The value of the expression [9]+[11] in

a2linetagadaW
2021-04-25
Answered

The value of the expression [9]+[11] in

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ensojadasH

Answered 2021-04-27
Author has **100** answers

Definition used:

Congruence.

Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x — y is divisible by m. It can be represented as$x=y\pm odm$ . This relation is called as congruence modulo m.

Calculation:

It is known that, [x] + [y] = [x+y].

Then, the value of the expression becomes as follows.

[9] + [11] = [9 + 11] = [20]

Since$20\equiv 5\pm od15$ . Thus [9] + [11] = [5] in $Z}_{15$ .

Therefore, the value of the expression is [5].

Congruence.

Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x — y is divisible by m. It can be represented as

Calculation:

It is known that, [x] + [y] = [x+y].

Then, the value of the expression becomes as follows.

[9] + [11] = [9 + 11] = [20]

Since

Therefore, the value of the expression is [5].

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For $1\le i\le 215$ let ${a}_{i}={\displaystyle \frac{1}{{2}^{i}}}$ and ${a}_{216}={\displaystyle \frac{1}{{2}^{215}}}$. Let ${x}_{1},{x}_{2},\dots ,{x}_{216}$ be positive real numbers such that $\sum _{i=1}^{216}{x}_{i}=1$ and

$\sum _{1\le i<j\le 216}{x}_{i}{x}_{j}={\displaystyle \frac{107}{215}}+\sum _{i=1}^{216}{\displaystyle \frac{{a}_{i}{x}_{i}^{2}}{2(1-{a}_{i})}}.$

Find the maximum possible value of ${x}_{2}.$

I simplified the condition to $\sum _{i=1}^{216}{\displaystyle \frac{{x}_{i}^{2}}{1-{a}_{i}}}={\displaystyle \frac{1}{215}},$ but I'm not sure what to do next.

$\sum _{1\le i<j\le 216}{x}_{i}{x}_{j}={\displaystyle \frac{107}{215}}+\sum _{i=1}^{216}{\displaystyle \frac{{a}_{i}{x}_{i}^{2}}{2(1-{a}_{i})}}.$

Find the maximum possible value of ${x}_{2}.$

I simplified the condition to $\sum _{i=1}^{216}{\displaystyle \frac{{x}_{i}^{2}}{1-{a}_{i}}}={\displaystyle \frac{1}{215}},$ but I'm not sure what to do next.