Rita the dog

Find the smallest positive real number x such that

sin(355x/113) - sin(πx) = 1

Your answer should be accurate at least to 5 places after the decimal point.

Note: I have an answer to this question, but I would like to have it corroborated.

Your criteria doesn't [sic] match any questions

when I click on the "Mathematics" section. Is this a general problem with Y!A or is there something wrong because of my location or something else specific to me?

The series starts:

1 - 1/2 - 1/3 + 1/4 - 1/5 + 1/6 - 1/7 - 1/8 + 1/9 + 1/10 - 1/11 - 1/12 - 1/13 + 1/14 + ...

The sign in front of 1/n is negative if the prime factorization of n has an odd number of factors (counted with multiplicity) and is positive otherwise. For example 12 = 2 * 2 * 3, so the prime factorization of 12 has 3 factors, so it is -1/12 in the sum.

Please provide proof that the infinite sum is zero, or explain why it is not zero.

What is the equation of the tangent line to the curve x^4 + xy + y^2 + x = 7 that is tangent to the curve at a point in the first quadrant and is also tangent to the curve at a point in the second quadrant?

[this question is motivated by a question of Andrew]

Show that

π/2 = arcsin(12/37) + arcsin(57/185) + arcsin(3232/10457) + arcsin(7735/25033) + arcsin(385468067/1308850405)

exactly.

2π =

arctan(35 / 12) + arctan(176 / 57) + arctan(9945 / 3232) + arctan(23808 / 7735) + arctan(1250801244 / 385468067)

Pythagorean triangle: a right triangle with integer side lengths.

Golden triangle: a right triangle whose legs are in the golden ratio.

Golden ratio: (1+√5)/2 : 1

There is no golden Pythagorean triangle because the golden ratio is irrational.

However the right triangle (a,b,c) = (262353, 424496, 499025) is almost golden because b:a = 1.6180337:1 is almost the golden ratio.

Can you find a more golden right triangle than the one just given?

Either prove or give a counterexample.

For non-negative integers n define

a(0) = -1, a(1) = 1, a(n) = 4a(n-1) - a(n-2) and

b(0) = 1, b(1) = 1, b(n) = 4b(n-1) - b(n-2).

Each positive integer n, let T(n) be the tetrahedron whose base is the equilateral triangle each of whose sides is b(n), and whose other three edges each equal a(n). Show that each of these tetrahedra has the same length of altitude to its base.

[note -- I deleted an earlier version of this question, which was misstated]

For each positive integer n = 1, 2, 3, ..., 1000 the gcd of n^2+1000 and and (n+1)^2+1000 is 1. Is this always true?

[Note: I know the answer, but I was surprised. This is based on a similar question someone else asked. Sorry, I don't have the link.]

I know the answer, but I just thought it was a cute problem that some others might like, and perhaps the answer may surprise some.