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Master 1 Given the sequence 2, 5, 8, 11, … Which term will the number 443 be?

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Master 1 Given the sequence 2, 5, 8, 11, … Which term will the number 443 be?

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Ans:148

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Master 2  Daily, a clock tower chimes once at 1am and once at 1pm, twice at 2am and twice at 2pm, thrice at 3am and thrice at 3pm, and so on Following the pattern, how many chimes are there in total for the whole day? (1 day = 24 h)

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Master 2  Daily, a clock tower chimes once at 1am and once at 1pm, twice at 2am and twice at 2pm, thrice at 3am and thrice at 3pm, and so on Following the pattern, how many chimes are there in total for the whole day? (1 day = 24 h)

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Ans:156

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C1 Jane has 9 boxes with 9 accompanying keys. Each box can only be opened by its accompanying key. If the 9 keys have been mixed up, find the maximum number of attempts Jane must make before she can open all the boxes.

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C1 Jane has 9 boxes with 9 accompanying keys. Each box can only be opened by its accompanying key. If the 9 keys have been mixed up, find the maximum number of attempts Jane must make before she can open all the boxes.

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Ans:45

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Ans:501

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C3   In the sequence { 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, … }, the 10’s begin from the mth term and end at the nth term. Find the value of n-m.

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C3   In the sequence { 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, … }, the 10’s begin from the mth term and end at the nth term. Find the value of n-m.

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Ans:18

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C4  2, 4, 6, 8, 10, 12, … are called consecutive even numbers. If the sum of 5 consecutive even numbers is 360, what is the smallest possible number among the 5 numbers?

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C4  2, 4, 6, 8, 10, 12, … are called consecutive even numbers. If the sum of 5 consecutive even numbers is 360, what is the smallest possible number among the 5 numbers?

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C5  A 2013-digit number is formed by writing the whole numbers starting from 1, in a connected way, as follows:123456789101112……….What are the last 4 digits of this number?

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C5  A 2013-digit number is formed by writing the whole numbers starting from 1, in a connected way, as follows:123456789101112……….What are the last 4 digits of this number?

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M1 Given that a number 23465HH is divisible by 8, what digit does H represent?

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M1 Given that a number 23465HH is divisible by 8, what digit does H represent?

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M2 If a number 4961X2Y is divisible by both 5 and 8, find the possible values of X and Y.

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M2 If a number 4961X2Y is divisible by both 5 and 8, find the possible values of X and Y.

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C1 A four-digit whole number abcd is a multiple of 9, and the three-digit whole number abc is a multiple of 7. What is the smallest value of the four-digit abcd whole number ?

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C1 A four-digit whole number abcd is a multiple of 9, and the three-digit whole number abc is a multiple of 7. What is the smallest value of the four-digit abcd whole number ?

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C2 A three digit integer abc is a multiple of 7. What is abc with the largest value of a + b + c?

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C2 A three digit integer abc is a multiple of 7. What is abc with the largest value of a + b + c?

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C3  A six digit integer abcdef  is a multiple of 4. What is the largest possible value of a + b + c + d + e + f?

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C3  A six digit integer abcdef  is a multiple of 4. What is the largest possible value of a + b + c + d + e + f?

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C4 A 5-digit number written in the form 24abc  has the last three digits unknown. If this number is divisible by 3, 4 and 5 respectively, find the greatest possible value that abc can take.

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C4 A 5-digit number written in the form 24abc  has the last three digits unknown. If this number is divisible by 3, 4 and 5 respectively, find the greatest possible value that abc can take.

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C5 A 4-digit number has the following properties:(a) This number is a perfect square.(b) The first two digits of the number are equal.(c) The last two digits of the number are equal. What is the 4-digit number?

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C5 A 4-digit number has the following properties:(a) This number is a perfect square.(b) The first two digits of the number are equal.(c) The last two digits of the number are equal. What is the 4-digit number?

