# Bộ đề luyện thi Apmops khối 6, 7 – Toán tiếng anh

Bộ đề luyện thi Apmops khối 6, 7 – Toán tiếng anh

## ĐỀ ÔN LUYỆN APMOPS – SỐ 1

P1. Calculate $(6+7+8-9-10)+(11+12+13-14-15)+(16+17+18-19-20)+\ldots+(2006+$ $2007+2008-2009-2010)+(2011+2012+2013-2014-2015)$

P2. Mary writes down a three-digit number. William copies her number twice in a row to form a six-digit number. When William’s number is divided by the square of Mary’s number, the answer is an integer. What is the value of this integer?

P3. In the figure below, the large equilateral triangle is formed by 25 smaller equilateral triangles each with an area of $1 \mathrm{~cm}^{2}$. What is the area of triangle $\mathrm{ABC}$, in $\mathrm{cm}^{2}$ ?

P4. What is the number and the letter in the $1000^{\text {th }}$ column in the following pattern?

P5. The average of 20 numbers is 18 . The $1^{\text {st }}$ number is increased by 2 , the $2^{\text {nd }}$ number is increased by 4 , the $3^{\text {rd }}$ is increased by $6, \ldots$, the $20^{\text {th }}$ number is increased by 40 (that is, the $\mathrm{n}^{\text {th }}$ number is increased by $2 \mathrm{n}$ ). What is the average of the 20 increased numbers?

P6. $\mathrm{N}$ is a positive integer and $\mathrm{N} !=\mathrm{N} \times(\mathrm{N}-1) \times(\mathrm{N}-2) \times \ldots \times 3 \times 2 \times 1$. How many 0 ,s are there at the end of the simplified value of $\frac{2015 !}{1997 !}$

P7. A fruit company orders $4800 \mathrm{~kg}$ of oranges at $\$ 1.80$ per kg. The shipping cost is $\$ 3000$. Suppose $10 \%$ of the oranges are spoiled during shipping, and the remaining oranges are all sold, what should be the selling price per kg if the fruit company wants to make an $8 \%$ profit?

P8. Find the last digit of $7^{2015}$. (Note: $7^{2015}=\underbrace{7 \times 7 \times 7 \times \ldots \times 7}_{2015 \text { factors }}$ )

P9. How many two-digit numbers have the property of being equal to 7 times the sum of their digits?

P10. In the diagram shown, the number of rectangles of all sizes is …?

P11. Each of the numbers from 1 to 9 is placed, one per circle, into the figure shown. The sum along each of the 4 sides is the same. How many different numbers can be placed in the middle circle to satisfy these conditions?

P12. For admission to the school play, adult were charged $\$ 130$ each and students $\$ 65$ each. A total of $\$ 30225$ was collected, from fewer than 400 people. What was the smallest possible number of adults who paid?

P13. In the figure given below, the side of the square $\mathrm{ABCD}$ is $2 \mathrm{~cm}$. $\mathrm{E}$ is the midpoint of $\mathrm{AB}$ and $\mathrm{F}$ is the midpoint of $\mathrm{AD}$. $\mathrm{G}$ is a certain point on $\mathrm{CF}$ and $3 \mathrm{CG}=2 \mathrm{GF}$. What is the area of the shaded triangle $\mathrm{BEG}$, in $\mathrm{cm}^{2}$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-03.jpg?height=360&width=351&top_left_y=1479&top_left_x=927)

P14. A six – digit number $a b a b a b$ is formed by repeating a two-digit number $a b$ three times, e.g. 525252. If all such numbers are divisible by $p$, find the maximum value of $p$ ?

P15. A palindrome is a number that can be read the same forwards and backwards. For example, 246642, 131 and 5005 are palindromic numbers. Find the smallest even palindrome that is larger than 56789 which is also divisible by 7.

