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**Notice: **

**Individual students, nonprofit libraries, or schools are **

**permitted to make fair use of the papers and its **

**solutions. Republication, systematic copying, or **

**multiple reproduction of any part of this material is **

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**Solution to **

**Sixth International Mathematics Assessment for Schools **

**Round 1 of Upper Division**

### 1.

### What is the simplified value of

### 20 17

### 2

### 0 1 7

### ×

### + + +

### ?

### （

### A

### ）

### 340

### （

### B

### ）

### 34

### 2017

### （

### C

### ）

### 10

### （

### D

### ）

### 20

### （

### E

### ）

### 34

### 【

### Solution

### 】

### 20 17

### 20 17

### 2 17

### 34

### 2

### 0 1 7

### 10

### ×

_{=}

### ×

_{= × =}

### + + +

### .

### Answer:

### （

### E

### ）

### 2.

### What is the remainder when 2017 is divided by 9?

### （

### A

### ）

### 0

### （

### B

### ）

### 1

### （

### C

### ）

### 2

### （

### D

### ）

### 3

### （

### E

### ）

### 7

### 【

### Solution 1

### 】

### 2017

### = ×

### 9 224 1

### +

### .

### 【

### Solution 2

### 】

### The remainder of an integer divided by 9 equals to the remainder of the sum of its

### digits divided by 9. Thus, the remainder of 2017 is 2

### + + + =

### 0 1 7 10

### divided by 9,

### which is 1.

### Answer:

### （

### B

### ）

### 3.

### Positive integers are arranged in the array as shown below, what is the sum of all

### the integers located on the fifth row ?

### 1

### 2 3 4

### 5 6 7 8 9

### 10 11 12 13 14 15 16

### ⋮

### （

### A

### ）

### 91

### （

### B

### ）

### 164

### （

### C

### ）

### 172

### （

### D

### ）

### 189

### （

### E

### ）

### 215

### 【

### Solution

### 】

### According to the pattern, the sum of the fifth row is

### 17 18 19

### + + +

### 20

### +

### 21 22

### +

### +

### 23

### +

### 24

### +

### 25 189

### =

### .

### Answer:

### （

### D

### ）

### 4.

### Arrange the numbers

### 2.718

### ,

### 2.718

### ,

### 2.718

### and 2.71828 in increasing order.

### (Repeating decimals are denoted by drawing a horizontal bar above the recurring

### figures.)

### （

### A

### ）

### 2.718

### <

### 2.718

### <

### 2.71828

### <

### 2.718

### （

### B

### ）

### 2.71828

### <

### 2.718

### <

### 2.718

### <

### 2.718

### （

### C

### ）

### 2.718

### <

### 2.71828

### <

### 2.718

### <

### 2.718

### （

### D

### ）

### 2.71828

### <

### 2.718

### <

### 2.718

### <

### 2.718

### 【

### Solution

### 】

### Observe that 2.718

### =

### 2.718718

### ⋯, 2.718 2.71818

### =

### ⋯ , 2.718 2.71888

### =

### ⋯ and

### 2.71828. By comparing the ten thousandths digit, we get

### 2.718

### <

### 2.71828

### <

### 2.718

### <

### 2.718

### .

### Answer:

### （

### C

### ）

### 5.

### The figure below plots the body temperature records of one patient in a day. The

### records started at 00:00 AM and were taken every 4 hours. After how many hours

### did the patient recorded his highest temperature?

### （

### A

### ）

### 0

### （

### B

### ）

### 4

### （

### C

### ）

### 12

### （

### D

### ）

### 16

### （

### E

### ）

### 24

### 【

### Solution

### 】

### Reading from the plot, the temperature is the highest at 16 o'clock.

### Answer:

### （

### D

### ）

### 6.

### On a 5 5

### ×

### table below, place into each cell the sum of its row number and

### column number. For example, value of

*a*

### below is 2

### + =

### 3

### 5

### . How many odd

### numbers are filled into the table?

### 1 2 3 4 5

### 1

### 2

*a *

