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A point is chosen randomly in a square. 4 lines are drawn to connect this point with the 4 vertices of this square, such that 4 triangles are formed. Find the probability that all interior angles of the 4 triangles are less than 120°. Give your answer correct to the nearest 3 decimal places.
Let’s assume the length of the square to be 1. The square is labelled as ABCD while the random point is called P as shown above.
Consider the locus formed by P such that ∠APD = 120°. This should be a circle because of converse of angles in the same segment. E is the center and ∠AED = 120° (angles at the centre is twice the angle at the circumference). If the random point falls in the shaded region, an interior angle will be larger than 120°, which violates the required criteria.
Combining the criteria produced from the 3 other sides of the square, we should be able to see that the required probability is the area of the center unshaded region divided by that of the whole square. Denote the area of the red and yellow shaded area as shown above as R and Y respectively. Then the required probability = 1 - 4Y + 4R.
We may carry out simulations for a large number of times and take the empirical probability as our answer. Note that we will never be able to obtain the exact answer in this approach, but this approach would work here since we were asked to give our answer correct to the nearest 3 decimal places only.
Psuedocode:
count = 0
Loop for 10,000,000 times
x = random(0 to 1)
y = random(0 to 1)
Use cosine theorem to check all 4 internal angles
If all angles < 120 then count++
End loop
Output count / 10,000,000
If time is allowed, the number of loops can be set to an even higher value so as to ensure accuracy.
Among 30 participants, there are 15 people who got the correct answers. Thank you for your participation!
SBD payout of the the question post = 37.299 SBD
Besides, 
The winners and prizes are tabulated below:
| Winner | Prize | SBD |
|---|---|---|
| @leedslemon | First prize | 37.299 / 8 = 4.662 SBD + 7.5 SP |
| @armandocat | Second prize | 37.299 / 8 = 4.662 SBD + 5 SP |
| @tvb | Third prize | 37.299 / 8 = 4.662 SBD + 2.5 SP |
| @tensaix2j | Consolation prize | 37.299 / 8 = 4.662 SBD |
| @ghasemkiani | Consolation prize | 37.299 / 8 = 4.662 SBD |
| @shieha | Consolation prize | 37.299 / 8 = 4.662 SBD |
| @vincentyip | Consolation prize | 37.299 / 8 = 4.662 SBD |
| @challk | Consolation prize | 37.299 / 8 = 4.662 SBD |
Congratulations to the winners!
The steemSTEM project (
Designed by
在一個正方形內隨機選取一點，並將此點與四個頂點連上直線，從而將正方形分割為四個三角形。求四個三角形之中所有內角均不超過 120° 的概率，答案準確至小數點後 3 位。
設正方形的長度為1，並標記為ABCD，而隨機點則稱為P，如上所示。
若固定∠APD= 120°，P形成的軌跡是一個圓（同弓形內的圓周角的逆定理）。E為圓的中心，可知∠AED= 120°（圓心角兩倍於圓周角）。 如果隨機點落在陰影區域，內角將大於120°，這違反了題目的要求。
結合正方形由其餘三面所產生的限制，我們可以知道題目所求的概率是中心非陰影區域的面積除以整個正方形的面積。 將紅色和黃色陰影區域的區域分別表示為 R 和 Y，那麼所需的概率 = 1 - 4 Y + 4 R。
我們可以進行電腦模擬，在正方形內隨機抽出一點，並計算是否符合題目的要求，最後將實驗概率作為我們的答案。請注意，此方法並不能獲得確切答案，但此題目只要求將答案準確至最接近的3位小數位，因此當運行次數足夠大，答案正確的機會率將會相當高。
偽代碼:
count = 0
Loop for 10,000,000 times
x = random(0 to 1)
y = random(0 to 1)
Use cosine theorem to check all 4 internal angles
If all angles < 120 then count++
End loop
Output count / 10,000,000
如果時間充足，則可將迴圈次數設置為更高的值，以確保準確性。
在30個參加者之中，有15人答對。多謝大家的熱烈參與！
比賽題目帖文的SBD收入 = 37.299 SBD
另外，
下表顯示得獎者及其所得獎金：
| 得獎者 | 獎項 | SBD |
|---|---|---|
| @leedslemon | 一等獎 | 37.299 / 8 = 4.662 SBD + 7.5 SP |
| @armandocat | 二等獎 | 37.299 / 8 = 4.662 SBD + 5 SP |
| @tvb | 三等獎 | 37.299 / 8 = 4.662 SBD + 2.5 SP |
| @tensaix2j | 安慰獎 | 37.299 / 8 = 4.662 SBD |
| @ghasemkiani | 安慰獎 | 37.299 / 8 = 4.662 SBD |
| @shieha | 安慰獎 | 37.299 / 8 = 4.662 SBD |
| @vincentyip | 安慰獎 | 37.299 / 8 = 4.662 SBD |
| @challk | 安慰獎 | 37.299 / 8 = 4.662 SBD |
恭喜所有得獎者！
steemSTEM（