# Tag Info

### If two pairs of numbers have equal sums and products, what can we conclude about the pairs?

We have $a$ and $b$ are the roots of $X^2-(a+b)X+ab$. We have $c$ and $d$ are the roots of $X^2-(c+d)X+cd$. But these quadratics are equal.
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### Solve $\int_0^\infty\frac{\ln(2e^x-1)}{e^x-1}dx$

Substitute $t=\frac{1}{2e^x-1}$ \begin{align} & \int_0^\infty\frac{\ln(2e^x-1)}{e^x-1}dx\\ =&\int_0^1 \frac{\ln t^2}{t^2-1}dt =\int_0^1 \int_0^\infty \frac{2y}{(1+y^2)(1+t^2y^2)}dy \ dt\\ =&...
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### Symmetry in Probability (AMC 12A 2023)

The idea in Solution 1, although not explicitly stated, is that if the frog has not reached or passed $10$, irrespective of how much further Flora needs to go, the probability that the next jump ...
• 141k
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### If two pairs of numbers have equal sums and products, what can we conclude about the pairs?

We can conclude that $\{c,d\}=\{a,b\}$. Indeed substituting $d=a+b-c$ into the $cd=ab$ one obtains for $c$ the equation: $$0=c^2-(a+b)c+ab=(c-a)(c-b).$$
• 26.7k
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### Finding the real roots of an octic polynomial (degree eight).

You're right: the polynomial initially resists simple attempts at finding its real roots through elementary methods like RRT (as a side note; you really only needed to try out the negative factors of ...
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### How many $3$-element subsets of $\{1,2,3,...,19,20\}$ have product divisible by $4$?

We use complementary counting. For a set to have the product of its elements not divisible by $4$, there are two cases: All elements are odd. There are $10$ odd numbers, so there are $\binom{10}{3}$ ...
• 2,882
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### Show that $x<\dfrac{\sqrt [8]{128}}{2}$

collect $u = x^2 + 1$ as $\; \; \; x^{15} u^2 - u + x = 0$ so $$x^2 + 1 = \frac{1 \pm \sqrt{1 - 4 x^{16} \;}}{2 x^{15} }$$ and you cannot have $4 x^{16} > 1.$ Also $4 x^{16} = 1$ ...
• 141k
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### $e^{e^x} = 10^{10} x^{10}e^{10^{10} x^{10}}$ - Find $x$ (G.H. Hardy)

Hardy addresses the problem in this 1910 note, and I'll summarize his method. Taking the natural logarithm of both sides gives $$e^x = 10 \log 10 + 10 \log x + 10^{10} x^{10} ,$$ so we may as well ...
• 103k
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$$\color{red}{x\sqrt{x}=\frac{4}{x^{2}}}\to \color{purple}{\sqrt{x}=\frac{4}{x}}$$ I think you can understand your mistake.

The answer by MathStackexchangeisNotSoBad identifies the error in the method that you have used. Here is a simpler alternative approach to the one you have taken which uses the rules of exponentiation ...
• 4,351
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### MIT Bee Qualifiying Exam 2024, Question $20$: $\displaystyle \int _1^3 \dfrac{x+\dfrac{x+\dots}{1+\dots}}{1+\dfrac{x+\dots}{1+\dots}}\,dx$

Let $$\dfrac{x+\dfrac{x+\dots}{1+\dots}}{1+\dfrac{x+\dots}{1+\dots}}=y$$ $$\implies \frac{x+y}{1+y}=y$$ or it means that $$y=\sqrt{x}$$ we rejected the negative one cause $y$ can never be negative. ...
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• 16.9k
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### On a distant island, there is a class of 200 students; 33 are boys and 101 are girls. In what numerical system are they calculating?

The answer is base $4$, that is, assuming every student has to be a boy or a girl. When we write something like $123$ in base $b$, what we really mean is $1\cdot b^2+2\cdot b+3\cdot1$. So this ...
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