16 original competition-style problems: limits, derivatives, integrals and functional equations used cleverly rather than routinely. 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.
What is ∫01 ln x dx? (The integral is improper at 0.)
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards, Symmetry
What is the limit of (ex − 1 − x)/x2 as x tends to 0?
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 ∫0π/2 1/(1 + tan2026 x) dx?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
For x > 0, what is the largest value of x1/x?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle
How many real solutions does the equation ex = x2 have?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
What is the area enclosed between the curve y = 2 − x2 and the graph y = |x|?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
What is the limit of n sin(π/n) as n tends to infinity?
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 ∫01 x2(1 − x)3 dx?
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 smallest positive x (in radians) with sin x = cos 2x?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
What is the limit of (1 + 1/x)x2 e−x as x tends to infinity?
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 sum of the infinite series Σn=0∞ n2/n! = 0 + 1 + 4/2 + 9/6 + 16/24 + …?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
A straight line through the origin touches the curve y = ex. What is its gradient?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
The region between y = x and y = x2 (for 0 ≤ x ≤ 1) is rotated through a full turn about the x-axis. What is the volume of the solid?
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 smallest value of x ln x for x > 0?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle
y is a function of x with dy/dx = y2 and y = 1 when x = 0. What is y when x = 1/2?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
What is ∫0∞ e−x2 dx?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
More Olympiad-style problems: Number theory · Combinatorics · Geometry · Algebra · Probability · Logic
Calculus and functions at other levels: Senior (ages 16 to 18)
Problem-solving strategies · Where next · Extension & competition maths
IGCSE Math Revision