Problem 1

Let P be a point in the interior of triangle ABC. Let D, E, F be the intersections of AP with BC, BP with CA, and CP with AB, respectively. Prove that the area of triangle ABC is 6 if each of triangles PFA, PDB, and PEC has area 1.

Step 4 of 5: Force equality

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x\le y,z\Longrightarrow y=z\Longrightarrow x=y=z;\qquad (x-1):1=y:z=1:1\Longrightarrow x=2.

Detailed analysis

Assume x is the smallest. If y>z then (y+1)z>(z+1)x, contradicting the common-product equality; if y<z, the other comparison gives a contradiction. Thus y=z, and then the equations give x=z. Since y:z=1:1=(x-1):1, x=2, and all three are 2.