Study & Practice

Question Topics.

Dive into specific mathematical fields with theory notes and curated practice problems. Start with our featured topics below, pulled straight from the Olympiad worksheets.

Topic 1

Number Theory

Properties of integers, prime factorizations, divisibility, modular arithmetic, and analyzing digits. Master generating sets and bounding techniques with these guided notes.

Core Theory Summary

Generating Sets

Explore sets satisfying specific properties (like "funky" sets where pairs share divisors but triples are coprime). Approach algorithmically by building from smaller subsets and working iteratively.

Analyzing Digits

Write numbers in expanded form (aₙ × 10ⁿ + ... + a₀) to apply modular arithmetic. This is especially useful for divisibility rules (e.g., modulo 9 or 11).

Bounds & Alternative Tools

Use inequalities to restrict number sizes. Apply the Pigeonhole Principle and Euler's Theorem for existence proofs without needing to provide an explicit construction.

Practice Questions

Call a set of positive integers "funky" if every pair of elements has a greatest common divisor not equal to 1, but every triple of elements are co-prime. Show that there is a funky set of size n.

Hint: Try constructing this set iteratively. For n elements, you need exactly "n choose 2" distinct primes. Assign one unique prime to serve as the shared prime factor for each unique pair.

Is it possible to choose 2000 distinct non-negative integers which are all less than 100,000 such that no three of them form an arithmetic progression?

Hint: Think about numbers written in a base where digits are restricted (e.g., base 3 using only digits 0 and 1). Can three such numbers ever form an arithmetic progression?

Find all 5-digit natural numbers such that after deleting any one digit results in a 4-digit number which is divisible by 7.

Hint: Write the number as abcde. Deleting the first digit gives bcde ≡ 0 (mod 7), so the original number is a × 10000 + bcde ≡ a × 4 (mod 7). Analyze this pattern for deletions at all positions.

Show that for every positive integer n not divisible by 2 or 5, there exists a multiple of n all of whose digits are ones.

Hint: Consider the infinite sequence of numbers 1, 11, 111, 1111... and use the Pigeonhole Principle. Since there are infinitely many such numbers but only n possible remainders modulo n, two must have the same remainder. Subtract them!

Topic 2

Diophantine Equations

Techniques for finding integer solutions to polynomial equations, including factorisation, bounding, polynomial division, modular arithmetic, and infinite descent.

Core Theory Summary

Factorisation & Bounding

Rearrange equations into products of algebraic factors and equate them to divisors of a constant. Use inequalities and monotonicity (e.g., bounding between consecutive squares) to restrict the finite space of integer solutions.

Polynomial Division

Use polynomial long division or polynomial modulus to extract factors. Additionally, analyse quadratic discriminants—over the integers, non-negative perfect square discriminants often provide strict bounds.

Modular Arithmetic & Descent

Apply Fermat's Little Theorem and Euler's Phi Function to prove no solutions exist via specific moduli. Use Infinite Descent to prove only trivial solutions exist by showing a "minimal" solution can always be reduced.

Practice Questions

Find all pairs of integers (x, y) such that 3x + 4y = 2xy.

Hint: Try factorising. Multiply the entire equation by 2 and add/subtract a constant to complete the rectangle, forming a product of two factors equal to an integer. Then check its divisors.

What is the largest integer n for which n³ + 100 is divisible by n + 10?

Hint: Use polynomial division or polynomial modulus. Write n³ + 100 as (n + 10) × Q(n) + R. For divisibility, the linear term n + 10 must divide the constant remainder R.

Show that the equation x⁴ + 131y⁴ = 3z⁴ + 2000 has no integer solutions.

Hint: Use modular arithmetic. Fourth powers have very restricted remainders modulo certain numbers. Try taking the entire equation modulo 16 to find a contradiction.

Prove that the equation 6(6a² + 3b² + c²) = n² has no solutions other than a = b = c = n = 0.

Hint: Use infinite descent. Start by considering the equation modulo 3 or 6 to show that n must be a multiple of 6. Substitute n = 6k, divide out common factors, and prove a strictly smaller solution must exist.