Inequalities∗Math 294A: Problem Solving Seminar – Vera FurstMarch 21, 2007• Arithmetic-mean–geometric-mean inequality: For x1, . . . , xn> 0,(x1x2···xn)1/n≤x1+ x2+ ··· + xnnwith equality if and only if x1= x2= ··· = xn.• Cauchy-Schwarz inequality: For all real a1, . . . , anand b1, . . . , bn,nXi=1aibi≤ nXi=1a2i!1/2 nXi=1b2i!1/2,with equality if and only if a1/b1= a2/b2= ··· = an/bn≥ 0. In vector form:~a ·~b ≤ k~akk~bk,with equality if and only if ~a = c~b for some nonnegative scalar c.Example 1. (Bernoulli’s inequality) Prove that for 0 < a < 1,(1 + x)a≤ 1 + axfor all x ≥ −1. How would the inequality change if a > 1 or a < 0?Example 2. Suppose that f : R → R is a twice differentiable function with f00(x) ≥ 0 for all x. Provethat for all a < b,fa + b2≤f(a) + f(b)2.Deduce the arithmetic-mean–quadratic-mean inequality for two numbers.Example 3. If a, b, c, d are positive numbers such that c2+ d2= (a2+ b2)3, prove thata3c+b3d≥ 1,with equality if and only if ad = bc.Problem 1. Let a, b, c be the sides of a triangle. Show thatab + bc + ca ≤ a2+ b2+ c2≤ 2(ab + bc + ca).∗These problems (or some version of them) all appear either on previous Putnam exams or in the books Problem-SolvingThrough Problems by L.C. Larson and Problem-Solving Strategies by A. Engel.1Problem 2. A farmer with 1000 feet of fencing wishes to fence a rectangular field adjacent to a straightriver. Naturally, no fence is needed along the river. Use the arithmetic-mean–geometric-mean inequalityto find the dimensions of the field that maximize its area.Problem 3. Prove that for each positive integer n,1 +1nn<1 +1n + 1n+1.We proved this inequality last semester using the binomial theorem; this time, do it by showing thatf(x) = (1 + 1/x)xis an increasing function.Problem 4. (Jensen’s inequality) Let f : R → R be a twice differentiable function with f00(x) ≥ 0 forall x. If λ1, . . . , λnare positive real numbers such that λ1+ ··· + λn= 1, thenf nXi=1λixi!≤nXi=1λif(xi)for any x1, . . . , xn∈ R. Deduce the general arithmetic-mean–geometric-mean inequality.Problem 5. For which real numbers k does the inequality cosh x ≤ ekx2hold for all real x?Problem 6. Show that if a, b, c are positive numbers with a + b + c = 1, thena +1a2+b +1b2+c +1c2≥1003.Problem 7. Show that if Ck=nkfor n > 2 and 1 ≤ k ≤ n, thennXk=1pCk≤pn(2n− 1).Problem 8. Show that for any integer n > 1,1e−1ne<1 −1nn<1e−12ne.Problem 9. Given n ≥ 2, which of the two numbers is larger:(a) An exponential tower of n 2’s or an e xponential tower of (n − 1) 3’s?(b) An exponential tower of n 3’s or an e xponential tower of (n − 1) 4’s?Problem 10. Let x, y > 0, and let s be the smallest of the numbers x, y + 1/x, 1/y. Find the greatestpossible value of s. For which x, y is this value assumed?Problem 11. Suppose twenty disjoined squares lie inside a square of side 1. Prove that there are foursquares among them such that the sum of the lengths of their sides does not exceed 2/√5.Problem 12. Find all positive integers n such that3n+ 4n+ ··· + (n + 2)n= (n +
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