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2017-10-30 Advanced

         

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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