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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
2and 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
