# Contest Problem Book No 1: Annual High School Mathematics by Charles T. Salkind By Charles T. Salkind

An excellent many scholars have participated each year within the Annual highschool arithmetic exam (AHSME) subsidized by way of the Mathematical organization of the USA (MAA) and 4 different nationwide agencies 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 colleges. when you consider that 1950, while the 1st of those examinations was once given., American highschool scholars have validated 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 educate in miles in line with hour is nearly the variety of clicks heard in what percentage seconds? etc, in accordance with the highschool curriculum in arithmetic.

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8. (Pointwise convergence) Let f , f1 , f2 , . . ∈ C ([0, 1]), the space of all continuous functions on the interval [0, 1]. We say that the sequence (fn ) is pointwise convergent to f if |fn (t) − f (t)| → 0, as n → ∞, for every t ∈ [0, 1]. We will show that there is no norm on C ([0, 1]), which defines pointwise convergence. Suppose, on the contrary, that · is a norm on C ([0, 1]) such that fn − f → 0 if and only if fn (t) → f (t) for every t ∈ [0, 1]. 7) Consider the sequence of functions g1 , g2 , .

We can say that a mapping L from a subset S = D (L) of a vector space E1 into a vector space E2 is linear if L(αx + βy) = αLx + βLy for all x, y ∈ S and all scalars α, β such that αx + βy ∈ S. On the other hand, such a mapping L has a unique extension to a linear mapping from the vector space span D (L) into E2 . Indeed, if y ∈ span D (L), then y = λ1 x1 + · · · + λn xn , for some x1 , . . , xn ∈ D (L) and some scalars λ1 , . . , λn , and we can define Ly = λ1 Lx1 + · · · + λn Lxn . For this reason, we can always assume, without loss of generality, that the domain of a linear mapping is a vector space.

For n = 1, 2, . . , define an = qn and bn = qn + ε/2n+1 . The intervals [an , bn )Q will not be disjoint and need not be contained in [0, 1), but we can easily define new intervals [an , bn )Q that have those properties. The details of such a construction are left as an exercise. The next theorem describes an important property of the integral of step functions. 6. Let (fn ) be a nonincreasing sequence of non-negative step functions such that limn→∞ fn (x) = 0 for every x ∈ R. Then limn→∞ fn = 0.