Download e-book for kindle: Contest Problem Book III: Annual High School Contest by Charles T. Salkind, James M. Earl

By Charles T. Salkind, James M. Earl

ISBN-10: 0883856255

ISBN-13: 9780883856253

The once a year highschool contests were subsidized on the grounds that 1950 by way of the Mathematical organization of the US and the Society of Actuaries, and extra lately via Mu Alpha Theta (1965), the nationwide Council of academics of arithmetic (1967) and the Casualty Actuarial Society (1971). difficulties from the contests throughout the interval 1950-1960 are released in quantity five of the recent Mathematical Library, and people for 1961-1965 are released in quantity 17. the recent Mathematical Library will proceed to put up those contest difficulties from time-to-time; the current quantity includes these from the interval 1966-1972. The questions have been compiled by way of Professor C.T. Salkind until eventually his dying, and because 1968 via Professor J.M. Earl, who died on November 25, 1972 after filing difficulties for the 1973 contest. Professors Earl and Salkind additionally ready the recommendations for the competition difficulties. In getting ready this and the sooner Contest challenge Books the editors of the NML have accelerated those recommendations and additional substitute ideas.

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Extra resources for Contest Problem Book III: Annual High School Contest 1966-1972 : Of the Mathematical Association of America : Society of Actuaries : Mu Alpha Theta (New Mathematical Library)

Example text

The letter s always represents a complex scalar. The letter z may or may not represent a complex scalar. The inequalities c ≤ a ≤ d and c ≤ b ≤ d are written simultaneously as c≤ a b ≤ d. The prefix “non” means “not” in the words nonconstant, nonempty, nonintegral, nonnegative, nonreal, nonsingular, nonsquare, nonunique, and nonzero. ” “Increasing” and “decreasing” indicate strict change for a change in the argument. The word “strict” is superfluous, and thus is omitted. Nonincreasing means nowhere increasing, while nondecreasing means nowhere decreasing.

Let I ⊆ R be a finite or infinite interval, let f : I → R. Then, the following statements are equivalent: i) f is convex. ii) f is continuous, and, for all x, y ∈ I, 2 1 1 1 3 (f [ 2 (x+y)]+f [ 2 (y+z)]+f [ 2 (x+z)] ≤ 13 [f (x)+f (y)+f (z)]+f [ 31 (x+y+z). (Remark: This result is Popoviciu’s inequality. See [1039, p. 4. 7. Let [a, b] ⊂ R, let f : [a, b] → R, and assume that f is convex. Then, b f [ 12 (a + b)] ≤ 1 b−a a f (x) dx ≤ 12 [f (a) + f (b)]. (Proof: See [1039, pp. 1. Let x and α be real numbers, and assume that x ≥ −1.

Iv) The symmetric hull of G is the graph sym(G) = (X, sym(R)). v) The transitive hull of G is the graph trans(G) = (X, trans(R)). vi) The equivalence hull of G is the graph equiv(G) = (X, equiv(R)). vii) G is reflexive if R is reflexive. viii) G is symmetric if R is symmetric. In this case, the arcs (x, y) and (y, x) in R are denoted by the subset {x, y} of X, called an edge. ix) G is transitive if R is transitive. x) G is an equivalence graph if R is an equivalence relation. xi) G is antisymmetric if R is antisymmetric.

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Contest Problem Book III: Annual High School Contest 1966-1972 : Of the Mathematical Association of America : Society of Actuaries : Mu Alpha Theta (New Mathematical Library) by Charles T. Salkind, James M. Earl


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