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

Question 1

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.

In a pack of 10 watches, 3 are known to be defective. If 2 watches are selected at random from the pack, what is the probability that at least one is defective?

$\frac2{11}$
$\frac1{13}$
$\frac3{13}$
$\frac7{15}$
$\frac8{15}$

Question 2

How many four-digit positive integers are divisible by 12 and consist entirely of prime digits?

$12$
$15$
$16$
$20$
None of the above

Question 3

Acute triangle $A_1B_1C_1$ is inscribed in a circle. Points $A_2,B_2,C_2$ are the midpoints of the minor arcs $B_1C_1,A_1C_1,A_1B_1$, respectively. Points $A_3,B_3,C_3$ are the midpoints of the minor arcs $B_2C_2,A_2C_2,A_2B_2$, respectively. If $\angle A_3B_3C_3=\angle A_1B_1C_1$, find $\angle A_1B_1C_1$ in degrees.

Question illustration

$15$
$30$
$45$
$60$
None of the above

Question 4

Let $S=1+\dfrac1{2^2}+\dfrac1{3^2}+\dfrac1{4^2}+\cdots+\dfrac1{2023^2}$. Find the integer part $\lfloor S\rfloor$.

Question 5

Suppose $112^n+223^n+334^n+445^n+556^n$ is divisible by 10. How many possible two-digit positive integer values of $n$ are there?

Enter the number only.

Question 6

Let $m,n$ be positive integers with $m\gt n$. If the last three digits of $2012^m$ and $2012^n$ are identical, find the smallest possible value of $m+n$.

$98$
$100$
$102$
$104$
None of the above

Question 7

In triangle $ABC$, $AB=AC$. Points $D,E$ lie on $AC,AB$, respectively. Given $BD=BC$ and $AD=DE=BE$, find $\angle A$.

Question illustration

$45^\circ$
$48^\circ$
$50^\circ$
$52^\circ$
$54^\circ$

Question 8

Given $(62)_n=4\times(14)_n$, find the value of $(36)_n$ in base ten.

$27$
$28$
$29$
$30$
None of the above

Question 9

A triangle has side lengths $a,b,c$ opposite angles $A,B,C$, respectively. Given $a^2=1$ and $b^2+c^2=2019$, find $\dfrac{\cot A}{\cot B+\cot C}$.

$1009$
$1010$
$2019$
$2020$
None of the above

Question 10

It is known that $\log_a(ab)=x$. What is $\log_b(ab)$ expressed in terms of $x$?

Use real logarithms with positive bases a and b, neither equal to 1. Enter an expression in x, using parentheses for the denominator.

Question 11

Let $A,B$ be the points on the parabola $y=x^2$ that are each at a distance of $8\sqrt2$ units from the line $y+x+4=0$. Find $|AB|^2$.

Enter the number only.

Question 12

For $r=1,2,3,\ldots$, let $u_r=1+2+\cdots+r$. Find the value of $\displaystyle\frac1{1/u_1}+\frac2{1/u_1+1/u_2}+\frac3{1/u_1+1/u_2+1/u_3}+\cdots+\frac{100}{1/u_1+1/u_2+\cdots+1/u_{100}}$.

$2575$
$2760$
$3120$
$3220$
None of the above

Question 13

A positive integer is called a good number if it equals four times the sum of its digits. Find the sum of all good numbers.

$120$
$121$
$125$
$129$
None of the above

Question 14

Trapezoid $ABCD$ is inscribed in a circle with centre $O$, as shown. Given $AB\parallel CD$, $\angle COD=3\angle AOB$ and $AB/CD=2/5$, find $\dfrac{\operatorname{area}(\triangle BOC)}{\operatorname{area}(\triangle AOB)}$.

Question illustration

$\frac32$
$\frac74$
$\frac{\sqrt3}{\sqrt2}$
$\frac{\sqrt5}{\sqrt2}$
$\frac{\sqrt7}{\sqrt2}$

Question 15

Given $x^2-8x+1=0$, find $x^4+\dfrac1{x^4}$.

$6$
$8$
$3838$
$3842$
None of the above

Question 16

Given $abc=1$, find $\dfrac{a}{ab+a+1}+\dfrac{b}{bc+b+1}+\dfrac{c}{ca+c+1}$.

Assume the displayed fractions are defined: every denominator is nonzero.

Question 17

Find the number of real solutions of $\dfrac{x}{100}=\sin x$, where $x$ is measured in radians.

$30$
$32$
$62$
$63$
$64$

Question 18

Find the remainder when $2^{2018}$ is divided by 33.

$19$
$22$
$23$
$24$
$25$

Question 19

Gunmetal consists of 74% copper, 16% zinc and 10% tin and is made by mixing brass and bronze. Bronze consists of 80% copper, 4% zinc and 16% tin. Brass consists only of copper and zinc. If the ratio of copper to zinc in brass is $m:n$, where $m,n$ are relatively prime positive integers, find $m+n$.

Enter the number only.

Question 20

A $2\times3$ rectangle is divided into six unit squares $A,B,C,D,E,F$. Each square is to be coloured using one of six colours so that no two adjacent squares have the same colour. How many distinct colourings are possible?

Adjacent means sharing a side. Colours may be reused, and it is not necessary to use all six colours. The six labelled positions remain fixed.

Question illustration

$12240$
$13230$
$13480$
$14400$
None of the above