### 3

### 4

### 5

### （

### A

### ）

### 5

### （

### B

### ）

### 10

### （

### C

### ）

### 12

### （

### D

### ）

### 18

### （

### E

### ）

### 25

### 【

### Solution 1

### 】

### All numbers filled in are as follows, there are 12 odd numbers.

### 1 2 3 4 5

### 1

### 2 3 4 5 6

### 2

### 3 4 5 6 7

### 3

### 4 5 6 7 8

### 4

### 5 6 7 8 9

### 5

### 6 7 8 9 10

### 0

### 4

### 8

### 12

### 16

### 20

### 24

### time

### 36.5

### 37

### 37.5

### 38

### 38.5

### 39

### 39.5

### 40

### 38

### 38.4

### 37.8

### 38.3

### 39.1

### 38.5

### 37.7

### ℃

### 【

### Solution 2

### 】

### Observe that the properties of pari are as follows: odd + odd = even, even + odd =

### odd and even + even = even, there are 2 odds on the first row, 3 on the second row,

### 2 on the third row, 3 on the fourth row and 2 on the fifth row. Thus, there is a total of

### 2

### + + + + =

### 3

### 2

### 3

### 2 12

### odd numbers.

### Answer:

### （

### C

### ）

### 7.

### There are 23 kids seated in a row. They call out the numbers from left to right as

### 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, … for the first round. They call out 1, 2, 3, 4, 1, 2, 3, 4,

### 1, 2, … from right to left for the second round. How many kids call out the same

### number in two rounds?

### （

### A

### ）

### 11

### （

### B

### ）

### 12

### （

### C

### ）

### 15

### （

### D

### ）

### 18

### （

### E

### ）

### 23

### 【

### Solution

### 】

### 23 kids call out the numbers as follows:

### 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

### First

### Call

### 1

**2 **

### 3

**4 **

### 1

**2 **

### 3

**4 **

### 1

**2 **

### 3

**4 **

### 1

**2 **

### 3

**4 **

### 1

**2 **

### 3

**4 **

### 1

**2 **

### 3

### Second

### Call

### 3

**2 **

### 1

**4 **

### 3

**2 **

### 1

**4 **

### 3

**2 **

### 1

**4 **

### 3

**2 **

### 1

**4 **

### 3

**2 **

### 1

**4 **

### 3

**2 **

### 1

### 11 kids call out the same number in two rounds.

### Answer:

### （

### A

### ）

### 8.

### Salted water with 3.2% concentration weights 500 g. How many salt, in grams, is

### left if the water is vaporized completely?

### （

### A

### ）

### 16

### （

### B

### ）

### 32

### （

### C

### ）

### 64

### （

### D

### ）

### 100

### （

### E

### ）

### 128

### 【

### Solution

### 】

### The salt is 500 3.2% 16

### ×

### =

### g.

### Answer:

### （

### A

### ）

### 9.

### In the figure below, Tom combined some squares of the same size into a shape of

### umbrella. Find the least number of squares he would use.

### （

### A

### ）

### 5

### （

### B

### ）

### 9

### （

### C

### ）

### 12

### （

### D

### ）

### 15

### （

### E

### ）

### 20

### 【

### Solution

### 】

### The larger the square is, the less number Tom needs.

### The square size is determined by the shortest length of

### the broken line segment enclosing the umbrella. As in

### the figure, the least number is using 15 squares

### .

### 10.

### The shape enclosed by solid lines in the figure below is composed of unit squares.

### What is the maximum area of a rectangle that can be cut from the shape along

### grid lines?

### （

### A

### ）

### 80

### （

### B

### ）

### 96

### （

### C

### ）

### 100

### （

### D

### ）

### 112

### （

### E

### ）

### 128

### 【

### Solution

### 】

### Compute the area of rectangles of following shapes: 2 22

### ×

### =

### 44

### , 4 20

### ×

### =

### 80

### ,

### 6 16

### × =

### 96

### , 8 14 112

### × =

### , 10 10 100

### × =

### , 12 8

### × =

### 96

### , 14 4

### × =

### 56

### and 16 2

### × =

### 32

### .

### The maximal area is 112.

### Answer:

### （

### D

### ）

### 11.

