Norway Math Olympiad | A Very Nice Geometry Challenge

[00:00:00]Can you find the length X?

[00:00:05]If we extend CB to point Q such that BQ is X.

[00:00:23]And join AQ then in triangle ABQ It is a B Q AB is X. BQ is X.

[00:01:02]That means these two angles they will be equal. Suppose alpha and this angle is two theta.

[00:01:14]So alpha plus alpha is two theta.

[00:01:19]That means alpha is theta.

[00:01:23]So this angle is theta.

[00:01:27]And this angle is theta.

[00:01:35]And now angle ACQ is theta.

[00:01:53]And angle AQC is also theta.

[00:02:00]That means AQ and AC, they will be equal.

[00:02:07]And AC is 8.

[00:02:09]That means AQ it will be also 8.

[00:02:17]And now in any isosceles triangle ABC if it is an isosceles triangle from point A if we make a perpendicular on BC suppose this point is P then it will bisect the base.

[00:03:01]And also it will bisect angle A.

[00:03:12]And we have AB is X and PB is X.

[00:03:19]That means ABP is an isosceles triangle.

[00:03:26]So from point B if we make a perpendicular on AP suppose this point is M then AP is 6.

[00:03:44]That means AM it will be 3 and PM it will be three and also angle ABP is two theta.

[00:03:59]That means this angle is theta and this angle is theta.

[00:04:08]And also AB is X and BQ is X.

[00:04:18]So from point B if we make a perpendicular on AQ we have AQ is eight.

[00:04:34]And suppose this point is N then AN and QN they will be AQ divided by two.

[00:04:50]So AN is four and QN is also four.

[00:04:58]And now in triangle ABM angle at M is 90° angle at B is theta.

[00:05:13]So this angle it will be 90° minus theta.

[00:05:24]And now in ANBM it is A N B M this angle is 90° This angle is also 90° and angle at A is theta minus 90° minus theta that is also 90° That means this angle it will be also 90° That means it is a rectangle and AN is four so BM it will be also four BM is four and now X is squared it will be four is squared minus three is squared X is squared is four is squared minus three is squared so X is squared is 25 That means X is five >> [music]