By Akiva Moiseevič Âglom; Isaak Moiseevič Âglom; James McCawley; Basil Gordon
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Additional resources for Challenging mathematical problems with elementary solutions [Vol. II]
For notions of probability used in the next three problems, refer to Section VII (page 27) in Volume I. 155. * * * A rod is broken into three pieces; the two break points are chosen at random. What is the probability that an acute-angled triangle can be formed from the three pieces? 156. ** * A rod is broken in two at a point chosen at random; then the larger of the two pieces is broken in two at a point chosen at random. What is the probability that the three pieces obtained can be joined to form a triangle?
A" A 5• Ae. A 7• For I. Points and lines a. 49 b. Fig. 34 this it is sufficient to note that through the three points which lie on one of the lines there pass all the remaining lines (two through each of the points). Let us start now with three points A}t Aa. As and the three lines. say Plo Pa. Ps, which join these points in pairs (fig. 34). By the conditions of the problem, one of the seven given Jines (other than the sides of this triangle) must pass through each vertex of the triangle AIAaAs.
Prove that for any positive ZI and F(ZIZJ = F(z}) Z2 + F(zJ. 153. Prove that the function F(z) assumes the value 1 at some point between 2 and 3. In what follows we shall always use the letter e to denote the value of z for which F(z) = I. Thus the conclusion of 153 is that the number e exists and that 2 < e < 3. This number e plays an important part in mathematics and often appears in contexts that at first glance have nothing to do with its definition as the area under a hyperbola. See, for example, problems 158, 163, 164, and 80 (in vol.