Question

Consider the numbers 624 and 847.

Show that these numbers can be written as the sum of three squares OR show that they cannot be written as the sum of three squares.

Answer #1

Find three numbers whose sum is
3333
and whose sum of squares is a minimum.

Find three positive numbers whose sum is 12, and whose sum of
squares is as small as possible, (a) using Lagrange multipliers
b)using critical numbers and the second derivative test.

Three numbers x, y, and z that sum to 99 and also have their
squares sum to 99. By Lagrange method, find x, y, and z so that
their product is a minimum.

The product of two numbers is 1 and the sum of their squares is
2. Find the numbers.

1.) Suppose a pair of dice are rolled. Consider the sum of the
numbers on the top of the dice and find the probabilities. (Enter
the probabilities as fractions.)
(a) 5, given that the sum is odd
(b) odd, given that a 5 was rolled
2.) Suppose a pair of dice are rolled. Consider the sum of the
numbers on the top of the dice and find the probabilities. (Enter
the probabilities as fractions.)
(a) 8, given that exactly one...

Find a formula for the sum of the squares of the numbers 1, 2,
...., n and prove it by induction.

Three faces of a tetrahedron are perpendicular to each other.
Show that the sum of squares of the areas of these three faces is
equal to the area of the third face

The
sum of two positive numbers is 32. What js the smallest possible
value of the sum of their squares?

Consider the simple linear regression model for which the
population regression equation can be written in conventional
notation as: yi= Beta1(xi)+
Beta2(xi)(zi)2+ui
Derive the Ordinary Least Squares estimator (OLS) of beta
i.e(BETA)

Consider the following numbers 3, 6, 9, 12, . . . , 75. Show
that if we pick 15 arbitrary numbers from them, then we will find
two that have sum equal to 81. I understand that there are 12
distinct sets containing pairs that sum to 81 plus a singleton
subset {3}. but wouldn't this mean that there are 2 remaining
"empty holes" that need to be filled? Not sure how to apply the
pigeonhole principle here.

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