SEAMO self-study edition 2027; not an official 2027 paper. Original source years are retained. 25 questions, 90 minutes; Q1–10: 3 points, Q11–20: 4 points, Q21–25: 6 points, total 100. Original response types and all subparts are preserved. Verified is an editorial convention; source defects, necessary assumptions and unresolved mathematical uncertainty are disclosed.
Given that 2016 is a leap year and Sonia’s birthday fell on Sunday, 24 June 2012, in which year will her birthday next fall on a Sunday?
In a 100-metre sprint, Austin finishes 20 metres ahead of Bob and 60 metres ahead of Carson. If each sprinter runs at their own constant speed, how many metres will Bob finish ahead of Carson?
The quadratic equation $ax^2+bx+c=0$ has no real roots. Adam obtained roots 2 and 4 because he wrote down a wrong value of $a$. Ben obtained roots $-1$ and 4 because he wrote the sign of one term wrongly. Find $\dfrac{2b+3c}{a}$.
A batch of berries has mass 100 kg. Of its mass, 1% is flesh and 99% is water. After a few hours in the sun, the water content is reduced to 98%. How much does the batch weigh now?
Assume only water is lost and the flesh mass stays unchanged.
Terry answered 70%, 80% and 90% of the problems correctly on a 20-, 25- and 30-problem quiz, respectively. What percentage of all problems did Terry answer correctly? Round to the nearest whole number.
$ABC$ is a right triangle with $\angle ACB=90^\circ$. A circle of radius $r$ is inscribed in it. Given $AB=27$ and the area of $ABC$ is 90, find the shaded area outside the circle but inside the triangle. Leave your answer in terms of $\pi$.
An equilateral triangle $ABC$ has area $\sqrt3$ and side length 2. The point $P$ is anywhere in its interior. What is the sum of the distances from $P$ to $AB$, $AC$ and $BC$?
A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. The ratio of the area of the square to the area of the triangle is $\dfrac{a}{b\sqrt c}$, where $a,b,c$ are positive integers, $a,b$ are coprime and $c$ has no square factor greater than 1. Find $a+b+c$.
In triangle $ABC$, $\angle A=60^\circ$ and the lengths $AB,AC$ are the roots of $x^2-5x+3=0$. The point $D$ is on $BC$ and $AD$ bisects $\angle A$. The lines $AB,AC$ are tangent to a circle with center $D$. Its radius is $\dfrac{a\sqrt b}{c}$, where $a,c$ are coprime positive integers and $b$ is a positive integer with no square factor greater than 1. Find $a+b+c$.
Tangency is to the lines AB and AC, extended beyond the side endpoints if necessary, as in the source diagram.
In right triangle $ABC$ with hypotenuse $AB$, $CF$ is the median to $AB$, $CE$ bisects $\angle ACB$, and $CD$ is perpendicular to $AB$. Prove that $\angle DCE=\angle ECF$.
How many positive integers $n\le2020$ satisfy $\left\lfloor n/2\right\rfloor+\left\lfloor n/3\right\rfloor+\left\lfloor n/6\right\rfloor=n$? Here $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.
Two circles with centers $O_1,O_2$ are externally tangent at $M$. The lines $AB$ and $CD$ are common external tangents, with contact points as shown. Given $O_2E\perp O_1A$, $\angle AO_1C=120^\circ$ and $O_2B=1$ cm, find the shaded area between the two circles and the tangent segment $AB$. Leave your answer in terms of $\pi$.
