Remember \(x^2 \geq 0\)
In general, even powers of real numbers must be positive, which can help you prove something is \(\geq 0.\) This is a helpful quirk of the real numbers worth keeping near the forefront of your mind.
Cauchy-Schwarz Inequality
For any collection of real numbers \((a_1, a_2, \dotsc, a_n)\) and \((b_1, b_2, \dotsc, b_n),\) \[ \Biggl(\sum_{i=1}^{n}a_i b_i\Biggr)^2 \leq \Biggl(\sum_{i=1}^{n}a_i^2 \Biggr)\Biggl(\sum_{i=1}^{n}b_i^2\Biggr). \] Phrased more generally, if you’ve ever seen an inner product space before, for vectors \(\bm{x}\) and \(\bm{y},\) \( |\langle \bm{x}, \bm{y} \rangle| \leq \|\bm{x}\|\,\|\bm{y}\|\,. \) The specific version of an inner product you may have seen before is the dot product on \(\mathbf{R}^n,\) in which case the Cauchy-Schwarz inequality boils down to the fact that \[ \bm{x}\cdot\bm{y} = \|\bm{x}\|\|\bm{y}\|\cos\big(\theta\big) \leq \|\bm{x}\|\|\bm{y}\| \,.\]
Arithmetic Mean, Geometric Mean Inequality
For nonnegative real numbers \(a_1, a_2, \dotsc, a_n,\) \[ \sqrt[n]{a_1a_2\dotsb a_n} \leq \frac{a_1+a_2+\dotsb+a_n}{n}\,, \] where equality holds only when all the \(a_i\) are equal. The expression on the left-hand side of the inequality is called the geometric mean of those numbers, and the expression on the right-hand side is called the arithmetic mean. Occasionally this inequality will be extended to include another expression, the harmonic mean, too, \[ \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_n}} \leq \sqrt[n]{a_1a_2\dotsb a_n} \leq \frac{a_1+a_2+\dotsb+a_n}{n}\,. \] which altogether constitute the three Pythagorean means of antiquity.
Expanding Binomials and Trinomials
The fact that a binomial expands as \[(a+b)^n = \sum_{i=0}^{n} \binom{n}{i}a^ib^{n-i}\,\] and that these binomial coefficients can be thought of as entries in Pascal’s triangle, but it’s less commonly known as a fact how the trinomial \((a+b+c)^n\) expands. The expansion will still be a summation of terms of the form \(q_{ijk} a_ib_jc^k\) where \(i+j+k = n\) and the coefficients \(q_{ijk}\) can be read off of a three-dimensional analogue of Pascal’s triangle: Pascal’s pyramid? Do these coefficients have a combinatorial description?