## Reduce the index on a radical and multiply or divide radicals of different index

Knowing that a radical has the same properties as exponents (written as a ratio) allows us to manipulate radicals in new ways. One thing we are allowed to do is reduce, not just the radicand, but the index as...

## Pigeonhole Principle

The Pigeonhole Principle, also known as the Dirichlet Principle (after its inventor, the famous mathematician Peter Gustav Dirichlet, 1805–1859). This simple principle does wonders. It is amazing how easy it is to understand this idea, and how difficult it...

## Solution Shortlist problem about geometry in IMO 2014 ( Problem 6)

Problem 6: Let ABC be a fixed acute-angled triangle. Consider some points E and F lying on the sides AC and AB, respectively, and let M be the midpoint of EF. Let the perpendicular bisector of EF intersect the...

## Proving that the circumcircle of triangle is tangent to the line

Problem 5: Let ABCD be a convex quadrilateral with . Point H is the foot of the perpendicular from A to BD. The points S and T are chosen on the sides AB and AD, respectively, in such a...

## How to proving That as P varies and point Q lies on a fixed circle (Solution Problem 4 : geometry shortlist IMO 2014)

Problem 4: Consider a fixed circle with three fixed points A, B, and C on it. Also, let us fix a real number . For a variable point on , let M be the point on the segment CP...

## Proving two line is Parallel from the circumcircle and circumcentre (Solution Problem 3 Shortlist IMO 2014)

Problem 3: Let and O be the circumcircle and the circumcentre of an acute-angled triangle ABC with AB a BC. The angle bisector of intersects at . Let be the circle with diameter BM. The angle bisectors of and...

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