Problem 1

Let

ABC

be a triangle with

\angle A<60^\circ

. Let

X

and

Y

be the points on the sides

AB

and

AC

, respectively, such that

CA+AX=CB+BX

and

BA+AY=BC+CY

. Let

P

be the point in the plane such that the lines

PX

and

PY

are perpendicular to

AB

and

AC

, respectively. Prove that

\angle BPC<120^\circ

Step 5 of 5: Finish the angle bound

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\angle BPC<\angle BOC=2\angle A<120^\circ

Detailed analysis

For P outside the circumcircle of BOC on the relevant side of BC, the inscribed angle subtending BC is smaller than the corresponding angle at O. Therefore angle BPC<angle BOC=2 angle A<120 degrees.