Teorema Heron

Diketahui suatu segitiga mempunyai sisi-sisi sepanjang , dan maka luasnya adalah

 

ini disebut semiperimeter (setengah keliling).

Contoh 1 :
Diketahui segitiga siku-siku a = 3, b = 4, c = 5 cm
s = 1/2(3 + 4 + 5) = 6
Luas = V(6.3.2.1) = V(36) = 6

Dengan pythagoras :
Luas = 1/2(3 x 4) = 6

Contoh 2 :
Diketahui segitiga sama sisi a = b = c = 20 cm
s = 1/2(20 x 3) = 30
Luas = V(30.10.10.10) = V(30000) = 100V3

Dengan pythagoras :
t = V(20^2 - 10^2) = V300 = 10V3
Luas = 1/2(20 x 10V3) = 100V3

Elementary Mathematics International Contest (IMC 2008) Solutions

Solution 2 :
3x + 2y = 32, y = (32 - 3x) / 2
4x + 3y = 44
4x + 3(32 - 3x) /2 = 44
8x + 96 - 9x = 88
x = 8
y = (32 - 24) / 2 = 4
2x + y = 16 + 4 = 20

Solution 3 :
sit + stand = 100, stand = 100 - sit.
0.75 stand + 0.25 sit = 70
0.75(100 - sit) + 0.25 sit = 70
75 - 0.75 sit + 0.25 sit = 70
5 = 0.50 sit
sit = 10

Solution 4 :
(110000 - 100000) / 3600 x t = (17 + 3)
t = 20 x 3600 / 10000 =  7.2 seconds

Elementary Mathematics International Contest (IMC 2008)

1. Starting from the central circle, move between two tangent circles. What is the number of ways of covering four circles with the numbers 2, 0, 0 and 8 inside, in that order?

2. Each duck weighs the same, and each duckling weighs the same. If the total weight of 3 ducks and 2 ducklings is 32 kilograms, the total weight of 4 ducks and 3 ducklings is 44 kilograms, what is the total weight, in kilograms, of 2 ducks and 1 duckling?

3. If 25% of the people who were sitting stand up, and 25% of the people who werestanding sit down, then 70% of the people are standing. How many percent of the people were standing initially?

4. A sedan of length 3 metres is chasing a truck of length 17 metres. The sedan is
travelling at a constant speed of 110 kilometres per hour, while the truck is travelling at a constant speed of 100 kilometres per hour. From the moment when the front of the sedan is level with the back of the truck to the moment  when the front of the truck is level with the back of the sedan, how many seconds would it take?

5. Consider all six-digit numbers consisting of each of the digits ‘0’, ‘1’, ‘2’, ‘3’, ‘4’ and ‘5’ exactly once in some order. If they are arranged in ascending order, what is the 502^nd number?

6. How many seven-digit numbers are there in which every digit is ‘2’ or ‘3’, and
no two ‘3’s are adjacent?

7. How many five-digit multiples of 3 have at least one of its digits equal to ‘3’?

8. ABCD is a parallelogram. M is a point on AD such that AM=2MD, N is a point on AB such that AN=2NB. The segments BM and DN intersect at O. If the area of
ABCD is 60 cm2, what is the total area of triangles BON and DOM?

9. ABCD is a square of side length 4 cm. E is the midpoint of AD and F is the midpoint of BC. An arc with centre C and radius 4 cm cuts EF at G, and an arc with centre F and radius 2 cm cuts EF at H. The difference between the areas of the region bounded by GH and the arcs BG and BH and the region bounded by EG, DE and the arc DG is of the form m? ?n cm², where m and n are integers. What is the value of m+n?

10.  In a chess tournament, the number of boy participants is double the number of girl participants. Every two participants play exactly one game against each other. At the end of the tournament, no games were drawn. The ratio between the number of wins by the girls and the number of wins by the boys is 7:5. How many boys were there in the tournament?

11. In the puzzle every different symbol stands for a different digit.

What is the answer of this expression which is a five-digit number?

12. In the figure below, the positive numbers are arranged in the grid follow by the arrows’ direction.

For example,
“8”is placed in Row 2, Column 3.
“9” is placed in Row 3, Column 2.

Which Row and which Column that “2008” is placed?

13. As I arrived at home in the afternoon. The 24-hour digital clock shows the time as below (HH:MM:SS). I noticed instantly that the first three digits on the platform clock were the same as the last three, and in the same order. How many times in twenty four hours does this happen?

Note: The clock shows time from 00:00:00 to 23:59:59.