P16. Ben and Josh together have to paint 3 houses and 20 fences. It takes Ben 5 hours to paint a house and 3 hours to paint a fence. It takes Josh 2 hours to paint a house and 1 hour to paint a fence. What is the minimum amount of time, in hours, that it takes them to finish painting all of the houses and fences? P17. Arrange the numbers 1 to 9 , using each number only once and placing only one number in each cell so that the totals in both directions (vertically and horizontally) are the same. How many different sums are there?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-04.jpg?height=362&width=374&top_left_y=470&top_left_x=904)

P18. Find the value of $S$ where $S=\frac{1}{2}+\frac{1}{2 \times 7}+\frac{1}{7 \times 5}+\frac{1}{5 \times 13}+\ldots+\frac{1}{11 \times 25}$

P19. In the following 8-pointed star, what is the sum of the angles $\mathrm{A} ; \mathrm{B} ; \mathrm{C} ; \mathrm{D} ; \mathrm{E} ; \mathrm{F} ; \mathrm{G} ; \mathrm{H}$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-04.jpg?height=363&width=468&top_left_y=1132&top_left_x=885)

P20. The pages of a book are numbered consecutively: 1, 2, 3, 4 and so on. No pages are missing. If in the page numbers the digit 3 occurs exactly 99 times, what is the number of the last page?

P21. In the figure below, $A$ and $B$ are the centres of two quarter-circles of radius $14 \mathrm{~cm}$ and $28 \mathrm{~cm}$, respectively. Find the difference between the areas of region $\mathbf{I}$ and $\mathbf{I I}$ in $\mathrm{cm}^{2}$. (Use $\left.\pi=\frac{22}{7}\right)$

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-04.jpg?height=357&width=369&top_left_y=1986&top_left_x=935)

P22. In the right-angled triangle $P Q R, P Q=Q R$. The segments $\mathrm{QS} ; \mathrm{TU}$ and $\mathrm{VW}$ are perpendicular to $P R$, and the segments $S T$ and UV are perpendicular to $\mathrm{QR}$, as shown. What fraction of triangle PQR is shaded?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-05.jpg?height=346&width=300&top_left_y=218&top_left_x=958)

P23. How many ways can we select six consecutive positive integers from 1 to 999 so that the tailing of the product of these six consecutive positive integers end with exactly four 0 ‘s?

P24. Eleven consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2012 , what number did Maria erase?

P25. A ‘Lucky number’ has been defined as a number which can be divided exactly by the sum of its digits. For example: 1729 is a Lucky number since $1+7+2+9=19$ and 1729 can be divided exactly by 19. Find the smallest Lucky number which is divisible by 13.

P26. Given a non-square rectangle, a square-cut is a cutting-up of the rectangle into two pieces, a square and a rectangle (which may or may not be a square). For example, performing a square-cut on a $2 \times 7$ rectangle yields a $2 \times 2$ square and a $2 \times 5$ rectangle, as shown.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-05.jpg?height=212&width=988&top_left_y=1533&top_left_x=594)

You are initially given a $40 \times 2011$ rectangle. At each stage, you make a square-cut on the non-square piece. You repeat this until all pieces are squares. How many square pieces are there at the end?

P27. You must color each square in the figure below in red, green or blue. Any two squares with adjacent sides must be of a different color. In how many different ways can this coloring be done?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-05.jpg?height=257&width=260&top_left_y=2239&top_left_x=978)

P28. Given the number pattern:

Row 1 1

Row 2 3 5

Row 3 $7 \quad 9 \quad 11$

Row 4

Row 5 $212^{13}$ $\sum_{25^{7}}^{17} 27$ 19 29

A triangle of three numbers A $-\cdots-B$ and $B$ are two successive number in the $i^{\text {th }}$ row and $C$ is in the $(i+1)^{\text {th }}$ row just below $\mathrm{A}$ and $\mathrm{B}$. If $\mathrm{A}+\mathrm{B}+\mathrm{C}=2093$, find the value of $\mathrm{C}$ ?

