Identities & Inequalities

  1. Putnam & Beyond

    Prove that for all real numbers \(x,\) \( 2^x + 3^x -4^x + 6^x -9^x \leq 1.\)

  2. Putnam & Beyond

    Prove that if \(a_1 + a_2 + \dotsb + a_n = n\) then \(a_1^4 + a_2^4 + \dotsb + a_n^4 \geq n.\)

  3. Putnam & Beyond

    For distinct real numbers \(a_1, a_2, \dotsc, a_n,\) find the maximum value of \(a_1a_{\sigma(1)} + a_2a_{\sigma(2)} + \dotsb + a_na_{\sigma(n)}\) over all possible permutations \(\sigma\) of the set \(\{1,2,\dotsc,n\}.\)

  4. Putnam & Beyond

    For a triangle with perimeter \(2\) and side-lengths \(a,\) \(b,\) and \(c,\) prove that \[ 1 \lt ab + bc + ca - abc \leq \frac{28}{27}. \]

  5. MIT

    Prove the “base case” of the Pythagorean mean inequality, that \[\frac{2xy}{x+y} \leq \sqrt{xy} \leq \frac{x+y}{2},\] with equality holding only in the case that \(x=y.\)

  6. MIT

    Suppose \(x\) and \(y\) are real numbers such that \(x^2 + y^2 = x + y.\) What is the largest possible value of \(x\,?\)

  7. Putnam 2019 A1

    Determine all possible values of the expression \[ A^3+B^3+C^3-3ABC \] where \(A,\) \(B,\) and \(C\) are nonnegative integers.

  8. Putnam 2003 A2

    Let \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\) be nonnegative real numbers. Show that \[ (a_1 a_2 \cdots a_n)^{1/n} + (b_1 b_2 \cdots b_n)^{1/n} \leq \Bigl((a_1+b_1) (a_2+b_2) \cdots (a_n + b_n)\Bigr)^{1/n}. \]

  9. Dylan McKnight

    Show that for all real numbers \(a\) and \(b\) and all real numbers \(p\) and \(q\) such that \(p^{-1} + q^{-1} = 1,\) \[ ab \leq \frac{a^p}{p}+ \frac{a^q}{q}\,. \]

  10. Putnam 2002 B3

    Show that, for all integers \(n \gt 1,\) \[ \frac{1}{2n\mathrm{e}} \lt \frac{1}{\mathrm{e}} - \biggl(1 - \frac{1}{n}\biggr)^n \lt \frac{1}{n\mathrm{e}}\,. \]

  11. Putnam 2021 B2

    Determine the maximum value of the sum \[ S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n} \] over all sequences \(a_1, a_2, a_3, \cdots\) of nonnegative real numbers satisfying \( \sum_{k=1}^\infty a_k = 1. \)

  12. Putnam & Beyond

    Prove that for all real numbers \(x,\) \(y,\) and \(z,\) \[ \frac{y^2-x^2}{2x^2+1} + \frac{z^2-y^2}{2y^2+1} + \frac{x^2-z^2}{2z^2+1} \geq 0. \]

  13. MIT

    Let \(x,y \gt 0\) and \(p\neq 0.\) The \(p\)-th power mean of \(x\) and \(y\) is defined to be \[M_p(x,y) = \biggl(\frac{x^p+y^p}{2}\biggr)^{1/p}.\] Note that \(M_{-1}(x,y)\) is the harmonic mean and \(M_1(x,y)\) is the arithmetic mean.
    1. For \(p \lt q,\) show that \(M_p(x,y) \leq M_q(x,y),\) with equality only when \(x = y.\)
    2. Compute the values of the following limits:
      \(\displaystyle \lim_{p \to \infty} M_p(x,y)\)
      \(\displaystyle \lim_{p \to 0} M_p(x,y)\)
      \(\displaystyle \lim_{p \to -\infty} M_p(x,y)\)

  14. MIT

    For \(p \gt 1\) and \(a_1, a_2, \dotsc, a_n\) all positive reals, show that \[ \sum_{k=1}^{n} \biggl(\frac{a_1+a_2+\dotsb+a_k}{k}\biggr)^p \lt \biggl(\frac{p}{p-1}\biggr)^p \]

  15. MIT

    For \(a_1, a_2, \dotsc \) all positive reals, provided that \(\sum_{n=1}^{\infty} a_n\) converges, show that \[ \sum_{n=1}^{\infty} \sqrt[n]{a_1a_2\dotsb a_n} \leq \mathrm{e} \sum_{n=1}^{\infty} a_n \]

  16. MIT

    For real numbers \(a_1, a_2, \dotsc, a_n\), show that \[ \min_{i\lt j}(a_i - a_j)^2 \leq \frac{12}{n(n^2-1)}\bigr(a_1^2 + \dotsb + a_n^2\bigl)\,. \]

  17. MIT

    For a real number \(t\) and positive integer \(n,\) define the function \(w\) by the formula \(w(t) = (t-1)(t-2)\dotsb(t-n+1).\) Prove that \[ \frac{1}{(n-1)!}\int\limits_{n}^{\infty} w(t)\mathrm{e}^{-t} \,\mathrm{d}t \;\lt\; \frac{1}{(\mathrm{e}-1)^n}.\]