This is a free diagnostic for the AIMO Preparation course (Australian Intermediate Mathematics Olympiad). 15 real AIMO-style problems (integer answers, 0–999) sample Number Theory, Geometry, Algebra and Combinatorics.
It's an exam: answer all the questions first (you can change any answer freely), then submit the whole exam to see your report. The worked solutions and hints unlock only after you submit. Use the jump menu at the top to move around.
At the end you get a knowledge-point report: a bar for every topic you attempted, your weak spots flagged, and the exact lessons to revise.
Year 8–12 students preparing for AIMO selection.
Let x denote a single digit. The tens digit in the product of 2x7 and 39 is 9. Find x .
❓ Enter your answer (a whole number):
An AIMO Number Theory problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Number Theory pathway in the full course.
If n is a positive integer and \(n^{2}\) equals the 4-digit number aabb , find n .
❓ Enter your answer (a whole number):
An AIMO Number Theory problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Number Theory pathway in the full course.
A 3-digit number abc is multiplied by 3 to give the 4-digit number c0ba . Find the number abc .
❓ Enter your answer (a whole number):
An AIMO Number Theory problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Number Theory pathway in the full course.
The n-th triangular number is the sum of the first n positive integers. Let T n denote the sum of the first n triangular numbers. Derive a formula for T n . For the input below: compute T 10 (the sum of the first 10 triangular numbers).
❓ Enter your answer (a whole number):
An AIMO Number Theory problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Number Theory pathway in the full course.
The $n$th triangular number is $t_n = \tfrac{n(n+1)}{2}$. Notice that $t_1 + t_9 = 1 + 45 = 46 = 10 + 36 = t_4 + t_8$, so $46$ is a sum of two triangular numbers in two different ways. Submit this common value $t_1 + t_9 = t_4 + t_8$.
❓ Enter your answer (a whole number):
An AIMO Number Theory problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Number Theory pathway in the full course.
Determine the number of non-negative integers x that satisfy the equation ⌊x/44⌋ = ⌊x/45⌋ .
❓ Enter your answer (a whole number):
An AIMO Number Theory problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Number Theory pathway in the full course.
Triangles \(ABC\) and \(XYZ\) are congruent right-angled isosceles triangles. A square \(KLMB\) is inscribed in \(\triangle ABC\) with two sides along the legs (one corner at the right angle); a square \(PQRS\) is inscribed in \(\triangle XYZ\) with one side along the hypotenuse. If the area of \(KLMB\) is \(189\), find the area of \(PQRS\).
❓ Enter your answer (a whole number):
An AIMO Geometry problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Geometry pathway in the full course.
A rectangle has sides of length \(28\) and \(15\). One diagonal is divided into \(7\) equal parts by \(6\) points, and every division point is joined to the two opposite corners. This cuts the rectangle into \(7\) quadrilaterals. Find the area of one of them, the quadrilateral \(DEBF\).
❓ Enter your answer (a whole number):
An AIMO Geometry problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Geometry pathway in the full course.
A triangle \(ABC\) is divided into four regions by three lines parallel to \(BC\). The lines divide \(AB\) into four equal segments. If the second largest region has area \(225\), what is the area of \(\triangle ABC\)?
❓ Enter your answer (a whole number):
An AIMO Geometry problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Geometry pathway in the full course.
Consider a circular sector of radius $360$ which is one-sixth of a circle. A circle is drawn inside this sector so that it is tangent to the two radii and to the circular arc. Calculate the radius of this smaller circle.
❓ Enter your answer (a whole number):
An AIMO Geometry problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Geometry pathway in the full course.
$ABCD$ is a trapezium for which $AB \parallel DC$, $AB = 84$ and $DC = 25$. A circle can be drawn in the trapezium so that it just touches all four sides. Find the perimeter of the trapezium.
❓ Enter your answer (a whole number):
An AIMO Geometry problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Geometry pathway in the full course.
A circle is inscribed in a hexagon $ABCDEF$ so that each side of the hexagon is tangent to the circle. Find the perimeter of the hexagon if $AB = 6$, $CD = 7$, and $EF = 8$.
❓ Enter your answer (a whole number):
An AIMO Geometry problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Geometry pathway in the full course.
Asha, Bree and Cala are three robots that are programmed to run athletic track races. When Asha runs a 400 m race she catches Bree at the finish line, if Bree starts 20 m ahead of Asha. Asha catches Cala at the finish line of a 1500 m race, if Cala has a 246 m start. Assuming each robot runs at constant speed, how many metres must Cala start ahead of Bree in an 800 m race, if they are to finish at the same time?
❓ Enter your answer (a whole number):
An AIMO Algebra problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Algebra pathway in the full course.
Gaston and Jordon always misread cooking times. If the required time is “1:32” (1 hour 32 minutes), Jordon reads it as 132 minutes while Gaston reads it as 1.32 hours. For one particular recipe, the difference between Jordon's and Gaston's misread times is exactly 90 minutes. What is the actual cooking time, in minutes?
❓ Enter your answer (a whole number):
An AIMO Algebra problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Algebra pathway in the full course.
A school adds 5 extra classrooms enabling 5 more classes and reducing the average class size by 6. Two months later another 5 classrooms are added, again enabling 5 more classes, this time reducing average class size by 4. The number of students did not change. How many students were at the school?
❓ Enter your answer (a whole number):
An AIMO Algebra problem (integer answer 0–999). Set up the relationship carefully and compute — AIMO rewards a clean, checkable method.
Work the problem to a single whole number, then check it against the conditions.
Revise: the AIMO Algebra pathway in the full course.
Go back and change any answers you like. When you're ready, submit the whole exam to lock it in and see your knowledge report — then you can revisit every question to read the full worked solution.
The report stays locked until you submit — you can't open it early.
Here is how you performed on every knowledge area you attempted. Bars show your accuracy per topic — green is strong, amber is shaky, red needs work. Only questions you answered are scored.
Want to turn the red and amber bars green? The full AIMO Preparation course builds AIMO problem-solving across Number Theory, Geometry, Algebra and Combinatorics.
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