Real Analysis

  1. Putnam & Beyond

    Consider a function \(f \colon \mathbf{R} \to \mathbf{R}\) defined as \[ f(x) = (x-a)(x-b) + (x-b)(x-c) + (x-a)(x-c) \] for some real numbers \(a,\) \(b,\) and \(c.\) Prove that \(f(x) \geq 0\) for all real numbers \(x\) if and only if \(a=b=c.\)

  2. MIT (Exercise)

    Show that \( \int_{0}^{\infty}\frac{\cos(ax)}{1+x^2}\,\mathrm{d}x \) converges for all \(a \in \mathbf{R}\) and compute its value.

  3. MIT (Exercise)

    For each continuous function \(f \colon [0,1] \to \mathbf{R},\) let \(I(f)\) denote \(\int_0^1 x^2f(x)\,\mathrm{d}x \) and let \(J(f)\) denote \(\int_0^1 xf(x)^2\,\mathrm{d}x.\) Find the maximum value of \(I(f) - J(f)\) over all such functions \(f.\)

  4. MIT

    Suppose that \(f\) and \(g\) are non-constant, differentiable, real-valued functions defined on \((-\infty, \infty).\) Furthermore suppose that for each pair of real numbers \(x\) and \(y\) we have \[ f(x+y) = f(x)f(y) - g(x)g(y) \qquad g(x+y) = f(x)g(y) + g(x)f(y)\,. \] If \(f'(0) = 0,\) prove that \(\bigl( f(x)\bigr)^2 +\bigl( g(x)\bigr)^2 = 1\) for all \(x.\)

  5. Putnam & Beyond

    Let \(f \colon \mathbf{R} \to \mathbf{R}\) be a continuous function satisfying \(f(x) = f\bigr(x^2\bigr)\) for all \(x \in \mathbf{R}.\) Prove that \(f\) is constant.

  6. MIT

    Let \(f\) be a twice-differentiable real-valued function such that \[ f(x) + f''(x) = -xg(x)f'(x) \,. \] for some function \(g(x) \geq 0\) for all real \(x.\) Prove that \(|f(x)|\) is bounded.

  7. Putnam 2011 B1

    Let \(h\) and \(k\) be positive integers. Prove that for every \(\epsilon \gt 0\), there are positive integers \(m\) and \(n\) such that \[ \epsilon \lt \bigl|h \sqrt{m} - k \sqrt{n}\bigr| \lt 2\epsilon. \]

  8. Putnam 1973

    Suppose that on the domain \([0,1]\) a function \(f\) has a continuous derivative satisfying \(0 \lt f'(x) \leq 1.\) Further suppose that \(f(0) = 0.\)

    1. Prove that \( \int_0^1 f(x)^3 \,\mathrm{d}x \;\leq\; \Bigr( \int_0^1 f(x)\,\mathrm{d}x\Bigl)^2 \)
    2. Find an example for which equality of the previous inequality occurs.

  9. Putnam 2010 B5

    Is there a strictly increasing (differentiable) function \(f\colon \mathbf{R} \to \mathbf{R}\) such that \(f'(x) = f\bigl(f(x)\bigr)\) for all \(x?\)

  10. Putnam 2008 A4

    Define \(f\colon \mathbf{R} \to \mathbf{R}\) by \[ f(x) = \begin{cases} x & \quad\text{if } x \leq \mathrm{e} \\ x f\bigl(\ln(x)\bigr) & \quad\text{if } x \gt \mathrm{e}. \end{cases} \] Does \(\sum_{n=1}^\infty \frac{1}{f(n)}\) converge?

  11. Putnam 2000 B4

    Let \(f(x)\) be a continuous function such that \(f(2x^2-1)=2xf(x)\) for all \(x.\) Show that \(f(x)=0\) for \(-1\leq x\leq 1.\)

  12. Putnam 2014 B2

    Suppose that \(f\) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \(x\) and \(\int_1^3 f(x)\,\mathrm{d}x = 0.\) How large can \(\int_1^3 \frac{f(x)}{x}\,\mathrm{d}x\) be?

  13. Putnam 2018 A5

    Let \(f\colon \mathbf{R} \to \mathbf{R}\) be an infinitely differentiable function satisfying \(f(0) = 0,\) \(f(1)= 1,\) and \(f(x) \geq 0\) for all \(x \in \mathbf{R}\). Show that there exist a positive integer \(n\) and a real number \(x\) such that \(f^{(n)}(x) \lt 0.\)

  14. Putnam 2012 A3

    Let \(f\colon [-1, 1] \to \mathbf{R}\) be a continuous function such that

    • \(f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)\) for every \(x\) in \([-1, 1],\)
    • \(f(0) = 1,\) and
    • \(\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}\) exists and is finite.

    Prove that \(f\) is unique, and express \(f(x)\) in closed form.