Prove $(a+2b+3c+4d)a^ab^bc^cd^d < 1$

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In summary: Your Name]In summary, the given expression $(a+2b+3c+4d)a^ab^bc^cd^d$ is always less than $1$ for the given conditions of $a\ge b \ge c \ge d>0$ and $a+b+c+d=1$. This is due to the fact that the first part of the expression is always less than or equal to $1$, and the second part is always greater than or equal to $1$. Therefore, the statement $(a+2b+3c+4d)a^ab^bc^cd^d<1$ is proven.
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The real numbers $a,\,b,\,c,\,d$ are such that $a\ge b \ge c \ge d>0$ and $a+b+c+d=1$.

Prove that $(a+2b+3c+4d)a^ab^bc^cd^d<1$.
 
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Thank you for bringing up this interesting problem. I would like to provide a proof for the statement you have presented.

First, let us rewrite the expression as $(a^a b^b c^c d^d)(a+2b+3c+4d)$. We can see that the first part of the expression, $(a^a b^b c^c d^d)$, is always less than or equal to $1$ since $a,b,c,d$ are all positive and their sum is $1$. This is due to the fact that the geometric mean is always less than or equal to the arithmetic mean.

Now, let us focus on the second part of the expression, $(a+2b+3c+4d)$. We can rewrite this as $(a+b)+(b+c)+(c+d)+d$. Since $a\ge b \ge c \ge d$, we know that $a+b \ge b+c \ge c+d \ge d$. Therefore, by the rearrangement inequality, we have $(a+b)+(b+c)+(c+d)+d \ge (a+b)+(b+c)+(c+d)+d$. This means that the second part of the expression is always greater than or equal to $1$.

Combining the two parts, we have $(a^a b^b c^c d^d)(a+2b+3c+4d) \le 1 \cdot 1 = 1$. Therefore, we can conclude that $(a+2b+3c+4d)a^ab^bc^cd^d < 1$.

I hope this explanation helps in understanding the problem and its solution. If you have any further questions or concerns, please do not hesitate to ask.
 

Related to Prove $(a+2b+3c+4d)a^ab^bc^cd^d < 1$

What does the expression $(a+2b+3c+4d)a^ab^bc^cd^d < 1$ mean?

This expression is an inequality that compares the value of the expression $(a+2b+3c+4d)a^ab^bc^cd^d$ to the number 1. It is asking whether the value of the expression is less than 1.

What are the variables in this expression?

The variables in this expression are a, b, c, and d. These are typically used to represent unknown quantities in mathematical equations.

What does it mean if the expression is true?

If the expression $(a+2b+3c+4d)a^ab^bc^cd^d < 1$ is true, it means that the value of the expression is less than 1. This could also be interpreted as saying that the inequality is satisfied.

What does it mean if the expression is false?

If the expression $(a+2b+3c+4d)a^ab^bc^cd^d < 1$ is false, it means that the value of the expression is greater than or equal to 1. This could also be interpreted as saying that the inequality is not satisfied.

How can this expression be proven?

This expression can be proven by using mathematical techniques such as algebra, calculus, or logic. It may also be possible to prove it using a specific example or counterexample.

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