### Given six cards with numbers 1, 2, 3, 4, 5 and 6 one card for each number. Each

### time Lee takes 2 cards, he computes the difference (larger one minus small one)

### and discards the two cards. Find the maximum possible sum of the three

### differences after all cards are discarded.

### （

### A

### ）

### 3

### （

### B

### ）

### 5

### （

### C

### ）

### 7

### （

### D

### ）

### 8

### （

### E

### ）

### 9

### 【

### Solution

### 】

### The sum of the differences is the sum of three of the six numbers minus the sum of

### the remaining three. The maximum is 6

### + + − − − =

### 5

### 4 3 2 1 9

### .

### Answer:

### （

### E

### ）

### 12.

### The houses of Mary and Jerry are connected by a trail. One day, they started from

### their respective house at the same time, and walked towards the other's house.

### The speed of Mary is 1.5 times that of Jerry and they met 12 minutes later. On

### the next day, Mary left his house and walked to Jerry's house with the same speed.

### How long would he take to reach Jerry's house?

### （

### A

### ）

### 15

### （

### B

### ）

### 18

### （

### C

### ）

### 20

### （

### D

### ）

### 24

### （

### E

### ）

### 30

### 【

### Solution 1

### 】

### Since Mary's speed is 1.5 times of Jerry. It will take Mary 12 1.5

### ÷

### =

### 8

### minutes from

### the place they met the first day to Jerry's house. Mary will take 12 8

### + =

### 20

### minutes to

### reach Jerry's house.

### 【

### Solution 2

### 】

### Assume the speed of Jerry is 1, then the speed of Mary is 1.5. The distance between

### two houses is (1 1.5) 12

### +

### × =

### 30

### . It will take Mary 30 1.5

### ÷

### =

### 20

### minutes to walk.

### Answer:

### （

### C

### ）

### 13.

### In the figure below, the area of square

* ABCD*

### is

### 36 cm

2### , the area of rectangle

*CDEF*

### is 18 cm

2### and the area of

*ADGH*

### is 48 cm

2### . What is the

### perimeter, in cm, of

*BFEDGH*

### ?

### （

### A

### ）

### 18

### （

### B

### ）

### 36

### （

### C

### ）

### 46

### （

### D

### ）

### 48

### （

### E

### ）

### 56

### Mary

### Jerry

*A *

*E *

*B *

*C *

*D *

*H *

*G *

*F *

### 【

### Solution

### 】

### The area of square

*ABCD*

### being 36 cm

2### , its side length is

*AB*

### =

*BC*

### =

### 6

### cm. Rectangle

*CDEF has area 18 cm*

2### , one half of area of

*ABCD, so *

### 1

### 3

### 2

*CF*

### =

*BC*

### =

### cm; Rectangle

*ADGH has area 48 cm*

2### ,

### 4

### 3

* of area of ABCD, *

### 4

### 8

### 3

*CF*

### =

*AB*

### =

### cm. Thus, the

*perimeter of BFEDGH is (6*

### + + + × =

### 3 6 8) 2

### 46

### cm.

### Answer: （C）

### 14.

### There are two routes starting in a bus stop. A bus departs for the first route every

### 8 minutes and departs the second route every 10 minutes. At 6:00 in the morning,

### two buses depart for the two routes at the same time. Among the choices below,

### when will the buses depart for the two routes simultaneously?

### （A）7:30

### （B）8:20

### （C）9:40

### （D）10:00

### （E）11:00

### 【Solution】

### Every [8, 10]

### =

### 40

### minutes the two buses depart at the same time. Among the

### options, only the difference between 10:00 and 6:00 is an integer multiple of 40

### minutes.

### Answer: （D）

### 15.

*Let a, b, c, d, e and f are distinct digits such that the expression *

*ab*

### +

*cd*

### =

*ef*

### .

### What is the least possible value of

*ef ? *

### （A）30

### （B）34

### （C）36

### （D）39

### （E）41

### 【Solution】

### Since

*a*

### + ≥ + =

*c*

### 1 2

### 3

### , one has

*e*

### ≥

### 3

### . Take

*e*

### =

### 3

*, one gets a and c are 1 and 2, *

*respectively. Moreover b and d are non-zero, otherwise f is equal to b or d. So *

