Maths Olympiad Exam

Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose
BC = AB + AI. Find BAC.

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Maths Olympiad Other Question

Let ABC be a triangle in which A = 60. Let BE and CF be the bisectors of the angles B and C with E on AC and F on AB. Let M be the reflection of A in the line EF. Prove that M lies on BC. A convex polygon is such that the distance between any two vertices of does not exceed 1.
(i) Prove that the distance between any two points on the boundary of does not exceed 1.
(ii) If X and Y are two distinct points inside, prove that there exists a point Z on the boundary of such that XZ + Y Z 1.