Win At Mathematics

One of our user asked: I am a competition math student, and I was doing a practice test when I found this problem: "The fourth degree polynomial equation $x^4 - 7x^3 + 4x^2 + 7x - 4 = 0$ has four real roots a, b, c and d. What is the value of the sum $1/a + 1/b + 1/c + 1/d$ ? Express your answer as a common fraction." First of all, I don't think this polynomial can be factored, so I don't know how to find the reciprocals of the roots. I need help and any help would be appreciated! Note: When they say "Express your answer as a common fraction.", it means that the answer IS a fraction. I tried asking this question already, but I worded it differently, and they didn't get a fraction so I knew it was wrong. This is the link to my other question: Link

One of our user asked: I could do this on a calculator but I thought there would be a more efficient way to do this than punching on my calculator the following: $3.5 \cdot 1.03 + 3.5 \cdot 1.03^2 + 3.5 \cdot 1.03^3 + \ldots + 3.5 \cdot 1.03^{10}$ So basically, $xy + xy^2 + xy^3 + ... + xy^{10}$ How can I simplify this to make it easier to calculate on a calculator?

One of our user asked: Write down the cubic equation given that $\alpha + \beta + \gamma = 4$ , $\alpha^2 + \beta^2 + \gamma^2 = 66$ , and $\alpha^3 + \beta^3 + \gamma^3 = 280$ Ok so, the sum of roots is given and I'm able to use the sum of the roots and the sum of the roots squared to get the sum of the combination of roots, but I'm unable to get the product of roots, because I can't seem to manipulate the sum of the cubes of roots to resemble the sum of roots and sum of squares of roots.

One of our user asked: When I type this equation ( $6x + 25 = 7y$ ) into WolframAlpha, it is able to tell me that the integer solution for this equation is: $x = 7n + 4$ , $y = 6n + 7$ , where n in the set of all integers How can I arrive at this solution on my own?

One of our user asked: A man want 762 by a number, but he trade it and its quotient result 13 and its rest was 13. What the quotient and rest of the original number? $$ q = ?; r=? $$ $$ 762 = q \cdot \overline{ ab } + r $$ $$ 762 = 13 \cdot \overline {ba} + 21 $$ Another way to write this is: $$ \frac{762-r}{ \overline{ba}}=r $$ $$ \frac{762-21}{ \overline{ba}}=13 $$

One of our user asked: We can consider $\mathbb{R}^+$ with two operation: +: $\mathbb{R}^+ \times \mathbb{R}^+\to \mathbb{R}^+$ that maps $(a,b)$ to $a+b:=ab$ and $ \cdot : \mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}^+$ that maps $(\lambda,a)$ to $a^\lambda$ . With respect to these operation we have that $\mathbb{R}^+$ is a vector space. We want find an example of linear application from $\mathbb{R}$ to $\mathbb{R}^+$ . We can fix $a\in \mathbb{R}^+$ and we can define $f_a:\mathbb{R}\to \mathbb{R}^+$ such that maps every $\lambda$ to $ f_a(\lambda):=a^\lambda$ . From the rules of powers, the map $f_a$ is a linear map for every $a\in \mathbb{R}^+$ . A natural question can be if all linear maps from $\mathbb{R}$ to $\mathbb{R}^+$ are an exponential maps. The answer is positive because for each linear map $F: \mathbb{R}\to \mathbb{R}^+$ we have that $F(\lambda)=F(\lambda \cdot 1)=(F(1))^\lambda=f_{F(1)}(\lambda)$ so $F=f_{F(1)}$ We can also observe that the m...