### 4 5

### 9

*b*

### + ≥ + =

*d*

### , that is,

*f*

### =

### 9

### . For example,

### 14

### +

### 25

### =

### 39

### and

### 15

### +

### 24

### =

### 39

### .

### Answer: （D）

### 16.

### Henry starts working at 9：00 in the morning and finishes at 5：00 in the afternoon.

### How many more degrees does the minute hand rotates than the hour hand does

### on the clock during this period?

### （A）120

### （B）1200

### （C）1320

### （D）2640

### （E）2880

### 【Solution】

### Henry works for 8 hours. In this period, the minute hand rotates 8 rounds, that is

### 8 360

### ×

### =

### 2880

### degrees. And the hour hand rotates

### 8

### 2

### 12

### =

### 3

### round, that is

### 2

### 360

### 240

### 3

### ×

### =

### degrees. The difference is

### 2880

### −

### 240

### =

### 2640

### degrees.

### Answer: （D）

### 17.

### The average score of a class in an exam is 70. Two students got 0 for absence and

### the average of remaining students is 74. What is the total number of students in

### the class?

### 【Solution 1】

### The average changes from 74 to 70 when two 0 score students are counted. Every

### other student gives 4 points of himself to the two zero score student to get a total

### 70 2 140

### × =

### . There are 140 4

### ÷ =

### 35

### other student and 35

### + =

### 2

### 37

### students in total.

### 【Solution 2】

*Assume there are x students in the class, then *

### 74(

*x*

### − =

### 2)

### 70

*x*

### which solves

*x*

### =

### 37

### .

### Answer: （E）

### 18.

### The price criteria of the subway ticket of a city is as follows: $2 for within 4 km,

### $1 more per 4 km for distances between 4 km and 12 km, $1 more per 6 km for

*distances over 12 km. It costs $8 to take subway from station A to station B. *

*Which of the following is closest to the distance between A and B? *

### （A）12 km （B）18 km

### （C）24 km

### （D）36 km （E）48 km

### 【Solution】

### The cost of a 12km trip is

### 2

### (12

### 4)

### 4

### 4

### −

### +

### =

* dollars, so the distance between A and B *

### is over 12 km, and the cost increases by $4 after 12 km. So the distance after 12 km is

### at least 6 4

### × =

### 24

### km but less than 6 5

### × =

### 30

### km. Then the distance between

*station A and station B is at least 36 km, and at most 42 km. Among the options, the *

*closest distance between A and B is 36 km. *

### Answer: （D）

### 19.

### The figure below shows the floor plan of a library. Each room is connected to

### adjacent rooms. One starts from room 1 and walks through all rooms without

### repetitions (going back). How many unique paths are there?

### （A）4

### （B）8

### （C）10

### （D）12

### （E）14

### 【Solution】

### We have all paths by enumeration:

### 1→2→3→4→5→6→7；1→2→3→4→5→7→6；1→2→3→4→6→5→7；

### 1→2→3→4→6→7→5；1→2→3→5→4→6→7；1→2→3→5→7→6→4；

### 1→2→4→3→5→6→7；1→2→4→3→5→7→6；1→3→2→4→5→6→7；

### 1→2→4→6→7→5→3；1→3→2→4→6→5→7；1→3→2→4→5→7→6；

### 1→3→2→4→6→7→5；1→3→5→7→6→4→2.

### There are totally 14 paths.

### Answer: （E）

### 1

### 2

### 3

### 4

### 5

### 6

### 7

*A *

*B *

### 20.

### In the figure below, 55 unit cubes were stacked in a pile.

### Paint the surface of the pile of cubes but the face on the

### ground is not painted. How many unpainted unit cubes are

### there when the stack is separated?

### （A）6

### （B）9

### （C）13

### （D）14

### （E）18

### 【Solution】

### The level on the ground has 3 3

### × =

### 9

### unpainted cubes, the second level has

### 2 2

### × =

### 4

### , the third level has 1. In total, there are 9

### + + =

### 4 1 14

### unpainted unit cubes.

### Answer: （D）

### 21.

### There are 50 mail boxes. One day 151 letters are distributed into these mail boxes.

### It turns out that one mail box has more letters than any other mail box. What is

### minimal number of letters this mail box can have?

### 【Solution】

### Since 151 3 50 1

### = ×

### +

### , at least one mail box has at least 3 1 4

### + =

### letters. When one

### mail box has 4 letters, other mail box has 3 letters, the conditions in the problem is

### satisfied. The minimum is 4.

### Answer: 004

### 22.

### In the figure below, a bottle consists of side faces of two identical cones. The

*radius of the bottom of the cone is 5 cm while the distance between A and B is 24 *

*cm. There is a hole at A and is sealed elsewhere. The bottle is now filled up with *

*water and placed on a horizontal table with BC along the table top face. What is *

### the volume, in cm

3### , of the water left, if the thickness of the bottle surface and size

### of hole is ignored (take π as 3.14)?

### 【Solution】

### Assume the diameter of bottom of the cone is

*CD. If BC is horizontal, it follows that AD is *

### also horizontal. There are no water leaking

### out. The volume of water is equal to the

### volume of the bottle, which is

2