SEAMO 2027 - Paper D - SET 05 - Self-study

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.

Find the last two digits of $9^{2022}$.

Trapezium $ABCD$ has parallel bases $AB=30$ cm and $CD=40$ cm, with $AD=BC$. Diagonals $AC$ and $BD$ intersect at $M$. If its area is $490$ cm$^2$, find the area of triangle $BCM$ in cm$^2$.

Evaluate
$$\frac1{2+\sqrt2}+\frac1{3\sqrt2+2\sqrt3}+\frac1{4\sqrt3+3\sqrt4}+\cdots+\frac1{100\sqrt{99}+99\sqrt{100}}.$$

Let $n=2021^2+2022^2$. Find $\sqrt{2n-1}$.

Find the four-digit number $\overline{abcd}$ satisfying
$$2\overline{abcd}+1000=\overline{dcba}.$$

It is now 3:15 PM. After how many minutes will the minute hand form an angle of $30^\circ$ with the hour hand?

Practice interpretation: find the first time after 3:15 PM when the smaller angle is 30 degrees. The source does not explicitly say “first”.

The hypotenuse of a right-angled triangle is 12 and its incircle has radius 1. Find the perimeter of the triangle.

Evaluate $\displaystyle\frac{16}{345678^2-345674\times345682}$.

Starting from any letter M in the figure, spell MADAM by moving up, down, left or right to an adjacent letter. How many ways are there?

Practice convention: a route may revisit a cell; different sequences of visited cells count as different routes.

Matchsticks form the triangular patterns shown. Figures 1, 2, 3 and 4 contain respectively 1, 4, 9 and 16 small equilateral triangles. In which figure will there be 16900 small equilateral triangles?

A positive integer $m$ has the property that $874m$ is a five-digit number ending in 92. Which of these are possible values of $m$?

(i) 48; (ii) 58; (iii) 68; (iv) 108.

Given $a=\sqrt6-2$ and $b=2\sqrt2-\sqrt6$, choose the correct statement.

Find the smallest positive integer $k$ such that $2^{69}+k$ is divisible by 127.

Two thirds of the journey from Lethbridge to Medicine Hat is highway; the remaining one third is country road. St. Paul joins the two sections. A van travels from Lethbridge towards Medicine Hat at 100 km/h on highway and 60 km/h on country road. A car travels in the opposite direction at 70 km/h on country road and 110 km/h on highway. They pass each other 44 km from St. Paul. Find the total distance, in km.

Practice assumption: both vehicles depart at the same time and maintain the stated speeds. Simultaneous departure is not expressly stated in the source.

Find the last two digits of $1+2+3+\cdots+2017+2018$.

Miss Mooney has three children. You see her with a little girl, whom she introduces as her middle child. Compute the probability that her other two children are girls, assuming that each child has probability $1/2$ of being a girl.

Practice probability model: the sexes of the three children are independent, each equally likely to be girl or boy; identifying the middle child introduces no further selection bias.

A fair die has faces numbered 1 to 6. It is thrown twice and the scores are added. Which event has the greatest probability? A perfect number is a positive integer equal to the sum of its positive proper divisors.

Original material: for digits $a_1,\ldots,a_n\in\{1,2,\ldots,9\}$, the source uses $\overline{a_1a_2\cdots a_n}$ for the decimal numeral and prints
$$10^{n-1}a_1+10^{n-2}a_2+\cdots+10a_{n+1}+a_n.$$
It gives the example $a=2,b=0$, for which $\overline{ab}=20$. Given $\overline{ab}=b^2$ and $\overline{acbc}=(\overline{ab})^2$, it asks for $\overline{ab}$.
Original options: (A) 123; (B) 246; (C) 280; (D) 369; (E) None of the above.

Practice task for manual assessment: use ordinary decimal place value and determine whether any digits $a,b,c\in\{1,\ldots,9\}$ satisfy both equations. Show every possibility arising from the first equation and check the second. Explain whether the printed key D can be justified.

Source warning: the numeral-definition index, digit domain, target/options and answer key are inconsistent. The original content is retained above. Answer the explicit diagnostic practice task with reasoning; there is no automatic numerical grading.

In the figure, $AD\perp AC$ and $\angle BAD=\angle DAE=15^\circ$. If $AB+AE=BC$, find $\angle ABC$ in degrees.

Which numbers in $\{1,6,7,9\}$ occur as the last digit of $n^n$ for infinitely many positive integers $n$?