中三數學問題急求解答

👨🏻‍🏫 學術台數學

[匿名]

2013-06-24 3:39 am

1) if x=∛4+∛2+1 ,find(1+ 1/x)^3

2) 10 people form a circle.each picks a number,and then each one announces the average of the numbers of the two people next to them.the numbers announces by the 10 people are :1,2,....10
what is the number picked by the people who announces the average 8?

3)in a bag there are 5 balls, each printed with a positive number,the teacher asked each student to pick four balls and add up the four numbers.It is known that the only possible answers are 20,21,23,25.
Find the five numbers on the five balls.

4) (x(y=1))/(x+y+1)=2 , (x(z=2))/(x+z+2)=3 ,((y+1)(z=2))/(y+z+3)=4
Find z.

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回答 (4)

50418129

2013-06-24 5:41 pm

✔ 最佳答案

(1)

x = ∛4 + ∛2 + 1
x = (∛2)^2 + (∛2) + 1
x = (∛2 - 1)[(∛2)^2 + (∛2) + 1] / (∛2 - 1)
x = (2 - 1) / (∛2 - 1) = 1 / (∛2 - 1)

thus, (1 + 1/x)^3
= [1 + (∛2 - 1)]^3
= 2

(2)

Let the 10 numbers picked by 10 people be a, b, c, d, e, f, g, h, i, j
without loss of generality, (a + c) / 2 = 1 => a + c = 2, b + d = 4, c + e = 6,
d + f = 8, e + g = 10, f + h = 12, g + i = 14, h + j = 16, i + a = 18, j + b = 20

we have 2(a + c + e + g + i) = 50
a + c + e + g + i = 25
2 + 10 + i = 25
i = 13

(3)

Let the five numbers be a, b, c, d, e, thus
a + b + c + d = 20
b + c + d + e = 21
c + d + e + a = 23
d + e + a + b = 25
e + a + b + c = x, where x can be 20, 21, 23 or 25

when x = 20, we have
4(a + b + c + d + e) = 89 + 20 = 109
a + b + c + d + e = 27.25
thus, a = 6.25, b = 4.25, c = 2.25, d = 7.25, e = 7.25

when x = 21, we have
4(a + b + c + d + e) = 89 + 21 = 110
a + b + c + d + e = 27.5
thus, a = 6.5, b = 4.5, c = 2.5, d = 6.5, e = 7.5

when x = 23, we have
4(a + b + c + d + e) = 89 + 23 = 112
a + b + c + d + e = 28
thus, a = 7, b = 5, c = 3, d = 5, e = 8

when x = 25, we have
4(a + b + c + d + e) = 89 + 25 = 114
a + b + c + d + e = 28.5
thus, a = 7.5, b = 5.5, c = 3.5, d = 3.5, e = 8.5

but i think the question should be positive integer, not positive number,
so the five number are 3, 5, 5, 7, 8

(4)

Let a = y + 1, b = z + 2,
xa / (x + a) = 2 , xb / (x + b) = 3 , ab / (a + b) = 4
(x + a) / xa = 1/2 , (x + b) / xb = 1/3 , (a + b) / ab = 1/4
1/a + 1/x = 1/2 , 1/b + 1/x = 1/3 , 1/b + 1/a = 1/4
1/a + 1/x + 1/b + 1/x + 1/b + 1/a = 1/2 + 1/3 + 1/4 = 13/12
2(1/a + 1/b + 1/x) = 13/12
1/a + 1/b + 1/x = 13/24
1/b = 13/24 - (1/a + 1/x) = 13/24 - 1/2 = 1/24
b = 24 => z = 22

參考: knowledge

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[匿名]

2013-06-24 7:48 pm

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