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C6 The 6-digit number 1082ab is a multiple of 12. How many possible different values of the last 2-digit number ab are there?

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C6 The 6-digit number 1082ab is a multiple of 12. How many possible different values of the last 2-digit number ab are there?

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M1 Reyes has one 10¢-coin, one 20¢-coin and one 50¢-coin. How many different values of money can these coin(s) form?

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M1 Reyes has one 10¢-coin, one 20¢-coin and one 50¢-coin. How many different values of money can these coin(s) form?

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M2 There are 4 number tiles below. Each tile has a different digit.

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M2 There are 4 number tiles below. Each tile has a different digit.

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If every digit can be used only once to form a number, how many different 3-digit numbers can be

formed?

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C1 Four boys A, B, C and D are passing a ball among them. Boy A starts passing the ball. After 3 passes, the ball returns to him. For example, the following is one of the different ways that the ball can be passed from one boy to another.A → C → D → A How many different ways are there altogether? (SMOPS/PartB’94/Q2)

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C1 Four boys A, B, C and D are passing a ball among them. Boy A starts passing the ball. After 3 passes, the ball returns to him. For example, the following is one of the different ways that the ball can be passed from one boy to another.A → C → D → A How many different ways are there altogether? (SMOPS/PartB’94/Q2)

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C2 In how many ways can 3 different numbers are selected from the set{2, 3, 4, 5, 6, 7} ,such that the sum of the 3 numbers is an even number? (SMOPS/PartA’97/Q22)

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C2 In how many ways can 3 different numbers are selected from the set{2, 3, 4, 5, 6, 7} ,such that the sum of the 3 numbers is an even number? (SMOPS/PartA’97/Q22)

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C3 How many ways are there in putting five different books into two different bags so that each bag contains at least one book? (SMOPS/PartA’90/Q28)

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C3 How many ways are there in putting five different books into two different bags so that each bag contains at least one book? (SMOPS/PartA’90/Q28)

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C4 Andrea wrote a 5-digit number on a piece of paper (represented by the shapes below). When she inverts this piece of paper, it shows the same 5-digit number. How many of such 5-digit numbers can Andrea write? (Hint: When you invert the digit 9, it becomes the digit 6.)

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C4 Andrea wrote a 5-digit number on a piece of paper (represented by the shapes below). When she inverts this piece of paper, it shows the same 5-digit number. How many of such 5-digit numbers can Andrea write? (Hint: When you invert the digit 9, it becomes the digit 6.)

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C5 Three chess pieces, each of the colour red, black and white, are to be placed on a 7 x 7 chessboard. If any two of the three pieces cannot be placed in the same row or the samecolumn, how many ways are there to place the three chess pieces? (APMOP’14/Round 1/Q7)

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C5 Three chess pieces, each of the colour red, black and white, are to be placed on a 7 x 7 chessboard. If any two of the three pieces cannot be placed in the same row or the samecolumn, how many ways are there to place the three chess pieces? (APMOP’14/Round 1/Q7)

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C6  Below is Singapore’s 28 District Code map.

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C6  Below is Singapore’s 28 District Code map.

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At most how many ways are there of colouring the central district (No. 1 - 10) using three different

colours such that adjacent districts do not share the same colour?

(AMPS’18/Prelim/Q20)

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M1 You are given digits 1 to 8.          (a) Find how many 5-digit numbers can be formed if each digit can only be used once in every number.

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M1 You are given digits 1 to 8.          (a) Find how many 5-digit numbers can be formed if each digit can only be used once in every number.

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M1 (b) Using only the odd digits, find how many 2-digit numbers can be formed if each digit can only be used once in every number.

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M1 (b) Using only the odd digits, find how many 2-digit numbers can be formed if each digit can only be used once in every number.

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M2 A basket of 5 fruits is to be selected from 6 apples and 4 oranges. Find the number of ways in which this can be done if

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M2 A basket of 5 fruits is to be selected from 6 apples and 4 oranges. Find the number of ways in which this can be done if

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(a) there are no restrictions,

(b) the basket has exactly 3 apples,

(c) the basket has at least 1 orange.

