2017-02-13 Intermediate Practice Problems Online

What mathematical fact is demonstrated above?

\sum_{n=1}^\infty \frac{1}{2^n} = 2

\sum_{n=1}^\infty \frac{1}{n} < 2

\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}

\sum_{n=1}^\infty \frac{1}{n^2} < 2

$\begin{array} { c c c c c } \square & + & \square & = & \square \\ \square & + & \square & = & \square \\ \square & + & \square & = & \square \\ \square & + & \square & = & \square \\ \square & + & \square & = & \square \\ \square & + & \square & = & \square \\ \end{array}$ Is it possible to fill the above squares with all the integers from 1 to 18 (using each number only once), such that each equation holds true?

For example, if we had only 4 rows and 12 numbers from 1 to 12 available, we could do the following:
$\begin{array} { c c c c c } 1 &+& 6 & =& 7 \\ 2 &+& 10 & =& 12 \\ 3 &+& 8 & =& 11 \\ 4 &+& 5 & =& 9. \end{array}$

The above shows a square ABCD with side length 8.
M and N are the midpoints of sides AB and CD, respectively.
A circle is inscribed in between lines DM and NB and the bottom side of the square.

Find the radius of this circle to 3 decimal places.

We label 10 points on the circumference of a circle as $P_1, P_2, \ldots, P_{10}.$ Find the number of ways to connect the points in pairs with non-intersecting chords, with no points left unconnected.

Hint: When the number of points is 6, the number of ways is 5, as shown below.

A square wire loop having side length $a$ , mass $m$ , and resistance $R$ is moving along the positive $x$ -axis at a speed of $v_0$ . It enters a uniform, steady magnetic field $\vec B = B_0 \big(-\widehat k \big)$ at $t = 0$ seconds, as shown in the figure.

Find the total amount of heat loss in the resistance.

Details and Assumptions:

The magnitude of $v_0$ is sufficient that the loop comes out of the region of magnetic field with some speed.
Neglect any type of energy loss other than the heat loss in resistance of the wires of the square.
Take $v_0 = \dfrac{3B^2 a^3}{mR}$ .

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2017-02-13 Intermediate