P29. The diagram below shows five circles, some pairs of which are connected by line segments. Five colors are available. Find the number of different ways of painting the circles if two circles connected by a line segment must be painted in different colours.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-06.jpg?height=388&width=448&top_left_y=1431&top_left_x=861)

P30. The figure is consist of three circle each of radius $1 \mathrm{~cm}$ with six identical shaded parts. Find the total area of the six shaded in $\mathrm{cm}^{2} ?$ ( in $\pi$ )

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-06.jpg?height=385&width=411&top_left_y=2046&top_left_x=937)

## ĐỀ ÔN LUYỆN APMOPS- SỐ 2

P1. Find the value of $999999 \times 222222+333333 \times 333334=$ ?

P2. How many rectangles are there in the following figure?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-07.jpg?height=248&width=383&top_left_y=493&top_left_x=928)

P3. Peter is ill. He has to take medicine A every 8 hours, medicine B every 5 hours and medicine $\mathrm{C}$ every 10 hours. If he took all three medicines at 7AM on Tuesday, when will he take them together again?

P4. In below figure, $\mathrm{E}, \mathrm{F}$ are midpoint of 2 squares of $20 \mathrm{~cm}$ side, $10 \mathrm{~cm}$ side respectively. Find the area of triangle $\mathrm{ABC}$, in $\mathrm{cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-07.jpg?height=432&width=571&top_left_y=1109&top_left_x=891)

P5. The four-digit number $\overline{3 A A 1}$ is divisible by 9. What digit does A represent?

P6. $\mathrm{X}$ and $\mathrm{Y}$ are two different numbers selected from the first 40 counting numbers from 1 to 40 inclusive. What is the largest value that $\frac{X+Y}{X-Y}$ can have?

P7. Six arrows land on the target shown at the below figure. Each arrow is in one of the regions of the target. Which of the following total score is possible: $16,19,26,31,41,44$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-07.jpg?height=246&width=262&top_left_y=2047&top_left_x=974)

P8. The result of multiplying a counting number by itself is a square number. For example, 1, 4, 9 , and 16 are each square numbers because $1 \times 1=1,2 \times 2=4$,

$3 \times 3=9,4 \times 4=16$. What year in the $20^{\text {th }}$ century (the years 1901 throught 2000 ) was a square number? P9. John and Mary went to a book shop and bought some exercise books. They had $\$ 100$ each. John could buy 7 large and 4 small ones. Mary could buy 5 large and 6 small ones and had $\$ 5$ left. How much was a small exercise book?

P10. Given that $\mathrm{A}^{4}=75600 \times \mathrm{B}$. If $\mathrm{A}$ and $\mathrm{B}$ are positive integers, find the smallest value of $\mathrm{B}$.

P11. Suppose five days after the day before yesterday is Friday. What of the week will tomorrow then be?

P12. A box contains over 100 marbles. The marbles can be divided into equal shares among 6, 7 or 8 children with 1 marbles left over each time.

What is the least number of marbles that the box can contain?

P13. Integer numbers are filled in a square grid in a pattern as shown at below figure. Which column and which row contain number 2017 ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-08.jpg?height=263&width=328&top_left_y=999&top_left_x=978)

P14. Square $\mathrm{ABCD}$, shown here, has side length 8 units and is divided into four congruent squares. One of these squares contains an inscribed circle, two other squares contain diagonals and the fourth square has perpendicular line segments drawn from the midpoints of adjacent sides to form a square in the interior. In square units, what is the total area of the shaded regions? Express your answer in terms of $\pi$ or take $\pi=22 / 7$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-08.jpg?height=294&width=263&top_left_y=1526&top_left_x=979)

P15. At each Stage, a new square is drawn on each side of the primeter of the figure in the previous stage. How many unit squares will be in Stage 10 ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-08.jpg?height=177&width=120&top_left_y=2033&top_left_x=885)

Stage 1 Stage 2

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-08.jpg?height=211&width=189&top_left_y=2025&top_left_x=1002)

Stage 3

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-08.jpg?height=292&width=237&top_left_y=1987&top_left_x=1206)

Satge 4

P16. Find the smallest value of $x+y+z$, where $x, y$ and $z$ are different positive integers that satisfy this equation: $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{10}$ P17. $40 \%$ of girls and $50 \%$ of boys in a class got ‘A’. If there are only 12 students in the class got ‘A’s and the ratio of boys and girls in the class is 4:5. How many student are there in the class?

