THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh March 9, 2013
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has at least n+ 1 distinct prime divisors. (b) When a = 3, determine all the positive integers n for which the assertion in (a) is false. 2. ABCD is a square; points U and V are situated on the respective sides BC and CD. Prove that the perimeter of triangle CUV is equal to twice the sidelength of the square if and only if ∠UAV = 45◦. 3. Let f(x) be a convex increasing realvalued function defined on the closed interval [0, 1] for which f(0) = 0 and f(1) = 1. Suppose that 0 < a < 1 and that b = f(a). (a) Prove that f is continuous on (0, 1). (b) Prove that 0 ≤ a− b ≤ 2 ∫ 1
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THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh
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THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh
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THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh
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THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh
3. Let n be a positive integer. A finite sequence {a1, a2, · · · , an} of positive integers ai is said to be tight if and only if 1 ≤ a1 < a2 < · · · < an, all ( n 2 ) differences aj − ai with i < j are distinct, and an is as small as possible. (a) Determine a tight sequence for n = 5. (b) Prove that there is a polynomial p(n) of degree not exceeding 3 such that an ≤ p(n) for every tight sequen...
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1. Determine the supremum and the infimum of (x− 1)x−1xx (x− (1/2))2x−1 for x > 1. 2. Let n and k be integers with n ≥ 0 and k ≥ 1. Let x0, x1, · · ·, xn be n+1 distinct points in R and let y0, y1, · · ·, yn be n + 1 real numbers (not necessarily distinct). Prove that there exists a polynomial p of degree at most n in the coordinates of x with respect to the standard basis for which p(xi) = yi ...
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