IPhO Math Tools Problem Set (Official Problems + Hints)
Curated IPhO problems chosen for heavy math usage: scaling, differential equations, vector calculus symmetry, and matrix methods.
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This set is not “math for its own sake”. It is a set of IPhO problems where scaling, ODE patterns, vector calculus, or matrix thinking is the real bottleneck.
If you have not done the hub path yet, start with the IPhO Math Tools Hub.
Archive mirror index: https://phoxiv.org/contests/ipho
Official documentations page: https://www.ipho-new.org/documentations/
Read the framework once: IPhO Problem-Solving Framework
Do the hub Entry and Core lessons first (then return here):
- Start with sections A and B below (starter problems).
- Finish the hub lessons, then do sections C and D (core).
- Do sections E and F (stretch) after you are comfortable.
Show the full hub lesson path (Math Tools | IPhO Prep)
A. Scaling and Dimensional Thinking
- IPhO 2022 T3 — “Scaling laws” (PDF)
Outline pattern: scaling that earns marks
- define the characteristic scales (L, T, V, m, rho) and nondimensionalise once - identify which terms must balance (dominant balance) - give a clear “why this term is small” ratio, not just intuition
B. ODE Patterns (Linear and Nonlinear)
- IPhO 2019 T1 — “Zero-length springs and slinky coils” (PDF)
- IPhO 2016 T2 — “Nonlinear Dynamics in Electric Circuits” (PDF)
Outline pattern: ODEs in olympiad mode
- reduce the system to 1 or 2 variables if possible (constraints/symmetry) - recognize the form (separable, linear first-order, small oscillations, energy integral) - if nonlinear, look for conserved quantities or phase-plane structure
C. Vector Calculus and Symmetry
- IPhO 2021 T2 — “Electrostatic Lens” (PDF)
Outline pattern: Gauss/Stokes without re-deriving everything
- write the symmetry and which components vanish - pick a surface/loop that makes the field constant or the integral trivial - check limiting cases (near-field, far-field) and units
D. Fourier/Signal Thinking and Estimation
- IPhO 2018 T1 — “LIGO-GW150914” (PDF)
Outline pattern: signals and waves with minimal machinery
- identify what sets the time scale and frequency scale - estimate first, then refine (do not start with exact integrals) - keep one bandwidth or resolution bound line as a check
E. Linear Algebra in the Wild (Matrices and Components)
- IPhO 2023 E2 — “Birefringence” (PDF)
Outline pattern: matrices as bookkeeping
- choose a basis and represent the system consistently - treat each element as a transformation (matrix or component rules) - verify with easy special cases (identity, swap axes, 90-degree rotation)
F. Diffusion/Transport Scaling (A PDE Problem That Behaves Like an ODE)
- IPhO 2024 E1 — “Heat Conduction in a Copper Rod” (PDF)
Outline pattern: diffusion without full PDE solving
- write the diffusion time scale t ∼ L²/α early - identify boundary conditions and what is being measured - extract parameters from slopes using a plot that linearises the model
After each attempt, rewrite your solution using 30 percent less algebra and 30 percent more structure (scales, assumptions, checks). Then continue via the IPhO Math Tools Hub.