P18. Find the value of:

$$

\mathrm{T}=\frac{1}{1 \times 2}+\frac{5}{2 \times 3}+\frac{11}{3 \times 4}+\ldots+\frac{89}{9 \times 10}

$$

P19. $(1,1,9)$ is a triple of counting numbers whose sum is 11 . We consider $(1,1,9),(1,9,1)$ and $(9,1,1)$ to be the same triple because each triple has the same three numbers. How many other triples of counting numbers have a sum of 11 ? (Counting number is greater than zero)

P20. In the addition problem at the right, different letters represent different digits. It is also given that $\mathrm{N}$ is 6 and $\mathrm{T}$ is greater than 1 . What four-digit number does THIS represent?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-09.jpg?height=177&width=217&top_left_y=1048&top_left_x=999)

P21. How many four-digit integers greater than 5000 are there for which the thousands digits equals the sum of the other three digits?

P22. During the rest hour, one of five students ( $A, B, C, D$ and $E)$ dropped a glass of water. The following are the responses of the children when the teacher questioned them:

A: It was $\mathrm{B}$ or $\mathrm{C}$ dropped it.

B: Neither E nor I did it.

$\mathrm{C}$ : Both $\mathrm{A}$ and $\mathrm{B}$ are lying.

$\mathrm{D}$ : Only one of A or B is telling the truth.

$\mathrm{E}: \mathrm{D}$ is not speaking the truth.

The class teacher knows that three of them NEVER lie while the other two ALWAYS lie. Who dropped the glass?

P23. As shown in the figure, $\mathrm{ABCD}$ is a right trapezoid. $\mathrm{AB}=10 \mathrm{~cm}, \mathrm{AD}=6 \mathrm{~cm}$. The shaded area is $6 \mathrm{~cm}^{2}$. What is the number of square centimeters in the area of trapezoid $\mathrm{ABCD}$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-09.jpg?height=232&width=388&top_left_y=2080&top_left_x=914)

P24. Let $a, b, c, d, e$ are integers satisfying the following expression:

$$

\frac{501}{2015}=\frac{1}{a+\frac{1}{b+\frac{1}{c+\frac{1}{d+\frac{1}{e}}}}}

$$

Find the value of $a+b+c+d+e$ ?

P25. The sum of 10 positive integers, not necessarily distinct, is 1001. If $\mathrm{d}$ is the greatest common divisor of the 10 numbers, find the maximum value of $d$ ?

P26. A staircase has 10 steps. If Peter can climb either 1 or 2 or 3 steps each time, in how many ways can he reach the tenth step?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-10.jpg?height=351&width=340&top_left_y=944&top_left_x=949)

P27. How many shortest ways to go from $\mathrm{A}$ to $\mathrm{B}$ in below figure?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-10.jpg?height=194&width=268&top_left_y=1391&top_left_x=994)

P28. In the multiplication on the right, each letter and each square represents a single digit. Different letters represent different digit but a square can represent any digit. What is the fivedigit number “HAPPY” stands for?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-10.jpg?height=217&width=226&top_left_y=1630&top_left_x=1532)

P29. As shown in the figure, two circles have the same radius of $2 \mathrm{~cm}$. The two shaded regions have the same area. What is the length of $\mathrm{AB}$ in centimeters? (A, B are center of two circles respectively)

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-10.jpg?height=206&width=306&top_left_y=2067&top_left_x=972)

P30. A pizza is cut into six pie-shaped pieces. Trung can choose any piece to eat first, but after that, each piece he chooses must have been next to a piece that has already been eaten (to make it easy to get the piece out of the pan). In how many different orders could he eat the six pieces?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-11.jpg?height=255&width=257&top_left_y=415&top_left_x=999)

