Algebraists' Homage: Papers in Ring Theory and Related by S. A. Amitsur, D. J. Saltman, George B. Seligman

By S. A. Amitsur, D. J. Saltman, George B. Seligman

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If the vertices of the triangle XY Z have coordinates X = (x, f (x)), Y = (y, f (y)), Z = (z, f (z)), then the area of the triangle is given by ⎞ ⎛ 1 x f (x) 1 Δ = det A, where A = ⎝ 1 y f (y) ⎠ . 2 1 z f (z) 4A function f : [a, b] → R is differentiable in a point c ∈ [a, b] if the function f (c) = [a, b] if it is differentiable in every point of A. f (x)−f (c) exists and f is differentiable in A ⊂ x−c x→c 5 Mean value theorem. For a continuous function lim f : [a, b] → R, which is differentiable in (a, b), there exists a number x ∈ (a, b) such that f (x)(b − a) = f (b) − f (a).

26. The lengths a, b and c of the sides of a triangle satisfy ab+bc+ca = 3. Prove that √ 3 ≤ a + b + c ≤ 2 3. 27. Let a, b, c be the lengths of the sides of a triangle, and let r be the inradius of the triangle. Prove that √ 1 1 1 3 + + ≤ . 28. Let a, b, c be the lengths of the sides of a triangle, and let s be the semiperimeter of the triangle. Prove that (i) (s − a)(s − b) < ab, (ii) (s − a)(s − b) + (s − b)(s − c) + (s − c)(s − a) ≤ ab + bc + ca . 29. If a, b, c are the lengths of the sides of an acute triangle, prove that a 2 + b 2 − c2 a 2 − b 2 + c2 ≤ a 2 + b 2 + c2 , cyclic where cyclic stands for the sum over all cyclic permutations of (a, b, c).

Conversely, we will observe that f is convex if U is convex. Let x1 , x2 ∈ [a, b] and let us consider A = (x1 , f (x1 )) and B = (x2 , f (x2 )). Clearly A and B belong to U , and since U is convex, the segment that joins them belongs to U , that is, the points of the form tB + (1 − t)A for t ∈ [0, 1]. Thus, (tx2 + (1 − t)x1 , tf (x2 ) + (1 − t)f (x1 )) ∈ U , but this implies that f (tx2 + (1 − t)x1 ) ≤ tf (x2 ) + (1 − t)f (x1 ). Hence f is convex. 6. A function f : [a, b] → R is convex if and only if, for each x0 ∈ (x0 ) [a, b], the function P (x) = f (x)−f is non-decreasing for x = x0 .

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