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Part of 2014 Putnam

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Putnam 2014 A2

Source:

12/7/2014
Let AA be the n×nn\times n matrix whose entry in the ii-th row and jj-th column is 1min⁡(i,j)\frac1{\min(i,j)} for 1≤i,j≤n.1\le i,j\le n. Compute det⁡(A).\det(A).
Putnamlinear algebramatrixinductioncollege contestsPutnam 2014Putnam matrices
Putnam 2014 B2

Source:

12/8/2014
Suppose that ff is a function on the interval [1,3][1,3] such that −1≤f(x)≤1-1\le f(x)\le 1 for all xx and ∫13f(x) dx=0.\displaystyle \int_1^3f(x)\,dx=0. How large can ∫13f(x)x dx\displaystyle\int_1^3\frac{f(x)}x\,dx be?
Putnamfunctionintegrationreal analysisabsolute valuePutnam calculusPutnam 2014