Đề thi Toán châu Á Thái Bình Dương APMOPS 2015

How many bottles of “Five Plus” must Jason purchase at the start so that he will be able to enjoy a total of 109 bottles, assuming he makes use of the exchange policy as much as possible[r]

Name of School : Date Of Birth : (DD/MM/YY)

April 2015

Hwa Chong Institution

Mathematics Learning and Research Centre

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Asia Pacific Mathematical Olympiad

for Primary Schools 2015

First Round 2 hours (150 marks)

Instructions to Participants :

Mathematical instruments, tables and calculators are not permitted Write your answers in the answer sheet provided,

Marks are Awarded for correct answers only

Problem 1: Kelly is years old and her father is 34 years old this year When the sum of their age is equal to 58, how old will be Kelly?

Problem 2: Find (1771+1717 7171+

1717171 717171 )÷

17171717 71717171 ;

Problem 3: Find 2× 9+17 × 99+81 ×999

Problem 4: Jenny and Chris started running from the same point on a 200-metre

circular track They ran in the same direction at m/s and m/s respectively How many times did overtaking take place within 16 minutes?

Problem 5: The length, breadth and height of a cuboid (rectangular box) is in the ratio of 4:3:2 and the total length of all edges is equal to 72 cm Find, in cm3, the volume of

the cuboid

Problem 6: There are 30 marbles in a bag, comprising 10 of each colour, red, yellow and green Each red, yellow and green marble weighs grams, grams and grams respectively Eight marbles are now selected randomly from the bag, with a total mass of 39 grams What is the maximum possible number of red marbles selected?

Problem 7: Using small wooden cubes, Bryan built a structure, which has the front view and side view as shown in the diagrams below What is the maximum possible number of wooden cubes used?

Problem 8: In a certain month with 31 days, there are as many Mondays as Fridays.

Which day of the week is the 10th of this month?

(1) Monday (2) Tuesday (3) Wednesday (4) Thursday

(5) Friday (6) Saturday (7) Sunday

Problem 9: If A is a whole number and 59<9

A<1 , how many possible values are there for A?

Problem 10: What is the remainder when 22015

+20152 is divided by 7?

Problem 11: Find

(101+234+567)×(234+567+89)−(101+234 +567+89)×(234 +567)

Problem 12: In the diagram, ABCD is a square with side length cm The point

E is on the extension of AB and it is given that BEFG is also a square Find, in cm2, the area of triangle AFC .

Problem 13: A logistic company is tasked to transport 89 tonnes of cargo The capacity of lorry and caravan is tonnes and tonnes respectively If each lorry consumes 14 litres of gasoline for the trip while each caravan only uses litres, what is the least total gas consumption (in litres) to complete the task?

Problem 14: There are 52 students in a class During sports festival, 30 students played basketball, 35 played football and 42 played table tennis What is the least possible number of students who have played all three sports?

Problem 15: As shown in the diagram, a soccer ball is made of 32 patches of leather. Each black patch is a pentagon adjacent to white patches Each white patch is a hexagon adjacent to black and white patches How many white patches are there altogether?

Problem 16: Alan was trying to find the sum of interior angles of an n-sided polygon. He made a small mistake by missing one angle and obtained a sum of 2015 ° Find the missing angle in degrees

Problem 17: In the diagram, A and B are the centres of two quarter-circles of radii 14 cm and 28 cm respectively Find the difference between areas of region I

and II in cm2 (Take π to be 22

7 )

Problem 18: It is given that 15 ABC 6´ is the smallest 6-digit number that starts with

“15”, ends with “6”, and divisible by 36 Find the value of A2+B2+C2

Problem 19: Find the total number of figures in a 6x6

chessboard

Problem 20: To promote a new type of isotonic drink “Five Plus”, the beverage company allows customer to exchange empty bottles for a new bottle of drink How many bottles of “Five Plus” must Jason purchase at the start so that he will be able to enjoy a total of 109 bottles, assuming he makes use of the exchange policy as much as possible?

Problem 21: When a whole number A is divided by another number B , the quotient Q is 15 and remainder R is If the sum of A , B , Q and R

is equal to 2169, find A

Problem 22: Four football teams participated in a round-robin tournament Every team plays against all other teams once each points are awarded for a win, point for a draw and point for a loss At the end of the tournament, if the points obtained by the teams are four consecutive numbers, what is the product of these four numbers?

Problem 23: Five water pipes A, B, C, D and E are connected to a water tank If A, B, C and D are turned on, it takes hours to fill the tank completely If B, C, D and E are turned on, it takes hours By using pipe A and E only, it would take 12 hours How many hours will it take if only pipe E is turned on?

Problem 24: In the diagram, ABCD is a square E and F are midpoints of

AB and BC respectively DE and DF intersect the diagonal AC at points M and N respectively If the area of square ABCD is 48 cm2, what is

the area, in cm2, of pentagon EBFNM ?

Problem 25: The diagram shows a 5 ×5 square with 25 unit squares Find the least number of unit squares to be shaded such that any 3 ×3 square in the diagram contains exactly four shaded unit squares.

Problem 26: Find the number of consecutive 0’s at the end of the product

5 ×1 0× 15 × 20× … ×2010 ×2015 ?

Problem 27: Alan and Betty started running towards each other at the same instant, from cities A and B respectively It is known that the ratio of Alan’s speed to Betty’s is 3:2 Given that station C is between the two cities, Alan and Betty reached C at 0900 hr and 1900 hr respectively At what time did Alan and Betty meet? (Write your answer using the 4-digit format For example, if your answer is 14:35, please write 1435)

Problem 28: What is the maximum number of integers that can be selected from

{1,2,3, , 48,49 }, such that when the selected integers are arranged around a circle,

the product of any two adjacent integers is less than 100?

Problem 29: Two overlapping squares with parallel sides are such that the part

common to both squares has an area of cm2 This is

9 the area of the larger

square and 14 the area of the smaller square What is the minimum perimeter, in cm, of the eight-sided figure formed by the overlapping squares?

Problem 30: There are five cards A , K , Q , J , 10 and five envelopes labelled ‘ A ’, ‘ K ’, ‘ Q ’, ‘ J ’, and ‘10’ In the dark, Amy randomly inserts one card into every envelope How many different ways are there such that every card is in the wrong envelope?