1. Two lines \ell and m intersect at O at an angle of 28^\circ. Let A be a point inside the acute angle formed by \ell and m. Let B and C be the reflections of A in lines \ell and m, respectively. Find the number of degrees in \angle BAC.

2.Equilateral triangle ABC has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC].

3.A sphere with radius 3 is inscribed in a conical frustum of slant height 10. (The sphere is tangent to both bases and the side of the frustum.) Find the volume of the frustum.

4.In tetrahedron ABCD, \angle ADB = \angle ADC = \angle BDC = 90^\circ. Let a = AD, b = BD, and c = CD.

(a) Find the circumradius of tetrahedron ABCD in terms of a, b, and c. (The circumradius of a tetrahedron is the radius of the sphere that passes through all 4 vertices of the tetrahedron.)

(b) Let O be the circumcenter of tetrahedron ABCD. Prove that \overline{OD} passes through the centroid of triangle ABC.

(The circumradius of a tetrahedron is the radius of the sphere that passes through all four vertices, and the circumcenter is the center of this sphere.)

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