Charles T. Salkind's Contest Problem Book No 1: Annual High School Mathematics PDF

By Charles T. Salkind

ISBN-10: 0883856050

ISBN-13: 9780883856055

A good many scholars have participated every year within the Annual highschool arithmetic exam (AHSME) backed via the Mathematical organization of the US (MAA) and 4 different nationwide organisations within the mathematical sciences.* In 1960, 150,000 scholars participated from approximately 5,200 excessive colleges. In 1980, 416,000 scholars participated from over 6,800 excessive faculties. considering 1950, while the 1st of those examinations was once given., American highschool scholars have demonstrated their abilities and ingenuity on such challenge as: The rails on a railroad are 30 ft lengthy. because the teach passes over the purpose the place the rails are joined, there's an audible click on. the rate of the teach in miles consistent with hour is nearly the variety of clicks heard in what percentage seconds? and so on, in keeping with the highschool curriculum in arithmetic.

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Additional info for Contest Problem Book No 1: Annual High School Mathematics Examinations 1950-1960 (New Mathematical Library) (Bk. 1)

Example text

The mapping (: N x N ~ N maps an ordered pair of natural numbers m onto a single natural number by the rule ([(m, n)] = 2 + l (2n + 1). Show that ( is an injective mapping. 7. Let E be the set of all even natural numbers. For each n E N, let ((n) 2n. Prove that the mapping ( : N -+ E is injective. = 34 Fundamental Concepts of Mathematics 8. Prove that the mapping ( : N ((n) ~ =n N defined by - 1 for n ~ 1 and ((0) = 0 is not injective. 9. Prove that the mapping ( : N ~ N defined by ((n) = n + 2 is not surjective.

That if a F b then Ha}, {a, b}} F Hb}, {b, an. Most importantly we can prove the following crucial theorem. 1. Theorem. Let a, b, c and d be objects. Then Ha}, {a,bn = {{c}, {c,d}} ifandonlyifa=candb=d. 9, if a = c and b = d, then Ha}, {a, b}} He}, {c, d}}. Thus the 'if' part of the theorem is trivial. Let us prove the 'only if' part. Assume that Ha}, {a, b}} = He}, {c, dD. By Proof. 9 again, there are the following two possible cases for the elements of these sets: (i) {a} = {c} and {a, b} = {c, d}, or (ii) {a} = {c, d} and {a, b} = {c}.

Example, Prove that 0 + 1 + 2 + ... + n natural numbers n = 2 (n + 1) holds for all n. Proof. Let S be the set of natural numbers n for which the formula o + 1 + 2 + ... + n =2"n (n + 1) holds: S = {n E NI 0 + 1 + 2 ... + n ="2n (n + 1)}. We want to prove that S = N. For this purpose we need only show that conditions 0) and Oi) are satisfied. Condition (i) is trivial. For Oi), let us suppose that k E S. Then k o + 1 + 2 + ... + k + (k + 1) ="2 (k + 1) + (k + 1) = (k ; 1) (k + 2), Therefore (k + 1) E S.

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Contest Problem Book No 1: Annual High School Mathematics Examinations 1950-1960 (New Mathematical Library) (Bk. 1) by Charles T. Salkind


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