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In triangle ABC, there exists a point K inside of it such that
∠ABK=∠BCK=∠CAK=30∘.
Is it necessarily true that triangle ABC is equilateral?
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The sequence (−1),(−1)(−1)1,(−1)(−1)(−1)11,… clearly converges to the integer −1.
The sequence (4),(4)(4)1,(4)(4)(4)11,… converges to the integer 2. (Can you prove it?)
Does there exist another value b=1,−1,4 such that the sequence {an} defined recursively by a0=b;an+1=ban1 also converges to an integer?
Note: Written out, this sequence is (b),(b)(b)1,(b)(b)(b)11,….
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Define a positive integer n to be totatively prime if the set of all positive integers less than n that are relatively prime to n contains no composite numbers. What is the largest totatively prime number?
For example, 9 is not totatively prime because 4 is less than 9 and is relatively prime to 9, but 4 is composite.
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Find the largest possible n such that ⌊1⌋+⌊2⌋+⌊3⌋+⋯+⌊n⌋is a prime number.
Clarification: ⌊x⌋ returns the largest integer less than or equal to x.
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Alice and Bob are having fun throwing a ball to each other on a merry-go-round. Charlie looks at the game from outside of the merry-go-round. From his perspective, the ball thrown by Alice flies straight along the y-axis at a constant velocity of v=v0ey to Bob. Bob can catch this ball after the flight time t0=T/4, because the merry-go-round has completed a quarter turn in the meantime.
But how does Alice observe (rotating reference system)? What average speed (average of absolute value of velocity vector) v′=t01∫0t0∣v′∣dt does the ball have from Alice's perspective?
Give the answer in units of v0 and with an accuracy of 3 decimal places.
Hints:
Bonus question: Which (fictitious) forces act on the ball in the rotating reference system? How can we explain the path of the ball?
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