primes p1$p_1$,, such that pj$p_j$ divides M+j$M+j$ for 1jn$1 \leq j \leq n$.

INMO 2019, Problem 5

Let AB$AB$ be a diameter of a circle Γ$\Gamma$ and let C$C$ be a point on Γ$\Gamma$ different from A$A$ and B$B$. let D$D$ be the foot of perpendicular from C$C$ on to AB$AB$. let K$K$ be a point of the segment CD$CD$ such that AC$AC$ is equal to the semiperimeter of the triangle ADK$ADK$. Show that the excircle of triangle ADK$ADK$ opposite A$A$ is tangent to Γ$\Gamma$.

INMO 2019, Problem 6

Let $$f$$ be a function defined from the set $$\{(x,y) : x,y$$ are real, $$xy \neq 0\}$$ to the set of all positive real number such that

(i)$f(x y, z)=f(x, z) f(y, z),$ for all $x, y \neq 0$
(ii)$\quad f(x, y z)=f(x, y) f(x, z),$ for all $x, y \neq 0$
(iii)$f(x, 1-x)=1,$ for all $x \neq 0,1$
Prove that
(a) $\quad f(x, x)=f(x,-x)=1,$ for all $x \neq 0$
(b) $f(x, y) f(y, x)=1,$ for all $x, y \neq 0$