Things quoted as "obvious" in various proofs I have read:

• The reflection of the orthocenter about any of a triangle's sides lies on the circumcircle of the triangle.
• An angle bisector of a triangle and the perpendicular bisector of the opposite side meet on the circumcircle of the triangle.
• If ABCD is cyclic, EF is parallel to AD, E is on AB or an extension thereof, F is on CD or an extension thereof, then EFBC is cyclic.

None of them are exactly hard to prove, but I just don't notice them, apparently. Statements like these: too easy to be a notable theorem but too hard to notice singularly. But even without item 3 up there I managed to finish the last problem.
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I really need to sort out what things are obvious and what things have to be explained algebraic manipulation after algebraic manipulation. Okay, fine. Next problem, and a hopefully really easy one until I stop being so lazy.

IMO 1968.1. Prove that every tetrahedron contains a vertex with three edges that can form a triangle.