Brush up before you prove it.

Arithmetic¹ Level 1

Rules

• Order of operations: brackets first, then multiplication and division (left to right), then addition and subtraction (left to right). So \(6+2\times3=12\), not 24.
• Division with a remainder: \(23\div4\) is 5 remainder 3, because \(4\times5=20\) and \(23-20=3\). The remainder is always smaller than the number you divide by.
• Making change: change = amount paid − total cost. Add up everything bought first.
• Rounding to the nearest ten: look at the ones digit. 5 or more rounds up (67 becomes 70), less rounds down (63 becomes 60).

Algebra¹ Levels 2–3

Fractions, decimals and percentages

• Fractions: to add or subtract, rewrite over a common denominator. To divide, multiply by the reciprocal.
• Percent means "per hundred": \(p\%\) of \(x\) is \(\tfrac{p}{100}\cdot x\).

Linear equations and inequalities

• Solve by doing the same inverse operation to both sides until \(x\) is alone.
• Multiplying or dividing an inequality by a negative number flips its direction.
• A line is \(y=mx+b\): slope \(m\), y-intercept \(b\).

Quadratics

• Factoring \(x^2+bx+c\): find \(p,q\) with \(p+q=b\) and \(pq=c\), then \(x^2+bx+c=(x+p)(x+q)\).
• Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
• Zero-product rule: if \(AB=0\) then \(A=0\) or \(B=0\). So \((x-5)(x-2)=0\) gives \(x=5\) or \(x=2\).
• The graph of \(y=ax^2+bx+c\) is a parabola: it opens up if \(a>0\), down if \(a<0\), with its vertex at \(x=-\tfrac{b}{2a}\).

Logarithms

Precalculus¹ Level 4

Functions and trigonometry

• Composition: \((f\circ g)(x)=f(g(x))\). Apply \(g\) first.
• Radians: \(\pi\text{ rad}=180^\circ\).
• Unit circle: \(\sin\tfrac{\pi}{6}=\tfrac12\), \(\cos\tfrac{\pi}{3}=\tfrac12\), \(\sin\tfrac{\pi}{4}=\cos\tfrac{\pi}{4}=\tfrac{\sqrt2}{2}\), \(\sin 0=0\), \(\cos 0=1\).

Complex numbers

• \(i^2=-1\). Powers of \(i\) repeat every four: \(i,\,-1,\,-i,\,1\).
• The conjugate of \(a+bi\) is \(a-bi\), and \((a+bi)(a-bi)=a^2+b^2\).

Calculus 1² Level 5

Limits and continuity

Limit:

\(\lim_{x\to a}f(x)=L\) means \(f(x)\) gets arbitrarily close to \(L\) as \(x\) approaches \(a\). It exists only if both one-sided limits exist and are equal.

Continuous at \(a\):

\(f(a)\) is defined and \(\lim_{x\to a}f(x)=f(a)\).

Derivatives

Theorems

Calculus 2² Level 6

Integrals

Sequences and series

Linear algebra³ Level 6

Linearly independent:

no vector in the set is a combination of the others.

Rank:

the number of linearly independent rows (equivalently, columns).

Eigenvector:

a nonzero \(v\) with \(Av=\lambda v\). The scalar \(\lambda\) is its eigenvalue.

Advanced topics^2,4,5 Level 7

Hard questions at this level are mostly conceptual: what a theorem says and what it implies.

Multivariable calculus and differential equations

Abstract algebra

Group:

a set with an operation that is closed and associative, with an identity element and an inverse for every element.

Field:

a set where you can add, subtract, multiply, and divide by anything nonzero, such as \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\).

Analysis and number theory

Computer basics⁴ Level 1

CPU:

the processor that executes program instructions.

RAM:

fast working memory for running programs. It's volatile: it loses its contents when power is off.

Storage (SSD, hard drive):

keeps files when the power is off (non-volatile), but is slower than RAM.

Operating system:

manages hardware, memory, files and running programs (Android, Windows, macOS, Linux).

Bit and byte:

a bit is a 0 or 1. A byte is 8 bits.

IP address and DNS:

every device on the internet has an IP address. DNS turns names like example.com into IP addresses.

