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.
The sum of $n$ real numbers is zero and the sum of their pairwise products is also zero. Prove that the sum of the cubes of the numbers is zero.
Each time the hands of a standard 12-hour clock form a 180° angle, a bell chimes once. From midnight today until midnight tomorrow, how many chimes will be heard?
Let $a_n$ be the number of ways to climb a flight of $n$ steps by taking either 1 or 2 steps at a time. Find the remainder when $a_{2019}$ is divided by 7.
Find the smallest whole number whose first digit is 4 and for which moving this digit to the last place produces a number equal to $\frac14$ of the original.
For an equilateral triangle, the ratio of the area of its circumscribed circle to the area of its inscribed circle is $m:n$, where $m,n$ are positive integers with no common factor. Find $m+n$.
$AB$ and $BC$ are diameters of semicircles $ADB$ and $CPB$, respectively. The line $AD$ is tangent to semicircle $CPB$ at $P$. Given $\angle PBC=28^\circ$, find $\angle PAC$.
A teacher wrote 4875, 4563 and a positive integer $N$ on the whiteboard and asked for their lowest common multiple. A student misread 4875 as 4275 but still obtained the correct answer. What is the least possible value of $N$?
In right triangle $ABC$, $BC$ is the hypotenuse and $AC=5$. Point $D$ is the midpoint of $BC$, and $E$ lies on $AC$ as shown. Given $DE=AE=2$, find $AB$.
