PLK competition selection quiz

已知道一個三角形其中兩邊的長度分別是12和35. 當三角形的面積爲最大時，求最後一條邊的長度。It is known that 2 of the sides of a triangle are of length 12 and 35 respectively. What is the length of the remaining side when the area of the triangle is the largest?

(a)  

一個三位數[abc] 滿足以下的條件：[abc] + 1000 是個平方數。如果 b = 2a, 求[abc]. （注：如果[xyz] = 135, 那麽 x=1, y=3, 和 z=5.）

A 3-digit number [abc] satisfies the condition where [abc] + 1000 is a square number. If b = 2a, find [abc]. (note: if [xyz] = 135, then x=1, y=3, and z=5.)

(a)  

已知道一個三角形其中兩邊的長度分別是12和35. 如果最後一條邊的長是整數，求周界的最小數值。

It is known that the length for 2 sides of a triangle are 35 and 12 respectively. If the length of the last side is an integer, find the smallest value of the perimeter.

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參考附圖。已知 CD:DA= 6:5, AF:FB = 2:3. 求CG:GF.

Refer to the figure. It is known that CD:DA = 6:5 and AF:FB = 2:3. Find CG:GF.

(a)  

某邊長為整數的正方形，它的面積為一個由兩個不同數字組成的四位數。求所有這種正方形的邊長。

For a square with integer side length, the value of the area is a 4-digit number composed of only 2 numbers. Find the number of squares satisfying this condition.

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4個人，A, B, C 和 D，傳球並必須在回合中把球傳給另一個人。已知在開始時A拿著球。經過5次傳球後，問有多少種傳球的路徑，使球回到 A 手中？

4 players, A, B, C and D, pass a ball, and they must pass the ball to another player in a round. It is known that initially A has the ball. How many different ways are there for the ball to return to A after 5 rounds of passing?

(a)