22 original competition-style problems: equations, sequences, functions and inequalities. Try each one before opening the hints; the second hint gives more away, and the full solution explains why the method works and where the idea leads.
Every question is free to read. The hints, answer checking and full solutions are included with every A Level, IB, IGCSE and CBSE plan. See plans.
Answer a problem to track what you have solved.
Press Project on any problem to show it full screen with a timer, the hints, the answer and the worked solution one step at a time (arrow keys move between problems; Space reveals the next step; F full screen; Esc closes). Switch on the ‘Add to worksheet’ buttons, pick problems, then print them from the worksheet builder or set them as homework for a class, with the full solutions as the mark scheme. Free problems are free for every class; problems marked ‘With a plan’ can be set by teachers with a plan or school licence. Ready-made sessions: maths club packs.
a, b and c are positive real numbers. What is the smallest possible value of a/b + b/c + c/a? Prove it.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
A function f from the real numbers to the real numbers satisfies f(x)f(y) − f(xy) = x + y for all real x and y. What is f(2026)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards, Proof techniques
a, b and c are the roots of x3 − 3x + 1 = 0. What is 1/(a + 1) + 1/(b + 1) + 1/(c + 1)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
x and y are positive with x + y = 1. What is the smallest possible value of (x + 1/x)(y + 1/y)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle
What is 1/(√1 + √2) + 1/(√2 + √3) + … + 1/(√99 + √100)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
A sequence has a1 = 1 and an+1 = an/(1 + an). What is a100?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise, Invariants
How many real numbers x satisfy x2 − 8⌊x⌋ + 7 = 0? (⌊x⌋ is the greatest whole number not more than x.)
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
Real numbers x, y, z satisfy x2 + y2 + z2 = 14. What is the largest possible value of x + 2y + 3z?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle
A function f on the positive whole numbers satisfies f(xy) = f(x) + f(y) for all x, y, with f(2) = 1 and f(3) = 2. What is f(72)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
How many different real numbers x satisfy x5 = 5x − 4?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
What is the sum of all real solutions of 4x − 3 × 2x+1 + 8 = 0?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
The quadratic x2 + px + q = 0 has roots p and q, where p and q are non-zero real numbers. What is p + q?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
x is a complex number with x + 1/x = √3. What is x12 + 1/x12?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
What is 1 × 1! + 2 × 2! + 3 × 3! + … + 10 × 10!?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
What is the largest constant k such that a2 + b2 + c2 ≥ k(ab + bc + ca) for all real numbers a, b, c?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
A polynomial P leaves remainders 4, 7 and 12 when divided by x − 1, x − 2 and x − 3 respectively. Its remainder on division by (x − 1)(x − 2)(x − 3) is r(x). What is r(10)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
A sequence has a1 = 1 and an+1 = 2an + 1. What is a20?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise, Invariants
A, B and C are the angles of a triangle. What is the largest possible value of sin A + sin B + sin C?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Symmetry
What is the sum of the infinite series 1/(1 × 2 × 3) + 1/(2 × 3 × 4) + 1/(3 × 4 × 5) + …?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
Non-zero real numbers a, b, c satisfy a + b + c = 0. What is (a3 + b3 + c3)/(abc)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
Real numbers x, y, z satisfy x + y + z = 6, xy + yz + zx = 11 and xyz = 6. What is the largest of x, y and z?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards, Symmetry
What is the smallest value of |x − 1| + |x − 2| + |x − 3| + … + |x − 9| over all real x?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Extremal principle
More Olympiad-style problems: Number theory · Combinatorics · Geometry · Probability · Logic · Calculus and functions
Algebra at other levels: Junior (ages 11 to 13) · Intermediate (ages 13 to 16) · Senior (ages 16 to 18)
Problem-solving strategies · Where next · Extension & competition maths
CBSE Math Revision