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2017-02-13 Intermediate

         

What mathematical fact is demonstrated above?

+=+=+=+=+=+= \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:
1+6=72+10=123+8=114+5=9. \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 P1,P2,,P10.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 aa, mass mm, and resistance RR is moving along the positive xx-axis at a speed of v0v_0. It enters a uniform, steady magnetic field B=B0(k^)\vec B = B_0 \big(-\widehat k \big) at t=0t = 0 seconds, as shown in the figure.

Find the total amount of heat loss in the resistance.

Details and Assumptions:

  • The magnitude of v0v_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 v0=3B2a3mR v_0 = \dfrac{3B^2 a^3}{mR} .

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