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In triangle $ABC,$ there exists a point $K$ inside of it such that
$\angle{ABK}= \angle{BCK}= \angle{CAK}=30^\circ.$
Is it necessarily true that triangle $ABC$ is equilateral?
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The sequence $(-1), {\large (-1)^ \frac{1}{(-1)} }, {\Large (-1) ^ \frac{1}{(-1)^ \frac{1}{(-1)}} }, \ldots$ clearly converges to the integer $-1.$
The sequence $(4), {\large (4)^ \frac{1}{(4)} }, {\Large (4) ^ \frac{1}{(4)^ \frac{1}{(4)}} }, \ldots$ converges to the integer 2. (Can you prove it?)
Does there exist another value $b \neq 1, -1, 4$ such that the sequence $\{a_n\}$ defined recursively by $a_0 = b; \quad a_{n+1} = {\large b^{\frac{1}{a_n}} }$ also converges to an integer?
Note: Written out, this sequence is $(b), {\large (b)^ \frac{1}{(b)} }, {\Large (b) ^ \frac{1}{(b)^ \frac{1}{(b)}} }, \ldots.$
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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 $\big\lfloor \sqrt{1} \big\rfloor + \big\lfloor \sqrt{2} \big\rfloor + \big\lfloor \sqrt{3} \big\rfloor + \cdots + \big\lfloor \sqrt{n} \big\rfloor$is a prime number.
Clarification: $\lfloor x \rfloor$ 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 $\vec v = v_0 \vec e_y$ to Bob. Bob can catch this ball after the flight time $t_0 = T/4,$ 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) $\overline{v'} = \frac{1}{t_0} \int_0^{t_0} |\vec v\,'| dt$ does the ball have from Alice's perspective?
Give the answer in units of $v_0$ 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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