### 注意：

### 允許學生個人、非營利性的圖書館或公立學校合理使用

### 本基金會網站所提供之各項試題及其解答。可直接下載

### 而不須申請。

### 重版、系統地複製或大量重製這些資料的任何部分，必

### 須獲得財團法人臺北市九章數學教育基金會的授權許

### 可。

### 申請此項授權請電郵

### ccmp@seed.net.tw

**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 **

**permitted only under license from the Chiuchang **

**Mathematics Foundation. **

**Requests for such permission should be made by **

**───────────────────────────────────────────────── **

**Solution Key to Second Round of IMAS 2015/2016 **

**Junior High School Division **

**───────────────────────────────────────────────── **
1. Michael bought 6 pens and 3 notebooks while Wallace bought 3 pens and 6

notebooks. The pens are identical and so are the notebooks. Michael’s bill is 6 dollars higher than Wallace’s. How many dollars is the price of a pen higher than the price of a notebook?

（A）1 （B）2 （C）3 （D）4 （E）5 【Suggested Solution】

Since 3 pens cost 6 dollars more than 3 notebooks, a pen costs 2 dollars more than a notebook.

Answer:（B） 2. If all the divisors of 2016 are arranged in decreasing order, by how much is the

third divisor larger than the fourth divisor?

（A）12 （B）48 （C）168 （D）672 （E）2016 【Suggested Solution】

Since 2016 is divisible by 2, 3 and 4, its third largest divisor is 2016 3 672 and its fourth largest divisor is 2016 4 504. Their difference is 672 504 168 .

Answer:（C）
3. Which of the following is equal to *xr**ys* if *r* 3*x*2*y* and *s**xy* *x* *y*?

（A） 2 2 2
2 2
*x y**x* *xy* *y* （B）*xy*2 3*x*2 *xy* *y*2 （C） 2 2
2
*x y* *x* *xy*
（D） 2 2
2 2
*xy* *x* *xy* （E） 2 2
*x y* *x* *y*
【Suggested Solution】
2 2 2 2 2 2
(3 2 ) ( ) 3 2 3
*xr* *ys**x x* *y* *y xy* *x* *y* *x* *xy**xy* *xy**y* *xy* *x* *xy**y* 。
Answer:（B）
*4. E is a variable point on the side BC of a square ABCD. DEFG is a rectangle with *
*FG passing through A. As the point E moves from B towards C, how does the *
*area of DEFG change? *

（A）Steadily increasing （B）Steadily decreasing

（C）Decreasing and then increasing （D）Increasing and then decreasing
（E）Remaining constant
*A * *D *
*E * *C *
*B *
*G *
*F *
*E*
*G*
*F*

【Suggested Solution】

*The area of triangle ADE is constant since it has a fixed base AD and a constant *
*altitude. The area of DEFG is always twice the area of ADE since they have a *
*common base DE and the same altitude. *

Answer:（E） 5. The diagram shows a 6 by 6 board with three barriers. An

ant is at the top left corner and wishes to reach the bottom right corner. It may only crawl between squares which share a common side, and only towards the bottom or the right. It cannot pass through any barrier. How many different paths can it follow?

（A）88 （B）90 （C）92 （D）96 （E）112 【Suggested Solution】

The diagram below shows the number of ways the ant can reach each square on the board according to the rules.

### 1 1 0 0

### 0

### 0

### 1 2 0 0

### 0

### 0

### 1 3 3 3

### 3

### 3

### 1 4 7 10 10 10

### 0 0 7 17 27 37

### 0 0 7 24 51 88

Answer:（A） 6. One square in a 3 by 3 board is to be painted black, a second squareblue and a third square red. If no two of these three squares are in the same row or in the same column, how many different ways of painting them are there?

【Suggested Solution】

There are 9 possible places for the black square, 4 for the blue square and only 1 for the red square. The total number of choices is 9 4 1 36 .

Answer: 36 7. The diagram shows a tile divided into regular hexagons of

side length 1 cm. What is the total area, in cm2, of the parts of the tile which are shaded?

【Suggested Solution】

The shaded parts consist of seven triangles of sides 1, 1 and .

The altitude is 1

2 and the area is thus

1 1 3

3

2 2 4 . The total area of the shaded

*A *

30 30

parts is therefore 3 7 7 3

4 4 .

Answer: 7 3

4 cm

2

*8. Let a, b, c and d be real numbers such that |a**b*|, |*a**b*|, |*c**d*| and
|*c**d*| are 6, 7, 8 and 9 in some order. What is the value of *a*2 *b*2 *c*2 *d*2?
【Suggested Solution】

From (*a**b*)2 (|*a**b*|)2 62 36and (*a b* )2 (|*a b* |)2 72 49, we have

2 2 1 2 2 1
(( ) ( ) ) (36 49)
2 2
*a* *b* *a**b* *a b* . Similarly,
2 2 1 2 2 1
(( ) ( ) ) (64 81)
2 2
*c* *d* *c**d* *c* *d* . It follows that
2 2 2 2 1 1 1
(36 49) (64 81) 230 115
2 2 2
*a* *b* *c* *d* .
Answer: 115
*9. A rectangle ABCD with * *BC*3cm and *AB* 3*cm is folded along AC so that *
*the point B lands on the point K symmetric to it about AC. What is the area, in *
cm2*, of triangle KDE, where E is the point on the extension of BC such that *

30
*CDE*

?

【Suggested Solution】

Since *CD**AB* 3cm and*CDE* 30 , we have *DE*2cm；

*From AB：BC = * 3：3 = 1： 3and*ABC* 90 , *ACB* 30 . Hence*ACK* 30 ,
so that*DCK* 90 30 30 30 .