## ĐỀ ÔN LUYỆN APMOPS- SỐ 3

1 If the following figure is folded into the shape of a cube, what is the number opposite to the face marked $P$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-12.jpg?height=177&width=288&top_left_y=414&top_left_x=447)

2 Find the value of

$$

\left(1-\frac{1}{2}\right) \times\left(1-\frac{1}{3}\right) \times\left(1-\frac{1}{4}\right) \times \cdots\left(1-\frac{1}{2010}\right)\left(1-\frac{1}{2011}\right) .

$$

3 A circle of radius $1 \mathrm{~m}$ has some points lying on its circumference. Find the minimum number of points such that at least two points are less than $1 \mathrm{~m}$ apart.

4 Three sides of a four-sided figure are of lengths $4 \mathrm{~cm}, 9 \mathrm{~cm}$ and $14 \mathrm{~cm}$ respectively. If the largest possible length of the fourth side is $x \mathrm{~cm}$ where $x$ is a whole number, find the value of $x$.

5 Find the largest prime number that divides the number

$$

(1 \times 2 \times 3 \times \cdots \times 97 \times 98)+(1 \times 2 \times 3 \times \cdots \times 98 \times 99 \times 100) .

$$

$6 A B C D$ is a square and $B C E$ is an equilateral triangle. If $B C$ is $8 \mathrm{~cm}$, find the radius of the circle passing through $A, E$ and $D$ in $\mathrm{cm}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-12.jpg?height=320&width=508&top_left_y=1539&top_left_x=343)

7 Find the value of

$$

\frac{19}{20}+\frac{1919}{2020}+\frac{191919}{202020}+\cdots+\underbrace{\frac{1919 \ldots 19}{2011 \text { of } 19 \text { ‘s }}}_{2011 \text { of } 20 \text { ‘s }} .

$$

8 The following diagram shows a circle of radius $8 \mathrm{~cm}$ with the centre $R$. Two smaller circles with centres $P$ and $Q$ touch the circle with centre $R$ and each other as shown in the diagram. Find the perimeter of the triangle $P Q R$ in $\mathrm{cm}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-13.jpg?height=349&width=352&top_left_y=411&top_left_x=378)

9 Peter and Jane competed in a $5000 \mathrm{~m}$ race. Peter’s speed was 4 times that of Jane’s. Jane ran from the beginning to the end, whereas Peter stopped running every now and then. When Jane crossed the finish line, Peter was $100 \mathrm{~m}$ behind. Jane ran a total of $x \mathrm{~m}$ during the time Peter was not running. Find the value of $x$.

10 If numbers are arranged in three rows $A, B$, and $C$ in the following manner, which row will contain the number 1000 ?

$$

\begin{array}{lllllllll}

A & 1 & 6 & 7 & 12 & 13 & 18 & 19 & \ldots \\

B & 2 & 5 & 8 & 11 & 14 & 17 & 20 & \ldots \\

C & 3 & 4 & 9 & 10 & 15 & 16 & 21 & \ldots

\end{array}

$$

11 The following diagram shows 5 identical circles. How many different straight cuts are there so that the five shaded circles can be divided into two parts of equal areas?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-13.jpg?height=260&width=334&top_left_y=1680&top_left_x=450)

12 A test with a maximum mark of 10 was administered to a class. Some of the results are shown in the table below. It is know that the average mark of those scoring more than 3 is 7 while the average mark of those getting below 8 is 5 . Given that none scored zero, find the number of pupils in the class.

\begin{tabular}{|l|l|l|l|l|l|l|l|}

\hline Score & 1 & 2 & 3 & $\ldots$ & 8 & 9 & 10 \\

\hline Number of pupils & 1 & 3 & 6 & $\ldots$ & 4 & 6 & 3 \\

\hline

\end{tabular}

13 A test comprises 10 true or false questions. Find the least number of answer scripts required to ensure that there are at least 2 scripts with identical answers to all the 10 questions. $14 P_{n}$ is defined as the product of the digits in the whole number $n$. For example, $P_{19}=1 \times 9=9$, $P_{32}=3 \times 2=6$. Find the value of

$$

P_{10}+P_{11}+P_{12}+\cdots+P_{98}+P_{99} .

