2016 IMO Problems/Problem 1

Pr♦❜❧❡♠ ✶✳ ❚r✐❛♥❣❧❡ BCF ❤❛s ❛ r✐❣❤t ❛♥❣❧❡ ❛t B✳ ▲❡t A ❜❡ t❤❡ ♣♦✐♥t ♦♥ ❧✐♥❡ CF s✉❝❤ t❤❛t FA = FB ❛♥❞ F ❧✐❡s ❜❡t✇❡❡♥ A ❛♥❞ C✳ P♦✐♥t D ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t DA = DC ❛♥❞ AC ✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ∠DAB✳ P♦✐♥t E ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t EA = ED ❛♥❞ AD ✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ∠EAC✳ ▲❡t M ❜❡ t❤❡ ♠✐❞♣♦✐♥t ♦❢ CF✳ ▲❡t X ❜❡ t❤❡ ♣♦✐♥t s✉❝❤ t❤❛t AMXE ✐s ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ✭✇❤❡r❡ AM k EX ❛♥❞ AE k MX✮✳ Pr♦✈❡ t❤❛t ❧✐♥❡s BD✱ FX✱ ❛♥❞ ME ❛r❡ ❝♦♥❝✉rr❡♥t✳