Problem 15 (trigonometry): Always observe the answer options before you start to work on a problem. For example, if the answer options show bounds, then don’t go ahead and try to calculate the actual value. Problem 16 (circles): Descartes’ Circle Theorem states that $2(a^2 + b^2 + c^2 + d^2)=(a+b+c+d)^2$,…
Algebra
AMC 12A 2018 Notes
Problem 11 (paper folding): When doing problems about folding something, just draw a perpendicular line across the midpoint and that is the line of reflection. Problem 15 (symmetry): Always consider symmetry, but also consider the edge cases too. Problem 17 (area geometry): Not everything is as it seems! Also, don’t…
Lemma Inequality For The Sum of Cubes of Two Terms
Lemma. If $a, b \geq 0$, then $a^3 +b^3 \geq a^2 b + a b ^ 2$. Proof. Since $a,b\geq 0$, $a+b\geq 0$. Also, by the trivial inequality, $(a-b)^2 \geq 0$. Thus, $(a-b)^2 (a+b)\geq 0$. Expanding, $a^3 + b^3 – a^2 b – a b^2 \geq 0$. So, $a^3 +…
Arithmetic Sequence Problem
If $p, q$ are distinct natural numbers, and in an arithmetic sequence {${a_n}$}, $S_p = S_q$, where $S_k$ denotes the sum of the first $k$ terms of the sequence {${a_n}$}. Prove that $S_{p+q}=0$. The formula for an arithmetic series is $S_n = n a_1 + \frac{n(n-1)}{2}d$. So, $S_p = p a_1+ \frac{p(p-1)}{2}d$ and $S_q = q a_1+ \frac{q(q-1)}{2}d$. Since $S_p…