$$

$15 A B C$ is a triangle with $B C=8 \mathrm{~cm} . D$ and $E$ line on $A B$ such that the vertical distance between $D$ and $E$ is $4 \mathrm{~cm}$. Find the area of the shaded region $C D E$ in $\mathrm{cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-14.jpg?height=391&width=539&top_left_y=590&top_left_x=362)

16 The points $A, B, C, D, E$ and $F$ are on the two straight lines as shown. How many triangles can there be formed with any 3 of the 6 points as vertices?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-14.jpg?height=337&width=766&top_left_y=1134&top_left_x=308)

17 A group of 50 girls were interviewed to find out how many books they had borrowed from the school library in April. The total number of books borrowed by the girls in April was 88, and 18 girls had borrowed only 1 book each. If each girl had borrowed either 1, 2, or 3 books, find the number of girls that had borrowed 2 books each.

18 The following diagram shows a triangle $P Q R$ on a 2 by 7 rectangular grid. Find the sum of the angle of $P Q R \underset{P}{\text { and angle }} P R Q$ in degrees.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-14.jpg?height=163&width=491&top_left_y=1889&top_left_x=360)

19 The following diagram shows a rectangle where the areas of the shaded regions are $5 \mathrm{~cm}^{2}, 6$ $\mathrm{cm}^{2}, 10 \mathrm{~cm}^{2}, 54 \mathrm{~cm}^{2}$ and $x \mathrm{~cm}^{2}$ respectively. Find the value of $x$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-14.jpg?height=294&width=446&top_left_y=2252&top_left_x=405)

20 The following diagram shows two squares $A B C D$ and $D G F E$. The side $C D$ touches the side $D G$. If the area of $D E F G$ is $80 \mathrm{~cm}^{2}$, find the area of the triangle $B G E \mathrm{in} \mathrm{cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-15.jpg?height=739&width=517&top_left_y=373&top_left_x=815)

21 Two points $P$ and $Q$ are $11 \mathrm{~cm}$ apart. A line perpendicular to the line $P Q$ is $7 \mathrm{~cm}$ from $P$ and $4 \mathrm{~cm}$ from $Q$. How many more lines, on the same plane, are $7 \mathrm{~cm}$ from $P$ and $4 \mathrm{~cm}$ from $Q$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-15.jpg?height=237&width=517&top_left_y=1286&top_left_x=358)

22 The greatest number of points of intersection of the grid lines that the diagonal of a rectangle with area $12 \mathrm{~cm}^{2}$ can pass through is 3 as shown.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-15.jpg?height=334&width=1132&top_left_y=1672&top_left_x=400)

Find the greatest possible number of points of intersection that the diagonal of a rectangle with area $432 \mathrm{~cm}^{2}$ can pass through.

23 Given that $(100 \times a+10 \times b+c) \times(a+b+c)=1926$ where $a, b$ and $c$ are whole numbers, find the value of $a+b+c$. 24 The following figure comprises 5 identical squares each of area $16 \mathrm{~cm}^{2} . A, B, C$ and $D$ are vertices of the squares. $E$ lies on $C D$ such that $A E$ divides the 5 squares into two parts of equal areas. Find the length of $C E$ in $\mathrm{cm}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-16.jpg?height=380&width=394&top_left_y=370&top_left_x=360)

25 Whole numbers from 1 to 10 are separated into two groups, each comprising 5 numbers such that the product of all the numbers in one group is divisible by the product of all the numbers in the other. If $n$ is the quotient of such a division, find the least possible value of $n$.

