By Yuan Y.
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32 Problems for investigation 27 (b) Is the converse true: If any plane meeting a surface S in more than one point intersects it in the points of a circle, is S necessarily a sphere? Part D: Solutions Section 1: Iterating Problem 1 (E) (a) Our first task is to investigate whether any natural number no with more than one digit is greater than the sum of its digits. , ... , aoare the digits of no. The coefficient am is greater than 0, and since no has more than one digit, m > 1. Denote by n. the sum am + am-.
Similarly, since AC, FG, HJ, and KL are all parallel, they can be produced to construct the parallelograms P 2 , P4 , P6 , and P8 (Fig. 49). The parallelograms P 2 and P 4 are in perspective from centre M, P 4 and P 6 are in perspective from centre P, and P 6 , and P 8 are in perspective from R. , 8 3 , 8 5 , • • • • Since the parallelograms PI and P 2 are congruent and their corresponding sides meet at right angles, their corresponding diagonals must also be perpendicular to one another. Thus f.
Investigate: Is the isosceles right-angled triangle the only exception? Can one inscribe rectangles of given perimeter 2s in right-angled triangles ABC, such that 2s =1= AC + CB, in the way shown in Fig. 21, in any other case? Problem 22 (E) (a) Are there any convex polygons other than obtuse-angled triangles in which one angle is greater than the sum of the remaining angles? (b) Are there convex n-gons with n acute angles for any value of n? Problem 23 (E) Prove that in any parallelogram the bisectors of the angles determine a rectangle R (Fig.