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2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, ...

What are the next five numbers, and why?

Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

After row 1, row n is obtained by inserting n between every pair of consecutive numbers that sum to n.

row 1:                  1 1
row 2:                 1 2 1
row 3:               1 3 2 3 1
row 4:            1 4 3  2  3 4 1
row 5:        1 5 4  3 5 2 5 3 4 5 1
row 6:   1  6  5 4  3 5  2  5  3  4 5  6 1
row 7: 1 7 6 5 4 7 3 5 7 2 7 5 3 7 4 5 6 7 1

The number of numbers in each row are 2, 3, 5, 7, 11, 13, 19...

Will they always be prime?

Consider two points on parabola y=x², (-a,a²) and (b,b²), where a and b are distinct real numbers.

If these two points are connected by a straight line, where does that line intersect the y-axis?

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Inspired by an interactive sculpture at the Museum of Mathematics, NYC.

Let ABC be a right-angled triangle with hypotenuse c = AB.
Let Wa and Wb be the lengths of the angle bisectors from A & B, respectively.

Prove that Wa + Wb ≤ 2c*sqrt( 2−√2).

Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Is 7013*2ⁿ+1 always composite?

Is 78557*2ⁿ+1 always composite?

After successfully solving an alphametics w1+w2=w3,where those three words spelled out the surnames of three famous Scottish writers WG sent the solution (216035+21754=237789) to his colleague, requesting him to reconstruct the original names and to mail his answer ASAP.
24 hours later a short message arrived: "A nice brush!"

Your comments are welcome...

"A B and A C are very rare".

Either decode the famous quotation or invent a meaningful one of your own.

A is an adjective, B and C are nouns.