26 Diagram I shows a ladder of length $4 \mathrm{~m}$ leaning vertically against a wall. It slides down without slipping to II, and then finally to a horizontal position as shown in III. If $M$ is at the mid-point of the ladder, find the distance travelled by $M$ during the slide in $\mathrm{m}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-16.jpg?height=368&width=958&top_left_y=1200&top_left_x=525)

27 A theme part issues entrance tickets bearing 5-digit serial numbers from 00000 to 99999 . If any adjacent numbers in the serial numbers differ by 5 (for example 12493), customers holding such a ticket could use the ticket to redeem a free drink. Find the number of tickets that have serial numbers with this property.

$28 S_{n}$ is defined as the sum of the digits in the whole number $n$. For example, $S_{3}=3$ and $S_{29}=$ $2+9=11$. Find the value of

$$

S_{1}+S_{2}+S_{3}+\cdots+S_{2010}+S_{2011} .

$$

29 Anthony, Benjamin and Cain were interviewed to find out how many hours they spend on the computer in a day. They gave the following replies.

– Anthony:

I spend 4 hours on the computer.

I spend 3 hours on the computer less than Benjamin.

I spend 2 hours on the computer less than Cain.

\section{- Benjamin:}

Can spends 5 hours on the computer.

The time I spend on the computer differs from Cain’s time by 2 hours.

The time I spend on the computer is not the least among the three of us.

\section{- Cain:}

I spend more time on the computer than Anthony.

I spend 4 hours on the computer.

Benjamin spends 3 hours on the computer more than Anthony.

If only two of the three statements made by each boy are true, find the number of hours that Anthony spends on the computer in a day.

$30 A B C$ is a triangle and $D$ lies on $A C$ such that $A D=B D=B C$. If all the three interior angles of triangle $A B C$, measured in degrees, are whole numbers, find the greatest possible value of angle $A B C$ in degrees.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-17.jpg?height=428&width=309&top_left_y=1431&top_left_x=471)

Number of correct answers for $Q 1$ to $Q 10$ :

Number of correct answers for $Q 11$ to $Q 20$ :

Number of correct answers for $Q 21$ to $Q 30$ : Marks ( $\times 4)$ :

Marks ( $\times 5)$ :

Marks $(\times 6)$ :

## ĐỀ ÔN LUYỆN APMOPS- SỐ 4

1 Find the value of $\left(1+\frac{2}{1}\right)\left(1+\frac{2}{2}\right)\left(1+\frac{2}{3}\right) \times\left(1+\frac{2}{4}\right) \times \ldots . . \times\left(1+\frac{2}{26}\right) \times\left(1+\frac{2}{27}\right)$.

2 The diagram shows 5 identical circles. On the answer sheet provided, draw a straight line to divide the figure into two parts of equal area.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-18.jpg?height=263&width=328&top_left_y=639&top_left_x=887)

3 The diagram shows two identical isosceles right-angled triangles. If the area of the shaded square in diagram $A$ is $50 \mathrm{~cm}^{2}$, what is the area of the shaded square in diagram $\mathrm{B}$ ?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-18.jpg?height=160&width=209&top_left_y=1096&top_left_x=495)

Diagram A

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-18.jpg?height=171&width=214&top_left_y=1094&top_left_x=1298)

Diagram B

4 A rectangular wooden block measuring $30 \mathrm{~cm}$ by $10 \mathrm{~cm}$ by $6 \mathrm{~cm}$ is cut into as many cubes of side $5 \mathrm{~cm}$ as possible. Find the volume of the remaining wood.

5 The diagram shows 2007 identical rectangles arranged as shown. What number does $A$ represent?

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-18.jpg?height=452&width=1605&top_left_y=1670&top_left_x=251)

6 The diagram shows a trapezium $A B C D$ with $A D=B C$. If $B D=7 \mathrm{~cm}$, angle $A B D=$ $45^{\circ}$, find the area of the trapezium.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-18.jpg?height=294&width=440&top_left_y=2269&top_left_x=845)

7 An organism reproduces by simple division into two. Each division takes 5 minutes to complete. When such an organism is placed in a container, the container is filled with organisms in 1 hour. How long would it take for the container to be filled if we start with two such organisms?

