- Determine the number of real solutions to the equation
√(2 -
*x*^{2}) =^{3}√(3 -*x*^{3}). - Evaluate
∫
_{0}^{a}*dx*/(1 + cos*x*+ sin*x*) for - π/2 <*a*< π. Use your answer to show that ∫_{0}^{π/2}*dx*/(1 + cos*x*+ sin*x*) = ln 2. - Let
*ABC*be a triangle and let*P*be a point in its interior. Suppose ∠*BAP*= 10^{o}, ∠*ABP*= 20^{o}, ∠*PCA*= 30^{o}and ∠*PAC*= 40^{o}. Find ∠*PBC*. - Let
*P*be an interior point of a triangle of area*T*. Through the point P, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let*a*,*b*and*c*be the areas of the three triangles. Prove that √*T*= √*a*+ √*b*+ √*c*. - Let
*f*(*x*,*y*) = (*x*+*y*)/2,*g*(*x*,*y*) = √*xy*,*h*(*x*,*y*) = 2*xy*/(*x*+*y*), and let*S*= {(*a*,*b*)∈ℕ×ℕ |*a*≠*b*and*f*(*a*,*b*),*g*(*a*,*b*),*h*(*a*,*b*)∈ℕ},*f*over*S*. - Let
*f*(*x*)∈ℤ[*x*] be a polynomial with integer coefficients such that*f*(1) = -1,*f*(4) = 2 and*f*(8) = 34. Suppose*n*∈ℤ is an integer such that*f*(*n*) =*n*^{2}- 4*n*- 18. Determine all possible values for*n*. - Find all pairs (
*m*,*n*) of nonnegative integers for which

*m*^{2}+2·3^{n}=*m*(2^{n+1}- 1).

Peter Linnell 2017-10-21