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13 votes

How can I learn how to solve hard problems like this Example?

Technique : rationalization When we are given $A/B$ or $1/B$ , we try to eliminate the Denominator. That will involve "conjugates" , that is , When Denominator is $x+\sqrt{y}$ , we could ...
Prem's user avatar
  • 12.1k
7 votes

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

You are estimating a little too roughly. Instead, you can divide your summands into three rough groups as @peterwhy suggests. Firstly, for $i \leq 10$ $$\frac{1}{10+i} \geq \frac{1}{20}$$ for $10 \leq ...
Noctis's user avatar
  • 214
5 votes

How can I learn how to solve hard problems like this Example?

And guess what, these problems are hard for me. Should I spend more time on hard ones before giving up? Should I memorise solutions by heart to get techniques? What could I do if I get stuck? I am ...
preferred_anon's user avatar
4 votes

How can I learn how to solve hard problems like this Example?

Let's start with some tough love: If you gave up after half an hour, you didn't really try. Prem and others have already posted some concrete tips on how this particular problem could be approached, ...
Einar Rødland's user avatar
4 votes

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

I suppose that a trick would be to use $$\log \left(1-2 x \cos (\phi)+x^2\right)=\sum_{n=1}^\infty \frac{ \cos (n \phi )}{n}\, x^n$$ Using Euler representation of the cosine function $$J_n=\Re\Big(\...
Claude Leibovici's user avatar
4 votes

Two numbers written on a board get replaced

The problem with considering the sum, is that this can either increase or decrease with each turn. If $a<b$, then the sum increases. If $a>b$, then the sum decreases. And so it is unclear ...
Adam Dougall's user avatar
3 votes

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

Let $f(x)=\dfrac1{10+\left \lceil x \right\rceil}$. We know $$\sum_{n=1}^{30}\frac1{10+n}=\int_0^{30}f(x) \text {d}x$$ For all $x$ in $[0,30]$, $f(x)=\dfrac1{10+\left \lceil x \right\rceil}>\dfrac1{...
Lucenaposition's user avatar
2 votes

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Restatement of the problem. Firstly, the logarithm can be split thanks to the given factorization, namely $(1 - 2x\cos\phi + x^2) = (1-xe^{i\phi})(1-xe^{-i\phi})$. Secondly, the factor $\phi^2$ can be ...
Abezhiko's user avatar
  • 10.2k
2 votes

When does $n$ divide $u_n$ if $u_1=1$, $u_n=(n-1)u_{n-1}+1$?

Not an answer but too long for a comment, this is an explanation for why, as @ChrisLewis pointed out, $$n\mathrel{|}u_n \iff n\mathrel{|}\sum_{k=0}^{n-1}(-1)^kk!$$ We have $u_n=(n-1)u_{n-1}+1$, ...
Zima's user avatar
  • 3,392
2 votes

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

${\displaystyle\sum_{i=1}^{30} \frac{1}{10+i}\\=\displaystyle\sum_{i=1}^{10} \frac{1}{10+i}+\displaystyle\sum_{i=11}^{20} \frac{1}{10+i}+\displaystyle\sum_{i=21}^{30} \frac{1}{10+i}\\>10\times \...
ican's user avatar
  • 77
1 vote

What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2?

COMMENT.-$$\sqrt{10}x^{\log(x)}=x^2\iff x^{\frac{\log(\sqrt{10})}{\log(x)}}\cdot x^{\log(x)}=x^2$$ equating exponents we get $$\frac{\log(\sqrt{10})}{\log(x)}+\log(x)=2$$ so the quadratic equation $$X^...
Piquito's user avatar
  • 30.2k
1 vote

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

The most likely intended method is what's suggested in peterwhy's comment, plus is also implemented in Noctis's answer and ican's answer, i.e., grouping the values into $3$ sets of $10$ consecutive ...
John Omielan's user avatar
  • 49.5k
1 vote

Finding the circumradius of a cyclic hexagon, given three non-consecutive sides and the fact that the midpoints of all sides are also cyclic

I am trying to solve the problem using the geometric intuition as much as possible, but at some point, algebra must come in. (A purely algebraic proof can also be given, writing equations for the ...
dan_fulea's user avatar
  • 34.2k
1 vote

Prove $\sum\limits_{\mathrm{cyc}} \sqrt{a+b} \ge \sum\limits_{\mathrm{cyc}} \sqrt{a + bc}$ for $ab + bc + ca + abc =4$

1. Restatement of Problem to Reference Its Equations Problem: Given non-negative real numbers $a,b,c$ satisfying $$ab+bc+ca+abc=4.\tag{Eq. 1.1} $$ Prove that $$\color{blue}{\sqrt{a+b}+\sqrt{b+c}+\...
Stephen Elliott's user avatar

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