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BBB 2022 - Secondary 3 - Preliminary Round

Question 1

How many integral solutions are there for $x$ if $-25\le\frac{9-2x}{3}\le64$?
  1. 133
  2. 134
  3. 135
  4. 136

Question 2

Given that $x$ is an acute angle and satisfies $\sin x=\frac8{17}$. Find the value of $\frac{\cos x}{\cos x-\sin x}$.
  1. $\frac8{17}$
  2. $\frac{15}{17}$
  3. $\frac87$
  4. $\frac{15}7$

Question 3

If $a$, $b$, $c$ and $d$ are all integers and $(x-1)(2x-3)(4x-5)=ax^3+bx^2+cx+d$ for all $x$, find the value of $a-b+c-d$.
  1. −90
  2. 90
  3. 80
  4. −80

Question 4

A straight line $L$ passes through the origin and $A(x,6)$ with the slope 3. Find $x$.
  1. 2
  2. 18
  3. 0.5
  4. 3

Question 5

Given $x>0$ and $x-\frac1x=3$, find the value of $x^4+\frac1{x^4}$.
  1. 47
  2. 51
  3. 123
  4. 119

Question 6

Find the maximum value of $-2x^2-16x-36$.
  1. −4
  2. 4
  3. −20
  4. 2

Question 7

There are 464 diagonals in a regular $n$-sided polygon. Find the value of $n$.
  1. 32
  2. 23
  3. 16
  4. 24

Question 8

Given a rectangular piece of paper ABCD. Then, the paper is folded along line segment MN as the figure below. If $\angle BMN=123^\circ$, find the measure of $\angle APN$.

Question illustration
  1. $112^\circ$
  2. $114^\circ$
  3. $116^\circ$
  4. $118^\circ$

Question 9

According to the pattern shown below, what is the 30th term in the sequence?

$2,\quad10,\quad24,\quad44,\quad70,\quad102,\quad140,\quad\ldots$
  1. 2730
  2. 2700
  3. 2640
  4. 2670

Question 10

$x,y$ are prime numbers and $3^x-5^y=y$. How many set(s) of solutions is / are there?
  1. 0
  2. 1
  3. 2
  4. 3

Question 11

$x,y$ are positive integers and $7x-3y=219$, find the maximum value of $x-y$.
  1. 27
  2. 28
  3. 29
  4. 30

Question 12

How many factors of 1200 are square numbers?
  1. 4
  2. 6
  3. 8
  4. 10

Question 13

Find the value of $3\times5+5\times7+\cdots+27\times29+29\times31$.
  1. 4249
  2. 4492
  3. 4924
  4. 4942

Question 14

The radius of a sector is 8 and its perimeter is 50. Find the area of this sector.
  1. 136
  2. 272
  3. 68
  4. 34

Question 15

Find the area of the region bounded by the $y$-axis and two lines with the equation $x+2y-1=0$ and $2x-y-7=0$.
  1. 5.25
  2. 9.75
  3. 11.25
  4. 3.75

Question 16

It is known that $x$ is rational, $x>0$ and $x=3+\frac{28}{3+\frac{28}{3+\cdots}}$. Find the value of $x$.
  1. 7
  2. 4
  3. 12
  4. 9

Question 17

Given that the side lengths of the triangle are 8, $a$ and $b$ ($a\ne b$). If $f(a)=f(b)=0$ and $f(x)=x^3-15x^2+71x-105$, find the smallest perimeter of that triangle.
  1. 16
  2. 18
  3. 20
  4. 15

Question 18

Numbers are drawn from 1000 integers 1 to 1000. At least how many numbers are drawn at random to ensure that there are two numbers whose product is 1200?
  1. 973
  2. 971
  3. 986
  4. 987

Question 19

How many rectangles are there in the figure below?

Question illustration
  1. 87
  2. 71
  3. 81
  4. 78

Question 20

There are 5 black balls and 4 white balls. If 3 balls are drawn randomly. Find the probability that at least one ball is white.
  1. $\frac13$
  2. $\frac{20}{21}$
  3. $\frac35$
  4. $\frac{37}{42}$