Programming basics⁴ Level 2

• Binary place values are powers of 2: \(1011_2=8+0+2+1=11\).
• Loops: in Python, range(n) gives \(0,1,\dots,n-1\), so the body runs \(n\) times.
• Arrays are usually indexed from 0: the last of \(n\) items is at index \(n-1\).
• Linear search checks items one by one: up to \(n\) comparisons.

x = 3
for i in range(4):
    x += i
print(x)

Data structures^1,2 Levels 2–6

• Load factor \(\alpha=n/m\) (keys per bucket). Collisions are handled by chaining (lists in buckets) or open addressing (probing for a free slot).
• In a heap stored in an array from index 0, node \(i\) has children \(2i+1\) and \(2i+2\), and parent \(\lfloor (i-1)/2\rfloor\).

Search algorithms^1,2 Levels 2–6

• Linear search: \(O(n)\), works on any list.
• Binary search: \(O(\log n)\), but only on sorted data. It halves the range each step, so it needs at most \(\lfloor\log_2 n\rfloor+1\) comparisons.
• Two pointers: on sorted data, move inward from both ends, \(O(n)\).
• BFS and DFS visit every vertex and edge of a graph: \(O(V+E)\).

Sorting^1,2 Levels 3–6

• Stable: equal keys keep their original order.
• Quicksort hits \(O(n^2)\) with consistently bad pivots, such as always picking the last element of already sorted input.
• Insertion sort is fast on nearly sorted data: its work is proportional to the number of out-of-order pairs (inversions).
• Counting sort \(O(n+k)\) and radix sort \(O(d(n+k))\) don't compare elements.

Trees^1,2 Levels 4–6

Height:

edges on the longest path from the root to a leaf (a single node has height 0).

Binary search tree (BST):

every key in the left subtree is smaller than the node, and every key in the right subtree is larger. Search and insert take \(O(h)\).

AVL tree:

a BST where subtree heights differ by at most 1, restored by rotations, so \(h=O(\log n)\).

• A tree with \(n\) nodes has \(n-1\) edges. A binary tree of height \(h\) has at most \(2^{h+1}-1\) nodes.
• Preorder: root, left, right. Inorder: left, root, right (sorted order for a BST). Postorder: left, right, root. Level order: breadth-first.
• Deleting a BST node with two children: replace it with its inorder successor (the smallest key in its right subtree).

Graphs¹ Levels 5–6

• Handshake lemma: the degrees add up to \(2|E|\). A complete graph on \(n\) vertices has \(\tfrac{n(n-1)}{2}\) edges.
• Adjacency list: \(O(V+E)\) space. Adjacency matrix: \(O(V^2)\) space, \(O(1)\) edge lookup.
• BFS finds shortest paths by edge count. Dijkstra finds shortest paths with non-negative weights, \(O((V+E)\log V)\) with a binary heap.
• Bellman-Ford allows negative weights and detects negative cycles, \(O(VE)\).
• A minimum spanning tree (Kruskal, Prim) of a connected graph has \(V-1\) edges.
• Topological sort orders a directed acyclic graph so every edge points forward. A graph is bipartite exactly when it has no odd cycle.

Theory of computation³ Level 7

P:

decision problems solvable in polynomial time.

NP:

decision problems whose "yes" answers can be verified in polynomial time, given a certificate. \(P\subseteq NP\).

NP-hard:

every NP problem reduces to it in polynomial time. It need not be in NP itself.

NP-complete:

both in NP and NP-hard. Examples: SAT, 3-SAT, Hamiltonian cycle, subset sum.

Decidable:

some Turing machine always halts with the right yes/no answer.

Coding¹

• Recursion needs a base case that stops it. Each call should move toward that case.
• Integer division and modulo in Python: 7 // 2 == 3, 7 % 2 == 1, and -7 // 2 == -4 (it rounds down).
• Short-circuiting: a and b skips b if a is false. a or b skips b if a is true.
• Off-by-one: range(1, 5) is 1, 2, 3, 4. The end isn't included.
• Object-oriented ideas: encapsulation (hide state behind methods), inheritance (a subclass reuses a parent), polymorphism (one call, behaviour chosen by the object's type).
• Git: a commit is a snapshot, a branch is a movable pointer to commits, and a merge joins two branches' histories.

def f(n):
    return 1 if n <= 1 else n * f(n - 1)

Object-oriented programming⁵

Class and object:

a class is a blueprint, and an object is one instance built from it.

Encapsulation:

bundle data with the methods that use it, and hide the data behind those methods (private fields, public methods).

Abstraction:

expose what an object does and hide how it does it.

Inheritance:

a subclass reuses and extends a parent class ("is-a": a Dog is an Animal).

Polymorphism:

one call, different behaviour depending on the object's actual type.

