Norway Math Olympiad Geometry Challenge

追加: 2026-05-28

[00:00:00]Can you find the length x given p is d center of triangle ABC and cq is x and If we extend a suppose this point is M.

[00:00:44]Then in any triangle A B C if P is midpoint of BC, Q is midpoint of AC and R is midpoint point of AB.

[00:01:16]And if we join A, BQ and C R then they will intersect at a single point that is centrid.

[00:01:41]Suppose this point is G.

[00:01:46]Then A G ratio GP is equal to BG ratio GQ is equal to CG ratio GR that is 2 ratio 1.

[00:02:16]So here P is centroidid that means B M and CM they will be equal.

[00:02:39]And also a ratio PM will be two ratio 1 and A is 4.

[00:02:57]A is 4. Ratio PM is 2 ratio 1.

[00:03:06]That means 4 over PM is 2 over 1. That means PM it will be 2.

[00:03:21]PM is two and we have B M is equal to CM.

[00:03:48]And now interangle A B C.

[00:04:04]It is A B C.

[00:04:09]This angle is 90°.

[00:04:14]And if we make a circle around ABC, then we know that in any circle diameter makes an angle of 90° on the circle.

[00:04:41]And angle B A C is 90°.

[00:04:47]That means BC it will be diameter and B M is equal to CM.

[00:05:02]That means M is midpoint of BC.

[00:05:08]That means m is center of the circle.

[00:05:15]So a m will be radius, b m will be radius and c m will be radius and a m is 4 + 2. It is 4 + 2 that is 6.

[00:05:36]So, CM it will be also six and now this angle is 90°.

[00:06:07]So, inter angle PQM PQs² + Q M².

[00:06:22]It will be ps² and pq is <unk>3 s² + q m² is 2 s² and 3 + q m² is 4.

[00:06:44]So q m² is 1. That means qm it will be one. QM is one.

[00:06:58]And now x is cq that is c m plus qm and cm is 6 + qm is 1 that is 7.

[00:07:18]So x is seven.