INMO 2019

Let $A B C$ be a triangle with $\angle B A C>90^{\circ}$. Let $D$ be a point on the segment $B C$ and $E$ be a point on the line $A D$ such that $A B$ is tangent to the circumcircle of triangle $A C D$ at $A$ and $B E$ is perpendicular to $A D$. Given that $C A=C D$ and $A E=C E$, determine $\angle B C A$ in degrees.

Let $A_{1} B_{1} C_{1} D_{1} E_{1}$ be a regular pentagon. For $2 \leq n \leq 11$,

let $A_{n} B_{n} C_{n} D_{n} E_{n}$ be the pentagon whose vertices are the midpoints of the sides of the pentagon $A_{n-1} B_{n-1} C_{n-1} D_{n-1} E_{n-1}$. All the 5 vertices of each of the 11 pentagons are arbitrarily coloured red or blue. Prove that four points among these 55 points have the same colour and form the vertices of a cyclic quadrilateral.

Let $m, n$ be distinct positive integers. Prove that

$$\operatorname{gcd}(m, n)+\operatorname{gcd}(m+1, n+1)+\operatorname{gcd}(m+2, n+2) \leq 2|m-n|+1$$

Further, determine when equality holds.

Let $n$ and $M$ be positive integers such that $M>n^{n-1}$. Prove that there are $n$ distinct primes $p_{1}, p_{2}, p_{3}, \ldots, p_{n}$ such that $p_{j}$ divides $M+j$ for $1 \leq j \leq n$.

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

Let $f$ be function defined from the set $\{(x, y): x, y$ reals, $x y \neq 0\}$ in to the set of all positive real numbers such that

(i) $f(x y, z)=f(x, z) f(y, z)$, for all $x, y \neq 0$;

(ii) $f(x, y z)=f(x, y) f(x, z)$, for all $x, y \neq 0$;

(iii) $f(x, 1-x)=1$, for all $x \neq 0,1$.

Prove that

(a) $f(x, x)=f(x,-x)=1$, for all $x \neq 0$;

(b) $f(x, y) f(y, x)=1$, for all $x, y \neq 0$.