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C4 In the first diagram (on the left) below, fill up 1, 2, 3, 4 and 5 in the circle without repetition. Next, write down on each arc the product of the two adjacent numbers. Please refer to an example at the diagram on the right. The sum of the numbers on the arcs of this diagram is 3+4+8+10+15 = 40. Find the largest possible sum of the numbers on the arcs. (NMOS’17/Prelim/Q10)

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C4 In the first diagram (on the left) below, fill up 1, 2, 3, 4 and 5 in the circle without repetition. Next, write down on each arc the product of the two adjacent numbers. Please refer to an example at the diagram on the right. The sum of the numbers on the arcs of this diagram is 3+4+8+10+15 = 40. Find the largest possible sum of the numbers on the arcs. (NMOS’17/Prelim/Q10)

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C3. Two distinct numbers from1 to 100 inclusive will form a pair if the sum of these two is a multiple of 5. How many different pairs are there? (NMOS’18/Special/Q5)

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C3. Two distinct numbers from1 to 100 inclusive will form a pair if the sum of these two is a multiple of 5. How many different pairs are there? (NMOS’18/Special/Q5)

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C2 How many ways are there to distribute 28 identical marbles into 3 different boxes such that no box is empty? (APMOPS/1stRound’08/Q28)

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C2 How many ways are there to distribute 28 identical marbles into 3 different boxes such that no box is empty? (APMOPS/1stRound’08/Q28)

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M1  The average of 7 numbers is 29. If the average of the first 3 numbers is 25 and the average of the last 5 numbers is 32, what is the 3rd number? 

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M1  The average of 7 numbers is 29. If the average of the first 3 numbers is 25 and the average of the last 5 numbers is 32, what is the 3rd number? 

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M2  The class average score of a Mathematics examination is 91. The average score of the girls is 89  and the average score of the boys is 92.5. There are 24 girls in the class. Find the number of boys. 

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M2  The class average score of a Mathematics examination is 91. The average score of the girls is 89  and the average score of the boys is 92.5. There are 24 girls in the class. Find the number of boys. 

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C2 4 classes A, B, C and D each has less than 50 students and the average number of students is 46. Class A and class B differ in number of students by 4, class B and class C by 3, and class C and class D by 2. If class A has the most number of students, find the number of students in A.  (APMOPS/1stRound’12/Q18) 

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C2 4 classes A, B, C and D each has less than 50 students and the average number of students is 46. Class A and class B differ in number of students by 4, class B and class C by 3, and class C and class D by 2. If class A has the most number of students, find the number of students in A.  (APMOPS/1stRound’12/Q18) 

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C3  There are 5 different whole numbers. The average of some 4 of these 5 numbers  is 36, 38, 39, 45 and 49. What is the largest number among the 5 numbers?                                                       

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C3  There are 5 different whole numbers. The average of some 4 of these 5 numbers  is 36, 38, 39, 45 and 49. What is the largest number among the 5 numbers?                                                       

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C4  Four distinct positive whole numbers are arranged in descending order. Given that the sum of the smallest  number  and  the  average  of  the  other  three  numbers  is  39,  and  the  sum  of  the  largest number and the average of the other three numbers is 51. What is the largest possible value of the largest number?  

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C4  Four distinct positive whole numbers are arranged in descending order. Given that the sum of the smallest  number  and  the  average  of  the  other  three  numbers  is  39,  and  the  sum  of  the  largest number and the average of the other three numbers is 51. What is the largest possible value of the largest number?  

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M1 In the figure below, FBCD is a rectangle with point A along FD and point E along BF. Using the area(s) of the three given portions or otherwise, find the area of the shaded part.

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M1 In the figure below, FBCD is a rectangle with point A along FD and point E along BF. Using the area(s) of the three given portions or otherwise, find the area of the shaded part.

