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BdMO Online Forum • View topic - IGO 2016 Medium/3

IGO 2016 Medium/3

For discussing Olympiad level Geometry Problems

IGO 2016 Medium/3

3. Find all positive integers \$N\$ such that there exists a triangle which can be dissected into \$N\$ similar quadrilaterals.
Last edited by Thamim Zahin on Wed Jan 11, 2017 4:11 pm, edited 1 time in total.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

Thamim Zahin

Posts: 97
Joined: Wed Aug 03, 2016 5:42 pm

Re: IGO 2016 Mediam/3

It is possible for all integer \$N \ge 3\$

It is obvious that a triangle can't be partitioned in \$1\$ quadrilateral.

Also if we divide a triangle into \$2\$ quadrilaterals, one is convex but other one is not.

For \$N=3\$, take a equilateral triangle. And divide the triangle into \$3\$ congruent quadrilaterals. Like the figure. \$O\$ is the circumcenter. We draw \$\angle OFB\$ such that it is equal to \$60^o\$. Also same thing for \$ \angle ODC \$ and \$ \angle OEA\$.

It is easy to prove that the quadrilaterals are congruent.

Now we can make a quadrilateral \$BCYX\$ such that it is similar to \$EOFB\$. And \$\angle ABX = \angle ACY = 60^o+120^o=180^o\$. So that means \$\triangle AXY\$ is a new triangle with \$5\$ similar quadrilaterals. We can make this process over and over. So it is possible for all \$N \ge 3\$
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I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

Thamim Zahin

Posts: 97
Joined: Wed Aug 03, 2016 5:42 pm