Can be solved by most 13-year-olds in China‘-2

Geometry Level 3

Sorry that the question seems confusing at first, since brilliant glitched and didn’t save my edits :/


Whole set
Carefully read assumptions before answering \small \color{#D61F06} \text{Carefully read assumptions before answering}
As in the figure, there are rays O A , O B , O C OA,OB,OC . B O C = 5 0 \angle BOC=50^\circ . Let O A OA be a rod which overlaps ray O C OC . As ray O A OA rotates counter-clockwise around point O O at a speed of 1 0 10^\circ per second, there are moments when one of O A , O B , O C OA,OB,OC bisects the angle formed by the other two.
It rotates COUNTER-CLOCKWISE :) It rotates COUNTER-CLOCKWISE :)

Denote time as t t in seconds. 0 t 36 0\le t\le 36 . If above scenario happens at t = 1 , t = 2 , t = 3 t=1,t=2,t=3 , submit your answer as the product of all possible values of t t , in this example submit 1 × 2 × 3 = 6 1\times 2\times 3=6 .

Assumptions:

  • There is no angle greater than 18 0 180^\circ or less than 0 0^\circ . No angle equals 18 0 180^\circ or 0 0^\circ .
    • This must be explained further:
    • Here O C OC bisects A O B \angle AOB while O D OD is not considered as a bisector at all.
  • Angle bisectors are rays. Rays, straight lines and line segments are to be STRICTLY DISTINGUISHED FROM ONE ANOTHER.


The answer is 775.

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1 solution

Saya Suka
Mar 14, 2021

OC bisects at 50° / 2 = 25°, then OB bisects at 50° × 2 = 100° and finally OA bisects at 360° - 50° = 310°.

Answer
= [25° / (10° per second)] × [100° / (10° per second)] × [310° / (10° per second)]
= 2.5 × 10 × 31
= 775


Thanks for posting a solution! :)

Jeff Giff - 2 months, 4 weeks ago

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Hahaha, sorry for the mistakes. Edited all the pluses (+) and changed them into multiplication signs (×). 😋 🤭🙏

Saya Suka - 2 months, 4 weeks ago

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