India Regional Mathematical Olympiad 2018
1
Let 
be a triangle with integer sides in which 
. Let the tangent to the circumcircle of triangle at 
intersect the line 
at 
. Suppose 
is also an integer. Prove that 
.
be a triangle with integer sides in which 
. Let the tangent to the circumcircle of triangle at 
intersect the line 
at 
. Suppose 
is also an integer. Prove that 
.
3
For a rational number 
, its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number 
has period 
. If 
denotes the set of all rational numbers of the form 
having period 
, find the sum of all elements in .
, its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number 
has period 
. If 
denotes the set of all rational numbers of the form 
having period 
, find the sum of all elements in .4
Let 
denote the set of 
points 
in the 
-plane, where 
are natural numbers, 
. Suppose the points of are arbitrarily coloured using two colours, red and blue. SHow that there always exist four points in the set of the form 
for some positive integer 
such that at least three of these four points have the same colour. (That is, there always exist four points in the set which form the vertices of a square with sides parallel to the axes and having at least three points of the same colour.)
denote the set of 
points 
in the 
-plane, where 
are natural numbers, 
. Suppose the points of are arbitrarily coloured using two colours, red and blue. SHow that there always exist four points in the set of the form 
for some positive integer 
such that at least three of these four points have the same colour. (That is, there always exist four points in the set which form the vertices of a square with sides parallel to the axes and having at least three points of the same colour.)5
Find all natural numbers 
such that ![$1+[\sqrt{2n}]~$](//latex.artofproblemsolving.com/e/6/a/e6aa1bf5d761758196027add39f2e3cd77db1c1c.png)
divides 
.
( For any real number
, ![$[x]$](//latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denotes the largest integer not exceeding . )
such that ![$1+[\sqrt{2n}]~$](http://latex.artofproblemsolving.com/e/6/a/e6aa1bf5d761758196027add39f2e3cd77db1c1c.png)
divides 
.( For any real number
, ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denotes the largest integer not exceeding . )
6
Let be an acute-angled triangle with . Let 
be the incentre of triangle , and let 
be the points where the incircle touches the sides 
respectively. Let 
meet the line 
at 
respectively. Further assume that both 
and 
are outside the triangle . Prove that
are concyclic.
is also the incentre of triangle 
.
be an acute-angled triangle with . Let 
be the incentre of triangle , and let 
be the points where the incircle touches the sides 
respectively. Let 
meet the line 
at 
respectively. Further assume that both 
and 
are outside the triangle . Prove that
are concyclic.
is also the incentre of triangle 
.Kerala Region
1
Let be an acute angled triangle and let be an interior point of the segment . Let the circumcircle of 
intersect 
at ( between and 
) and let circumcircle of 
intersect 
at 
( between and 
). Let 
be the circumcenter of 
. Prove that 
bisects 
.
be an acute angled triangle and let be an interior point of the segment . Let the circumcircle of 
intersect 
at ( between and 
) and let circumcircle of 
intersect 
at 
( between and 
). Let 
be the circumcenter of 
. Prove that 
bisects 
.
2
Find the set of all real values of 
for which the real polynomial equation 
has real roots, given that 
and 
form a geometric progression.
for which the real polynomial equation 
has real roots, given that 
and 
form a geometric progression.3
Show that there are infinitely many tuples 
of natural numbers such that 
.
of natural numbers such that 
.4
Suppose 
points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?
points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?5
In a cyclic quadrilateral 
with circumcenter , the diagonals and 
intersect at . Let the circumcircles of triangles 
and 
intersect at . Let the circumcircles of triangles 
and 
intersect at 
. If lies inside and if the points 
are all distinct, prove that lie on a circle.
with circumcenter , the diagonals and 
intersect at . Let the circumcircles of triangles 
and 
intersect at . Let the circumcircles of triangles 
and 
intersect at 
. If lies inside and if the points 
are all distinct, prove that lie on a circle.6
Define a sequence 
of real numbers by ![\[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\]](//latex.artofproblemsolving.com/1/f/b/1fb2327fcdb33383563ab5805cf9ffe24ac43a7d.png)
Prove that ![\[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\]](//latex.artofproblemsolving.com/8/2/6/8265605cb6de33f6c738117f4d4834c10b3dce7a.png)
for every natural number 
.
of real numbers by ![\[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\]](http://latex.artofproblemsolving.com/1/f/b/1fb2327fcdb33383563ab5805cf9ffe24ac43a7d.png)
Prove that ![\[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\]](http://latex.artofproblemsolving.com/8/2/6/8265605cb6de33f6c738117f4d4834c10b3dce7a.png)
for every natural number 
.
