Mathematical Problems －find best solutions－

Problem　１

Answer the following for △ABC on plane,

(1) Prove  sinA+sinB≦ sinC

(2) Prove the following inequality and answer the type of triangle when the equality sign holds.

sinA+sinB+sinC≦cosA/2+cosB/2+cosC/2

Problem 2

There is hyperbola x^2/a^2－ｙ^2/b^2＝１, where a>0, b>0 , on xy-plane with the origin O. Let A be a point at (p,q) on the hypabola. Let S and T be the points of intersection of the tangent line atpoint A and two asymptotes of the hypabola. Show that the area of △OST is constant, independent of point A.

Problem 3

Find all sets of integers, (x,y,z) satisfying the following equality.

x^2+ｙ^2+ｚ^2＝２xyz

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