18 original competition-style problems: angles, areas, circles, lattice points and solids. 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.
Triangle ABC has sides 13, 14 and 15. What is the radius of the circle that passes through the midpoints of its three sides?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Working backwards
A regular 12-sided polygon is drawn with all its corners on a circle of radius 1. What is its area?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
The three medians of a triangle have lengths 9, 12 and 15. What is the area of the triangle?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
A quadrilateral with sides 1, 2, 3 and 4 (in that order) has all four corners on a circle. What is its area?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
P is any point inside an equilateral triangle with sides of length 2. What is the sum of the perpendicular distances from P to the three sides?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Invariants
Two circles of radius 1 and 4 touch each other and both touch a straight line, on the same side of it. A third circle fits in the gap: it touches both circles and the line. What is its radius?
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 possible area of a triangle whose three corners lie on a circle of radius 1?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Symmetry
What is the area of the largest equilateral triangle that fits inside a square of side 1?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Symmetry
What is the shortest distance from the origin to the plane x + 2y + 2z = 9?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
What is the cosine of the acute angle between two of the space diagonals of a cube (lines joining opposite corners through the centre)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
A triangle has corners (0, 0), (4, 0) and (0, 3). What is the distance between its centroid (where the medians meet) and its incentre (the centre of the inscribed circle)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
A square of area 5/8 is drawn inside a unit square, with one corner on each side of the unit square. Each corner of the small square is the same distance d from the nearest corner of the unit square going clockwise, with d < 1/2. What is d?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
An equilateral triangle ABC is drawn in a circle. P is a point on the arc BC that does not contain A, with PB = 3 and PC = 5. What is PA?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
A triangle has sides 6, 7 and 8. What is the length of the median to the side of length 8?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
What is the area enclosed by the ellipse x2/9 + y2/4 = 1?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
A rectangle is drawn inside a 3-4-5 right-angled triangle with one side along the hypotenuse and the other two corners on the legs. What is its largest possible area?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle
Three edges of a tetrahedron meet at a corner O at right angles to each other, with lengths 3, 4 and 5. What is the distance from O to the opposite face?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
Two circles of radius 1 have their centres 1 unit apart. What is the area of the region inside both circles?
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 · Algebra · Probability · Logic · Calculus and functions
Geometry 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