Math
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 +…
Math Formulas
$(a-b)^3=a^3-3a^2 b+3a b^2-b^3$ (remember the 3’s) $(a+b)^3=a^3+3a^2 b+3a b^2+b^3$ (remember the 3’s) $1^2 + 2^2 + … + n^2 = \frac{n(n+1)(2n+1)}{6}$ $1^3 + 2^3 + … + n^3 = \frac{n^2 (n+1)^2}{4}$ $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ For an $m \times n$ grid of numbers satisfying: let $a_{i,j}$ = the entry in…
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…
CMOQR 2026 P3
Problem: Let point $P$ be outside circle $\Gamma$. The tangents from $P$ to $\Gamma$ hit $\Gamma$ at $A$ and $B$. A third line through $P$ hits $\Gamma$ at $C$ and $D$, such that $C$ is between $P$ and $D$. Point $Q$ is on chord $CD$ such that $\angle DAQ$ =…
Algorithm for Finding Primes
Using brute force isn’t particularly the best solution when it comes to determining primes. Brute Force Example: Sieve of Eratosthenes: This method tries all the numbers, but as soon as a prime number is found all the multiples of it is marked as not prime, meaning there are less options…
Python Simple and Compound Interest Calculator
Don’t have a calculator? Don’t worry. Just paste this code into Python and let Python do all the work.
SVB Bank
The Silicon Valley Bank Became Bankrupt Silicon Valley Bank, SVB for short, was ranked number 63 in 2021 according to the USA’s top banks. In 2022, it skyrocketed to number 16, an insane feat. In 2023 however, it became bankrupt within a span of 2 days. Here is how this…
Conjugate
Conjucates are very useful when it comes to simplifying fractions with an irrational denominater. A conjugate is where you change the signs between two terms. Take a look at the polynomial below: If you try to add this fraction to a rational fraction like 1/3, it will be very exhausting…
How to Find if a Number is Divisible by Another (Up to 20)
1: Yes. 2: Examine the last digit of N. If it is an even number, then N is divisible by 2. 3: Add the sum of the digits of N and call it P. If P is divisible by 3, than N is divisible by 3. 4: Take the last…