Hong Kong Mathematics Mathematics Olympiad (1983
84)
Sample Event (Individual) (1983 (
(i)
(iii)
一正
b 邊形之內角和為 a ,求 b。
If 8b
c 21 , find
8b
a
a , 求 a。
The sum of the interior angles of a regular b-sided polygon is a. Find b .
若
(iv)
)
In the given diagram, the sum of the three marked angles is a. Find a .
附圖所示三角之和為
(ii)
84)
c .
b
c
c 21 , 求 c。
If c logd 81 , find d . .
d d
81 ,求 d 。 若 c log d 81
P. 1
Hong Kong Mathematics Mathematics Olympiad (1983
84)
Event 1 (Individual) (1983 1 (
(i)
If 100a 352 152 , find a .
84) )
a
若 100a 352 152 ,求 a。
(ii)
2 If a 1
34b , find
b .
b
若 a 12 34b ,求 b。
(iii)
If b is a root of x 2 cx 5 0 , find c .
若
(iv)
c
b 為 x 2 cx 5 0 之一根,求 c 。
If x + x + c is a factor of 2 x 2 3x 4d , find d . . x + c 為 若 x +
2 x2 3x 4d 之因式,求 d 。
P. 2
d d
Hong Kong Mathematics Mathematics Olympiad (1983
84)
Event 2 (Individual) (1983 2 (
(i)
,
為
x 2 10 x 20 0 之根,且 a
1
1
.
a
,求 a。
The point A( A(b, c) lies on the line 2 y x 15 . Find c .
2 y x 15
If x2 cx 40 x k
若
1
b
(0 < < 90 90), 且 10cos2 b ,求 b。 sin a (0
A(b, c) 在直線 點 A(
(iv)
1
If sin a (0 (0 < < 90 90) , and 10cos2 b , find b .
若
(iii)
)
If , are roots of x2 10 x 20 0 , find a , where a
若
(ii)
84)
2
x2 cx 40 x k
上,求
d , find
2
d ,求
c
c。
d . .
d d
d 。
P. 3
Hong Kong Mathematics Mathematics Olympiad (1983
84)
Event 3 (Individual) (1983 3 (
(i)
One ball is taken at random from a bag containing b + 4 red balls and 2b 2
一袋內有紅球 b + 4 個,白球 球為白色之機會為 x, 求 x。
x
2b 2 個。若隨意於袋內取球一個,而該
If sin x (90 (90 < < 180 180) and tan 15 y , find y . y .
若
b
b cm2 為一邊長 (8 + a) cm 的立方體之總表面積,求 b。
white balls. If x is the probability that the ball is white, find x . x .
(iv)
a
a 為 2 x3 3x2 x 1 被 x + x + 1 除所得之餘數,求 a 。
If b cm2 is the total surface area of a cube of side (8 + a) cm , find b .
若
(iii)
)
If a is the remainder when when 2 x3 3x2 x 1 is divided divided by x + x + 1 , find a .
若
(ii)
84)
(90 < < 180 180) 及 tan 15 y ,求 y。 sin x (90
P. 4
y
Hong Kong Mathematics Mathematics Olympiad (1983
84)
Event 4 (Individual) (1983 4 (
(i)
84) )
In figure 1, DE // BC // BC .. If AD 4 , DB , DB 6 , DE DE 6 and BC BC a , find a .
a
BC ,若 DE 6 且 BC BC a,求 a。 在圖一中, DE //// BC ,若 AD 4, DB 6, DE A
4
E
D 6 6
Figure 1
圖一 C
B
(ii)
is the positive acute angle such that cos
為正銳角, cos
(iii)
If c3
若
(iv)
a
b2
c3
a 17
。若
tan
b 15
a 17
. If tan
b 15
, find b .
b
,求 b。
, find c .
c
b 2 ,求 c。
The area of an equilateral triangle is c 3 cm2 . If its perimeter is d cm cm , find d . .
一等邊三角形之面積為
c 3 cm2 。 若其周界長 d cm cm,求 d 。
P. 5
d d
Hong Kong Mathematics Mathematics Olympiad (1983
84)
Event 5 (Individual) (1983 5 (
(i)
84) )
In Figure 2 , find a .
a
在圖二,求 a。
a
100
50 110
(ii)
a , find 5
If b log2
Figure 2
圖二
b .
b
a 若 b log2 ,求 b。 5
(iii)
A piece of string , 20 m long , is divided into 3 parts in the ratio of b 2 : b : b + 2 . If N m m is the length of of the longest portion , find N . .
一繩長 20 m,依 求 N 。
(iv)
N N
b 2 : b : b + 2 之比例分成三段。若最長一段為 N m m,
Each interior angle of an N -sided -sided regular polygon is x x . Find x . x . x。求 x。 正 N 邊形之每一內角為 邊形之每一內角為 x
P. 6
x