Alan

I think you’ve missed quite a few!  What about 19635 or 87132 for example?

Notice that the integrand is an odd function i.e. if f(x) = sin(sinh(x)) then f(-x) = -f(x).  This means the integral of f(x) from -1 to +1 is zero. 

"Eric warms up 1kg milk in a pan on a gas stove from 20 °C to 80 °C. Of the energy the gas supplies, 50% is lost and 20% is absorbed by the pan. The remaining energy goes to the milk. The formula for the specific heat of the milk is 3,9 J/g · °C. The heat of combustion of the gas is 32 MJ/m3. How many cubic meters of gas has been burnt by Eric?"

This should help answer question 2:

I think there are 502 that satisfy the condition.

Only the odd numbers provide a₂ greater than a₁.  There are 1004 of them. 

All these a₂'s are even, so a₃ = a₂/2 = (3a₁ + 1)/2 which is greater than a₁.

The corresponding a₄'s must alternate even and odd.  The even ones will be a₃/2 = (3a₁ + 1)/4 which must be less than or equal to a₁ (equality when a₁ = 1).  The odd ones must be   3(3a₁ + 1)/2 + 1 = 9a₁/2 + 5/2 > a₁.  There are 1004/2 = 502 of these. 

As follows:

Different approach, same result:

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Since A is 90°, then cosA = 0.

Are you sure this is the question you meant to ask?

"Suppose a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,where a,b,c are integers whose greatest common divisor is 1. Determine |c|."

Here is the first equation:

I'll leave you to use the same technique on the second equation.