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# PROBLEMS FOR MAY

E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3.

no later then June 30, 2002.

145.
Let ABC be a right triangle with ÐA = 90°. Let P be a point on the hypotenuse BC, and let Q and R be the respective feet of the perpendiculars from P to AC and AB. For what position of P is the length of QR minimum?

146.
Suppose that ABC is an equilateral triangle. Let P and Q be the respective midpoint of AB and AC, and let U and V be points on the side BC with 4BU = 4VC = BC and 2UV = BC. Suppose that PV are joined and that W is the foot of the perpendicular from U to PV and that Z is the foot of the perpendicular from Q to PV.
Explain how that four polygons APZQ, BUWP, CQZV and UVW can be rearranged to form a rectangle. Is this rectangle a square?

147.
Let a > 0 and let n be a positive integer. Determine the maximum value of

 x1 x2 ¼xn(1 + x1)(x1 + x2) ¼(xn-1 + xn)(xn + an+1)
subject to the constraint that x1, x2, ¼, xn > 0.

148.
For a given prime number p, find the number of distinct sequences of natural numbers (positive integers) { a0, a1, ¼, an ¼} satisfying, for each positive integer n, the equation

 a0a1 + a0a2 + ¼+ a0an + pan+1 = 1 .

149.
Consider a cube concentric with a parallelepiped (rectangular box) with sides a < b < c and faces parallel to that of the cube. Find the side length of the cube for which the difference between the volume of the union and the volume of the intersection of the cube and parallelepiped is minimum.

150.
The area of the bases of a truncated pyramid are equal to S1 and S2 and the total area of the lateral surface is S. Prove that, if there is a plane parallel to each of the bases that partitions the truncated pyramid into two truncated pyramids within each of which a sphere can be inscribed, then

 S = ( __ÖS1 + __ÖS2 )(Ö[4 ]S1 +Ö[4 ]S2)2 .

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