# Tag Info

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### An interesting trigonometric integral

\begin{align*} I&= \int_0^\pi \cos^2\cos x+\sin^2\sin x \, dx\\ &= 2\int_0^{\pi/2} \cos^2\cos x+\sin^2\sin x \, dx\\ &= 2\int_0^{\pi/2} \cos^2\sin x+\sin^2\cos x \, dx \quad(x\mapsto\pi/2-...
Accepted

Accepted

### Any different way to solve an integral involving a trigonometric ratio?

Decompose the integrand as follows \begin{align} & \int_0^{2\pi}\frac{\sin x+1}{\cos x +2}\ dx\\ =& \int_ 0^{2\pi}\left( \frac{\sin x}{\cos x +2}- \frac{\cos x+2-\sqrt3}{\sqrt3(\cos x +2)}+\...

### Show that the polynomial $P(x):=x^4-6x+6$ has no real roots .

We can use for example $$x^4-6x+6 =(x^2-1)^2+2x^2-6x+5 >0$$ indeed $(x^2-1)^2\ge 0$ and $$36-4\cdot 2\cdot 5 =-4<0$$
Accepted

### Australian Mathematics Competition Year 7-8 Question 21

The answer should be 2. Place a "1" in the center, then a "2" in one of the corners. The black numbers are given while the red numbers are deduced. It is also easy to see that ...

### An interesting trigonometric integral

Let's take a completely different approach. Set $$f(x)=\cos^2\cos x+\sin^2\sin x-1.$$ We will show that $f$ integrates to $0$ over $[0,\pi]$. Observe that \begin{align*} f\left(x+\frac{\pi}{2}\right) &...

### What is the least number of concerts needed to be scheduled in order that each musician may listen, as part of the audience, to every other musician?

Given a schedule with $n$ concerts, define $S_1,\dots,S_6$ to be subsets of $\{1,\dots,n\}$, where $S_i$ is corresponds to the set of concerts that musician number $i$ played in. For example, if $n=6$,...
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### Show that $x<\dfrac{\sqrt {128}}{2}$

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 $4 x^{16} = 1$ ...

### Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(3x)-f(x)=x.$ If $f(8)=7$, then $f(14)$ is equal to

$$f(x)-f\big(\frac{x}{3}\big)=\frac{x}{3}.$$ Similarly, $$f\big(\frac{x}{3^n}\big)-f\big(\frac{x}{3^{n+1}}\big)=\frac{x}{3^{n+1}}.$$ Let us add this up over all $n$. This gives $f(x)=f(0)+\frac{x}{2}$ ...
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### Number of parallelograms in a hexagon of equilateral triangles

You can convince yourself that: The two points at the acute angles of a parallelogram uniquely determine that parallelogram; Any two points that don't lie on a common line in the grid can be the two ...
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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 $\binom{10}{3}$ ...
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### Show that $\forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c}$
Here's one way to solve the problem while avoiding algebra. The key idea is that $\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} = 1$. Thus, we will substitute $x = \frac{a}{b}, y = \frac{b}{c},$ and ...
### Can we arrange $\{1,...,16\}$ in $4\times 4$-grid so {products of rows} = {products of columns}?
RobPratt has answered the question for all $n\le 13$. I shall prove in this answer that when $n\ge 11$, there is no solution. Let $\pi$ be the prime counting function. I claim is impossible to form a ...