[VA] SRC #008 - 2021 is here !

[Valentin Albillo]

      
... and what better way to welcome 2021 than a little 2021-themed SRC ?

Welcome to my SRC #008 - 2021 is here !, a small nice one to commemorate 2021's arrival. See if you can deliver using your HP calculator, physical or emulated.

• Please NO Lua, NO Python, NO Excel, NO Mathematica, etc. This is the Museum of HP Calculators, so if you can't or won't use an HP calculator please post your results somewhere else, there are plenty of threads suitable for you to show off your non-HP achievements. Thank you.

That said, try your HP-hand with these three:

1) Let's partition 2021 into a set of positive integer numbers that add up to 2021. Find the set of such numbers whose product is maximum, and output that maximum in all its full glory.

For instance, we could have 2021 = 1 + 1 + ... + 1 (2021 1's) and their product would be 1 x 1 x ... x 1 = 1, which doesn't quite cut it. We could also have 2021 = 137 + 682 + 1202 and their product would be 137 x 682 x 1202 = 112307668, which is much better but still far from the maximum as well.


2) Numerically evaluate as accurately as possible this nice definite integral using your favorite HP calc (per standard notation, the | ... | vertical bars mean "absolute value"):



and for extra points, see if you can symbolically recognize the resulting value. You'll need a sufficiently accurate value to do it, though ...


3) It is trivially easy to express the fraction 4/2021 as the sum of 4 numbers of the form 1/N where N is a positive integer, namely:

            4/2021   =   1/2021 + 1/2021 + 1/2021 + 1/2021

See if you can do it with just 3 such numbers.


My own solutions and comments in a few days.

• P.S.: If you manage to solve them, do not spoil the fun for others: include your solutions in a CODE block leaving a number of blank lines at the beginning so that they can't be seen without willingly scrolling.

V.

P.S.: Edited to include a third case, see post #21 below.