8 Given that $a, b$ and $c$ are different whole numbers from 1 to 9 , find the largest possible value of $\frac{a+b+c}{a \times b \times c}$.

9 The diagram comprises a circle of radius $3 \mathrm{~cm}$, two semi-circles of radii $2 \mathrm{~cm}$ and two semi-circles of radii $1 \mathrm{~cm}$. Find the ratio of the areas of the regions $A, B$ and C.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-19.jpg?height=274&width=297&top_left_y=840&top_left_x=891)

10 In 2005, both John and Mary have the same amount of pocket money per month. In 2006, John had an increase of $10 \%$ and Mary a decrease of $10 \%$ in their pocket money. In 2007, John had a decrease of $10 \%$ and Mary an increase of $10 \%$ in their pocket money.

Which one of the following statements is correct?

(A) Both have the same amount of pocket money now.

(B) John has more pocket money now.

(C) Mary has more pocket money now.

(D) It is impossible to tell who has more pocket money now.

11 A set of 9-digit numbers each of which is formed by using each of the digits 1 to 9 once and only once. How many of these numbers are prime?

12 Water expands $10 \%$ when it freezes to ice. Find the depth of water to which a rectangular container of base $22 \mathrm{~cm}$ by $33 \mathrm{~cm}$ and height $44 \mathrm{~cm}$ is to be filled so that when the water freezes completely to ice it will fill the container exactly.

13 The diagram shows two circles with centre 0 . Given that line $A B$ is a chord $14 \mathrm{~cm}$ long and just touches the circumference of the shaded circle, find the area of the non-shaded region.

Take $\pi$ as $\frac{22}{7}$.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-19.jpg?height=316&width=234&top_left_y=2252&top_left_x=868)14 Joan could cycle $1 \mathrm{~km}$ in 4 minutes with the wind and returned in 5 minutes against the wind. How long would it take her to cycle $1 \mathrm{~km}$ if there is no wind? Assuming her cycling speed and the wind speed are constant throughout the journey.

15 Given that $\sqrt{1+1 \times 2 \times 3 \times 4}=5, \sqrt{1+2 \times 3 \times 4 \times 5}=11, \sqrt{1+3 \times 4 \times 5 \times 6}=19$ and $\sqrt{1+4 \times 5 \times 6 \times 7}=29$, find the value of $\sqrt{1+204 \times 205 \times 206 \times 207}$.

16 Peter walked once around a track and Jane ran several times around it in the same direction. They left the starting point at the same time and returned to it at the same time. In between, Jane overtook Peter twice. If she had run around the track in the opposite direction how many times would she have passed Peter? Assume that their speeds had been constant throughout the journey.

17 Three clocks, with their hour hands missing, have minute hands which run faster than normal. Clocks A, B and C each gains 2, 6 and 15 minutes per hour respectively. They start at noon with all three minute hands pointing to 12. How many hours have passed before all three minute hands next point at the same time?

$18 \mathrm{ABCD}$ is a rectangle and $A E F, B E H, H G C$ and FGD are straight lines. Given that the area of the 4-sided figure EFGH is $82 \mathrm{~cm}^{2}$, find the total area of the shaded regions.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-20.jpg?height=517&width=249&top_left_y=1409&top_left_x=869)

19 Four cards, each with a letter on one side and a number on the other side, are laid on a table.

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-20.jpg?height=240&width=149&top_left_y=2093&top_left_x=611)

(a)

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-20.jpg?height=228&width=146&top_left_y=2096&top_left_x=861)

(b)

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-20.jpg?height=234&width=165&top_left_y=2093&top_left_x=1091)

(c)

![](https://cdn.mathpix.com/cropped/2023_02_24_baf1a34d0f6a9d2a1a0bg-20.jpg?height=243&width=162&top_left_y=2083&top_left_x=1324)

(d)

John claims that any card with a letter $\mathbf{A}$ on one side always has the number “1” on the other side. Which two of the four cards would you turn over to check his statement?

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