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M2 ABCD is a right-angled trapezium. AE = ED, BC = 3FC = 18 cm, AD = 8 cm, CD = 6 cm. If the area of △EGD is equal to the area of △CGF, what is the area of the shaded part?  

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M2 ABCD is a right-angled trapezium. AE = ED, BC = 3FC = 18 cm, AD = 8 cm, CD = 6 cm. If the area of △EGD is equal to the area of △CGF, what is the area of the shaded part?  

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C1: The figure below is made up of six identical rectangles (with dimension 5 cm by 10 cm). Find the area (in cm2) of the shaded region. (NMOS/Prelim’14/Q11) 

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C1: The figure below is made up of six identical rectangles (with dimension 5 cm by 10 cm). Find the area (in cm2) of the shaded region. (NMOS/Prelim’14/Q11) 

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C2 The figure below shows a margin of 4 cm from 2 sides of a rectangle PQRS. Given that PQ = 2PS and the area of the shaded region is 116 cm2, find the perimeter (in cm) of the rectangle PTVU. (NMOS/Prelim’13/Q11) 

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C2 The figure below shows a margin of 4 cm from 2 sides of a rectangle PQRS. Given that PQ = 2PS and the area of the shaded region is 116 cm2, find the perimeter (in cm) of the rectangle PTVU. (NMOS/Prelim’13/Q11) 

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C3 The following diagram showed a square ABCD. The points E and F are mid-point of AD and DC respectively. BE and BF cut diagonal AC at M and N respectively. Given the area of Square is 48 cm², determine the area of the region EMNFD.                            (APMOPS’15/Round1/Q20) 

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C3 The following diagram showed a square ABCD. The points E and F are mid-point of AD and DC respectively. BE and BF cut diagonal AC at M and N respectively. Given the area of Square is 48 cm², determine the area of the region EMNFD.                            (APMOPS’15/Round1/Q20) 

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C4 In the figure, each side of the square ABCD is 12 cm. E and F are midpoints of AB and AD respectively, and FG = 𝟏/𝟑 FC. Find the area (cm²) of the shaded region. (NMOS’18/Prelim/Q26) 

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C4 In the figure, each side of the square ABCD is 12 cm. E and F are midpoints of AB and AD respectively, and FG = 𝟏/𝟑 FC. Find the area (cm²) of the shaded region. (NMOS’18/Prelim/Q26) 

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M1 Study the figure and find the respective values of a and b. 

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M1 Study the figure and find the respective values of a and b. 

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M2 The figure below is made up of a pair of parallel lines and a regular pentagon ABCDE. Find the value of x.

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M2 The figure below is made up of a pair of parallel lines and a regular pentagon ABCDE. Find the value of x.

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C1 In this figure, find the value of x. (NMOS/Prelim’14/Q12)

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C1 In this figure, find the value of x. (NMOS/Prelim’14/Q12)

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C2 The figure shows a regular octagon. Find the value of angle SRT in degrees, where R is the centre of the star. (APMOPS/1stRound’14/Q1)

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C2 The figure shows a regular octagon. Find the value of angle SRT in degrees, where R is the centre of the star. (APMOPS/1stRound’14/Q1)

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C3 In the following figure, B, D, E and C lie on straight line. Given LBAD = LDAE = 12°, AC is perpendicular to AD and BC = AB + AE. Find the LABC.

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C3 In the following figure, B, D, E and C lie on straight line. Given LBAD = LDAE = 12°, AC is perpendicular to AD and BC = AB + AE. Find the LABC.

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C4 In the figure below, ABC is a right-angled isosceles triangle with LBAC = 90°, and ACDIs an isosceles triangle with AC = AD. Find in degrees, LBDC. (NMOS’18/Prelim/Q19)

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C4 In the figure below, ABC is a right-angled isosceles triangle with LBAC = 90°, and ACDIs an isosceles triangle with AC = AD. Find in degrees, LBDC. (NMOS’18/Prelim/Q19)