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SEAMO 2027 - Paper D - SET 08 - 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.

The annulus shown is divided into 14 regions. Each region is painted one colour. What is the minimum number of colours needed so that no two adjacent regions share a colour? Regions are adjacent when they share a boundary segment.

Question illustration
  1. 3
  2. 4
  3. 5
  4. 6
  5. 7

Question 2

Given $\frac{a}{2b-a}=5$, find $\frac{b}{2b-a}$.
  1. 2
  2. 3
  3. 4
  4. 5
  5. None of the above

Question 3

Two circles with centres $M,N$ are tangent to a line $\ell$ at $A,B$, respectively, as shown. Given $AB=25$, $MN=24$, and quadrilateral $ABNM$ has area 300, evaluate the product of the two radii. Original choices: (A) 120; (B) 144; (C) 150; (D) 176; (E) None of the above.

The printed numerical data may be inconsistent. This item is assessed manually: give the product if the configuration exists, or state that no such configuration exists. The original choices and key are retained for source comparison only.

Question illustration

Question 4

Find the number of digits in $4^{22}\times5^{40}$.
  1. 22
  2. 23
  3. 40
  4. 42
  5. None of the above

Question 5

Let $\overline{abcdef}$ be a six-digit integer such that $\overline{fabcde}$ is 5 times $\overline{abcdef}$. Find $a+b+c+d+e+f$.
  1. 25
  2. 26
  3. 27
  4. 28
  5. 30

Question 6

Given $abc=1$, find $\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}$.
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 7

Line $AB$ intersects a clock face at 10 and 4. How many minutes after 4:00 PM do the hour and minute hands make equal angles with line $AB$?

Interpret the question as asking for the first time after 4:00 PM; angles are the smaller angles between each hand and the full line.
  1. $17\frac6{11}$
  2. $17\frac6{13}$
  3. $18\frac6{11}$
  4. $18\frac6{13}$
  5. None of the above

Question 8

The sum $\frac1{1\times2\times3}+\frac1{2\times3\times4}+\cdots+\frac1{18\times19\times20}$ is $m/n$ in lowest terms. Find $m+n$.
  1. 603
  2. 749
  3. 787
  4. 938
  5. 949

Question 9

If digits $a,b$ are chosen uniformly at random, compute the probability that the four-digit integer $\overline{1a7b}$ is divisible by 4.

Question 10

$AB$ is a chord of a circle with centre $O$ and radius 104 cm. Point $M$ lies on the chord, with $AM=126$ cm and $MB=66$ cm. Find $OM$ in centimetres.

Question illustration

Question 11

The inequality $\frac7{13}\lt \frac{n}{n+k}\lt \frac59$ holds for exactly one positive integer $k$. Find the largest possible value of $n$.

Question 12

$ABCDE$ is a regular pentagon. Point $O$ is inside it and triangle $DOE$ is equilateral. Find $\angle AOC$.
  1. $108^\circ$
  2. $128^\circ$
  3. $138^\circ$
  4. $168^\circ$
  5. None of the above

Question 13

Let $a,b,c$ be positive real numbers such that $a^2+ab+b^2=25$, $b^2+bc+c^2=49$, and $c^2+ac+a^2=64$. Find $(a+b+c)^2$.
  1. 120
  2. 125
  3. 129
  4. 130
  5. None of the above

Question 14

$x=-1$ is a root of $x^2+ax+b=0$, and $\frac{a-b}{a+b}=\frac17$. Find $ab$.
  1. 6
  2. 8
  3. 12
  4. 15
  5. 20

Question 15

In triangle $ABC$, point $D$ lies on segment $AC$ such that $AB=AD$ and $\angle ABC-\angle ACB=30^\circ$. Find $\angle CAD$.

The target is retained exactly as printed: angle CAD, not another angle in the figure. Answer that literal question; a discrepancy with the source key is discussed in the solution.

Question illustration
  1. $10^\circ$
  2. $15^\circ$
  3. $20^\circ$
  4. $30^\circ$
  5. None of the above

Question 16

Which is larger: (i) the product of 329 factors of 2, or (ii) the product of 141 factors of 5?
  1. $(i)>(ii)$
  2. $(i)=(ii)$
  3. $(i)<(ii)$
  4. Both (A) and (B)
  5. Both (B) and (C)

Question 17

Find the number of ordered pairs of integers $(x,y)$ such that $x+y+xy=x^2+y^2$.

Question 18

Adam has a triangular field $ABC$ with $AB=5$, $BC=8$, $CA=11$. He builds a straight fence from $A$ to a point $D$ on side $BC$ such that $AD$ bisects $\angle BAC$. Find the area of field $ABD$.
  1. $\frac{4\sqrt{21}}{11}$
  2. $\frac{4\sqrt{21}}5$
  3. $\frac{5\sqrt{21}}{11}$
  4. $\frac{5\sqrt{21}}4$
  5. None of the above

Question 19

Given $(a+b)^3=a^3+3a^2b+3ab^2+b^3$, find the second and third terms in the expansion of $(2x+3y)^3$.
  1. $18x^2y+54xy^2$
  2. $24x^2y+54xy^2$
  3. $36x^2y+54xy^2$
  4. $24x^2y+27xy^2$
  5. None of the above

Question 20

For real $x$, let $[x]$ be the largest integer not exceeding $x$, and let $\{x\}=x-[x]$. Let $a,b$ be real numbers with $b$ nonzero and $a=b[a/b]-b\{a/b\}$. Which statement is incorrect?

Notation clarification: the source omits braces in its fractional-part definition. Here {x}=x−[x]. The minus sign in the given equation is retained exactly.
  1. If b is an integer, then a is an integer.
  2. If a is a nonzero integer, then b is an integer.
  3. If b is rational, then a is rational.
  4. If a is nonzero and rational, then b is rational.
  5. If b is an even integer, then a is an even integer.