*It follows that KC is parallel to DE, so that the area of KCD is the same as that of *

*CDE, which is * 1 2 3 3
2 2 2 cm
2
Answer: 3
2 cm
2
*A * *D *
*E *
*C *
*B * _{3 }
*K *
3
30

10. In the expression (((10□2)□2)□2)□ , each □ is to be replaced by a 2 different one of +, −, × and ÷. How many different values can this expression take?

【Suggested Solution】

If the addition and the subtraction are performed consecutively, they will cancel out. The same applies to the multiplication and the division. To obtain a value other than 10, we must alternate the first pair with the second. Thus we only have to examine eight cases:

(((102)2)2) 2 11、 (((10 2) 2) 2) 2 8 、 (((10 2) 2) 2) 2 9 、 (((102)2)2) 2 12、(((10 2) 2) 2) 2 11 、(((10 2) 2) 2) 2 9 、 (((102) 2) 2) 2 8、 (((10 2) 2) 2) 2 12 .

Altogether, there are five possible values, namely, 8, 9, 10, 11 and 12.

Answer: 5
*11. Let a, b and c be real numbers such that * *abc*1, (*a*1)(*b*1)(*c* 1) 16 and

(*a*2)(*b*2)(*c*2)53. What is the value of (*a*1)(*b*1)(*c*1)?
【Suggested Solution】

From (*a*1)(*b*1)(*c* 1) *abc**ab**ac**bc* *a* *b* *c* 1, we have

16 1 14
*ab**ac**bc* *a* *b* *c* *abc* . From
(*a*2)(*b*2)(*c*2)*abc*2*ab*2*ac*2*bc*4*a*4*b*4*c*8, we have
53 8
2 2 2 22
2
*abc*
*ab**ac**bc* *a* *b* *c* ；

Subtraction yields*a* *b* *c* 22 14 8 . Hence*ab**ac bc* 14 8 6.

It follows that (*a*1)(*b*1)(*c* 1) *abc**ab**ac bc* *a* *b* *c* 1 1 6 8 1 2.
Answer: 2
*12. The area of triangle ABC is 120 cm*2 and *BC*16cm. What is the minimum

*length, in cm, of the perimeter of ABC? *
【Suggested Solution】

*The altitude of ABC to the base BC has length *
120 2

15 16

. When *AB* *AC, we have AC=*
2 2

15 8 17

*AB* *so that the perimeter of ABC *
is 16+17+17=50. We claim that this is minimum.
*Let B’ be the reflection of B across the line *

*through A parallel to BC. Then A lies on CB’and *

*B C* *AB* *AC* *AB* *AC*. For any other point

*A’on this line, *

*A B* *A C* *A B* *A C* *B C* *AB**AC*
Answer: 50 cm
*A *
8
15
*D * *C *
*B *
*A*
*B*
*L *

*13. Let a, b, c and d be four different non-zero digits. The greatest common divisor *
of the four-digit numbers *abcd* and *acbd is n. What is the largest possible *
*value of n? *

【Suggested Solution】

We may assume by symmetry that *b**c*. Then *abcd* *acbd* 90(*b c* ), which is
*divisible by the desired greatest common divisor n. Since * *d* 0*, n is not divisible by *
*10. If n is not divisible by 5, then it is at most * 90( ) 18( )

5

*b c*

*b c*

_{} _{} _{≥}

18 8 144 *. If n *
is divisible by 5, then it is a divisor of 90( ) 45( )

2

*b c*

*b c*

_{} _{}

. Since *b c* 9 1 8,

*n≤ * 45 7 315. If *n*315*,then d=5 and (b,c)=(9,2) or (8,1). Since 315 is divisible *
by 9, so is the digit-sum of *abcd*, which is therefore either 4815 or 2925. However,
*neither is divisible by 7. The next highest value for n is * 45 5 225*. As before, d=5. *
*If we take b=7 and c=2, then both 4725 and 4275 are divisible by 225. Hence this is *
*the maximum value for n. *

Answer: 225 14. The first diagram shows a 6 by 6 board, and

the second diagram shows an L-shaped piece consisting of four 1 by 1 squares. Paint as few of the squares of the 6 by 6 board black so that wherever the L-pieced piece is placed on the board covering four squares, at least one of the squares will be black. The L-shaped piece may be turned about or flipped over.

【Suggested Solution】

If we paint black all squares on three parallel diagonals of respective lengths 3, 6, 3, any placement of the L-shaped pieces must cover one of these 12 squares. (10 points) This is in fact minimum since the board may be divided into six 2 by 3 subboards, and within each subboard, at least 2 squares must be painted black. (10 points)

Answer: 12

*15. P and Q are points on the bisector of the exterior angle at A of triangle ABC, with *
*A between P and Q, such that BP is parallel to CQ. D is the point on BC such that *
*DP=DQ. Prove that AB is parallel to DQ. *

【Suggested Solution】

*We first assume that AB>AC, so that the extensions of PQ and BC meet at some point *

*X. Let E be the point on BC such that QE is parallel to AB. Then we have * *XQ* *XE*

*XA* *XB*

and *XQ* *XC*

*XP* *XB* *, that is XA×XE=XB×XQ=XP×XC, hence *

*XA* *XC*

*XP* *XE* *, so that PE is *
*parallel to AC. *(10 points)

*Now ∠EPQ=∠CAQ=∠PAB=∠EQP, so that EP=EQ. It follows that E coincides *
*with D, and DQ is parallel to AB. *(5 points)

*The case AB=AC is similar. *(5 points)