• Overriding: a subclass replaces a parent's method with the same signature. Overloading: several methods share a name but take different parameters.
• Interface: declares methods without implementing them. Abstract class: may mix implemented and unimplemented methods, and can't be instantiated.
• Composition ("has-a": a Car has an Engine) is often preferred to inheritance because it couples classes less.
• Access modifiers: public (anyone), protected (the class and its subclasses), private (the class only). Exact rules vary by language.
• SOLID: single responsibility, open-closed, Liskov substitution, interface segregation, dependency inversion.

class Shape:
    def area(self): return 0
class Square(Shape):
    def __init__(self, s): self.s = s
    def area(self): return self.s * self.s

shapes = [Shape(), Square(3)]
print(sum(x.area() for x in shapes))

Databases and SQL²

• WHERE filters rows before grouping. HAVING filters groups after GROUP BY.
• INNER JOIN keeps matching rows only. LEFT JOIN keeps every left row, with NULLs where there's no match.
• Primary key: unique and not null. Foreign key: must match a key in another table.
• NULL isn't equal to anything, even NULL: test it with IS NULL. COUNT(col) skips NULLs, and COUNT(*) counts every row.
• Indexes speed up reads but slow down writes. Normalization removes duplicated data.
• ACID: atomicity, consistency, isolation, durability. The isolation levels, from weakest: read uncommitted, read committed, repeatable read, serializable.²

SELECT customer, SUM(amount) FROM orders
GROUP BY customer HAVING SUM(amount) > 20;

Frontend³

• Box model: content, then padding, border and margin. With box-sizing: border-box, width includes padding and border.
• Specificity (strongest first): inline styles, IDs, then classes, attributes and pseudo-classes, then elements.³
• Flexbox: justify-content aligns along the main axis, align-items along the cross axis. Grid lays out rows and columns together.
• rem is relative to the root font size, em to the element's font size (the parent's, when setting font-size itself).
• Most DOM events bubble from the target up through its ancestors.
• In JavaScript, === compares without type conversion: 0 == "0" is true but 0 === "0" is false.
• Accessibility: use semantic elements, give images alt text, and label form fields.

Backend⁴

• Idempotent methods give the same result when repeated: GET, PUT and DELETE are, POST isn't. GET is also safe: it changes nothing.⁴
• Authentication proves who you are. Authorization decides what you may do.
• Store passwords as slow, salted hashes (bcrypt, Argon2), never as plain text or fast hashes.⁴
• Horizontal scaling adds machines behind a load balancer. Vertical scaling makes one machine bigger.
• Common web vulnerabilities: injection, cross-site scripting (XSS), broken access control.

System design basics^3,6 Medium–Hard

Load balancer:

spreads incoming requests across several servers, and stops sending to ones that fail health checks.

Reverse proxy:

sits in front of servers and receives clients' requests for them (for TLS, caching, routing). A forward proxy sits in front of clients and sends requests on their behalf.

Cache:

a fast copy of data close to where it's needed. A TTL (time to live) sets how long an entry stays. Invalidation removes entries that are out of date, and a stale cache serves old data.

CDN (content delivery network):

servers around the world that cache content near users, cutting latency and load on the origin server.

DNS:

turns domain names into IP addresses.

TLS and HTTPS:

TLS encrypts and authenticates a connection, and HTTPS is HTTP over TLS. SSL is TLS's older, deprecated predecessor, which is why people still say "SSL/TLS".

FTP and SFTP:

FTP transfers files with no encryption (port 21). SFTP transfers files over an encrypted SSH connection (port 22).

API gateway:

a single entry point that routes requests to back-end services and handles shared concerns like authentication and rate limiting.

Microeconomics¹

• Law of demand: when the price rises, the quantity demanded falls. Equilibrium is where supply equals demand.
• A price ceiling below equilibrium causes a shortage. A price floor above it causes a surplus.
• A tax drives a wedge between what buyers pay and what sellers keep, creating deadweight loss.
• Opportunity cost is the value of the next best alternative given up.
• Firms maximize profit where marginal revenue = marginal cost. In perfect competition, \(P=MC\). A monopoly prices above \(MC\).
• Public goods are non-rival and non-excludable. Externalities are costs or benefits that fall on others.

Macroeconomics¹

• Real GDP adjusts nominal GDP for price changes.
• The three types of unemployment are frictional, structural and cyclical.
• Fiscal policy uses government spending and taxes. Monetary policy uses interest rates and the money supply.
• The short-run Phillips curve shows a trade-off between inflation and unemployment.

