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2017-10-30 Advanced

         

In triangle ABC,ABC, there exists a point KK inside of it such that

ABK=BCK=CAK=30.\angle{ABK}= \angle{BCK}= \angle{CAK}=30^\circ.

Is it necessarily true that triangle ABCABC is equilateral?

The sequence (1),(1)1(1),(1)1(1)1(1), (-1), {\large (-1)^ \frac{1}{(-1)} }, {\Large (-1) ^ \frac{1}{(-1)^ \frac{1}{(-1)}} }, \ldots clearly converges to the integer 1.-1.

The sequence (4),(4)1(4),(4)1(4)1(4), (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 b1,1,4 b \neq 1, -1, 4 such that the sequence {an}\{a_n\} defined recursively by a0=b;an+1=b1ana_0 = b; \quad a_{n+1} = {\large b^{\frac{1}{a_n}} } also converges to an integer?

Note: Written out, this sequence is (b),(b)1(b),(b)1(b)1(b),. (b), {\large (b)^ \frac{1}{(b)} }, {\Large (b) ^ \frac{1}{(b)^ \frac{1}{(b)}} }, \ldots.

Define a positive integer nn to be totatively prime if the set of all positive integers less than nn that are relatively prime to nn 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 nn such that 1+2+3++n\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: x\lfloor x \rfloor returns the largest integer less than or equal to xx.

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 yy-axis at a constant velocity of v=v0ey\vec v = v_0 \vec e_y to Bob. Bob can catch this ball after the flight time t0=T/4,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) v=1t00t0vdt\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 v0v_0 and with an accuracy of 3 decimal places.


Hints:

  • The merry-go-round rotates with constant frequency ω=2π/T\omega = 2 \pi/T, so that Alice ((point rA)r_A) and Bob ((point rB)r_B) move on circular paths in the stationary reference system. In the rotating reference system, both points rAr_A^\prime and rBr_B^\prime are stationary.
  • Search for a 2×22\times 2 matrix D\mathbf{D} such that the transformation reads r=Dr\vec{r}^\prime = \mathbf{D} \cdot \vec{r}. For the calculation of the average velocity, you may use the integral: 1+x2dx=12(x1+x2+arcsinh(x)). \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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