HARD
IOQM - PRMO and RMO
IMPORTANT
Earn 100
Let be a triangle and let be its circumcircle. The internal bisectors of angles and intersect at and respectively, and the internal bisectors of angles and of the triangle intersect at and respectively. If the smallest angle of triangle is what is the magnitude of the smallest angle of triangle in degrees?
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Important Questions on Triangle
HARD
IOQM - PRMO and RMO
IMPORTANT
A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village, north of the north gate, and it can be just seen appearing on the horizon from a point east of the south gate. What is the diameter, in meters, of the wall that surrounds the village?
HARD
IOQM - PRMO and RMO
IMPORTANT
Let be a triangle with sides Let denote the incircle of Draw tangents to which are parallel to the sides of Let be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed
HARD
IOQM - PRMO and RMO
IMPORTANT
In a triangle the median (with on ) and the angle bisector (with on ) are perpendicular to each other. If and find the integer nearest to the area of triangle
HARD
IOQM - PRMO and RMO
IMPORTANT
Let be a triangle such that . Suppose the tangent to the circumcircle of at is perpendicular to . Find measured in degrees.
HARD
IOQM - PRMO and RMO
IMPORTANT
The centre of the circle passing through the midpoints of the sides of an isosceles triangle lies on the circumcircle of triangle . If the larger angle of triangle is and the smaller one is , then what is the value of ?
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
A friction-less board has the shape of an equilateral triangle of side length meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex towards the side The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form where is an integer. What is the value of
HARD
IOQM - PRMO and RMO
IMPORTANT
Let be an acute angled triangle with and Let be a point on such that Consider points on such that If denotes the product of all possible values of find , where is the integer part of
HARD
IOQM - PRMO and RMO
IMPORTANT
Let be a chord of a circle with centre Let be a point on the circle such that and lies inside the triangle . Let be a point on such that Find the measure of in degrees.