Banking^1,2

• Fractional reserve banking: banks lend out most deposits. The simple money multiplier is \(\tfrac{1}{\text{reserve ratio}}\).
• A bank balance sheet: assets = liabilities + equity. Deposits are the bank's liabilities, and loans are its assets.
• Central bank tools (the US Federal Reserve): open market operations, the policy rate target, and interest on reserve balances. The Fed cut reserve requirements to 0% in March 2020.²
• FDIC deposit insurance covers $250,000 per depositor, per insured bank, per ownership category.²
• APR is the yearly rate without compounding. APY includes compounding.

Trading and investing^1,3

• Stocks are ownership (equity). Bonds are loans (debt). When interest rates rise, existing bond prices fall.
• A market order fills right away at the best available price. A limit order fills only at your price or better.
• The bid-ask spread is the gap between the highest bid and the lowest ask.
• A call option is the right to buy at the strike price, and a put the right to sell. A future is an obligation to trade later at a set price.
• Short selling: sell borrowed shares, hoping to buy them back cheaper.
• Diversification reduces company-specific risk, but not market-wide risk.

Statics^1,2

• Equilibrium: \(\sum F=0\) and \(\sum M=0\). Start with a free-body diagram.
• Moment: \(M=F\cdot d\), with \(d\) the perpendicular distance to the point.
• A couple (two equal, opposite, offset forces) creates a pure moment and no net force.
• Supports in 2D: a roller gives 1 reaction, a pin 2, a fixed support 3.
• Friction: \(f\le\mu_s N\). A block about to slip on an incline has \(\mu_s=\tan\theta\).
• Trusses: solve with the method of joints (equilibrium at each pin) or the method of sections (cut and take moments).

Dynamics¹

• Momentum is conserved in every collision. Kinetic energy is conserved only in elastic ones.
• Doubling speed doubles momentum but quadruples kinetic energy.

Electrical circuits¹

• Kirchhoff's current law: the current into a node equals the current out. Voltage law: the voltages around any loop sum to zero.
• In series the current is the same through each part. In parallel the voltage is the same across each branch.
• Thevenin equivalent: \(V_{th}\) is the open-circuit voltage, and \(R_{th}\) is the resistance seen with independent sources turned off.
• After one time constant \(\tau=RC\), a charging capacitor reaches about 63% of its final voltage.

Thermodynamics¹

• Processes: isothermal (constant \(T\), so \(\Delta U=0\) for an ideal gas), isochoric (constant \(V\), \(W=0\)), isobaric (constant \(P\), \(W=P\Delta V\)), adiabatic (\(Q=0\)).
• Second law: the entropy of an isolated system never decreases. No engine beats the Carnot efficiency.¹
• During a phase change, added heat changes the internal energy, not the temperature.

Cell biology¹

• Osmosis: water moves across a membrane toward the side with more solute. Active transport moves substances against their gradient using ATP.
• Respiration runs glycolysis, then the Krebs cycle, then the electron transport chain.
• Mitosis makes 2 identical diploid cells. Meiosis makes 4 genetically different haploid cells, with crossing over.

Genetics¹

• DNA base pairing: A–T and G–C. In RNA, U replaces T.
• Replication is semi-conservative: each new double helix keeps one old strand.
• Central dogma: DNA → (transcription) → mRNA → (translation) → protein.
• There are 64 codons. AUG is the start codon, and 3 codons are stops.
• Genotype is the alleles (AA, Aa, aa). Phenotype is the observed trait.
• X-linked recessive traits show up more often in males, who have one X.
• PCR copies a DNA segment many times.

Atomic structure¹

• Atomic number \(Z\) = protons. Mass number \(A\) = protons + neutrons. A neutral atom has \(Z\) electrons.
• Isotopes have the same \(Z\) but different numbers of neutrons. Ions have gained or lost electrons.
• Electron configuration fills 1s, 2s, 2p, 3s, 3p, 4s, 3d… Valence electrons are those in the outermost shell.
• Quantum numbers: \(n\) (shell), \(l\) (subshell), \(m_l\) (orbital), \(m_s\) (spin).
• Periodic trends: atomic radius shrinks across a period and grows down a group. Ionization energy and electronegativity do the opposite. Fluorine is the most electronegative element.

Stoichiometry¹

• Balance equations so each element has the same number of atoms on both sides. Then the coefficients give mole ratios.
• The limiting reagent runs out first and sets the theoretical yield.
• An empirical formula is the simplest whole-number ratio (CH₂O). A molecular formula is the actual count (C₆H₁₂O₆).