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?

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 \( (b), {\large (b)^ \frac{1}{(b)} }, {\Large (b) ^ \frac{1}{(b)^ \frac{1}{(b)}} }, \ldots. \)

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.

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 \(x\).

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:**

- The merry-go-round rotates with constant frequency \(\omega = 2 \pi/T\), so that Alice \((\)point \(r_A)\) and Bob \((\)point \(r_B)\) move on circular paths in the stationary reference system. In the rotating reference system, both points \(r_A^\prime\) and \(r_B^\prime\) are stationary.
- Search for a \(2\times 2\) matrix \(\mathbf{D}\) such that the transformation reads \(\vec{r}^\prime = \mathbf{D} \cdot \vec{r}\). For the calculation of the average velocity, you may use the integral: \[ \int \sqrt{1 + x^2} d x = \frac{1}{2} \left( x \sqrt{1 + x^2} + \text{arcsinh}(x) \right).\]

**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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