Find $|f(0)|$.]]>

Prove that $\angle ABP=\angle ACP$.]]>

$a+b+c+d=4\\a^2+b^2+c^2+d^2=6\\a^3+b^3+c^3+d^3=\dfrac{94}{9}$

in $[0, 2]$.]]>

Evaluate $(a+c)(b+c)(a-d)(b-d)$.]]>

$x^3+3xy^2=-49\\x^2-8xy+y^2=8y-17x$]]>

Find the angle $\angle ADC$.]]>

This problem was posted on another site (I am paraphrasing):

A and B are initially 12 feet apart, and they wish to swap positions. A moves directly towards B's initial position without regard for social distancing, while B wishes to maintain a minimum distance of 6 ft. from A. Both move at the same constant speed/. What is the minimum distance B must travel?

I have worked on this for some 5 days now, to no avail. I have filled pages of calculations, all leading to a...

Social distancing problem]]>

where \(\displaystyle 0<k<5.\) Then value of \(\displaystyle k\) for which \(\displaystyle \displaystyle I_{k}\) is smallest.]]>

\[\int_{0}^{1}\frac{x^{n-1}+x^{n-\frac{1}{2}}-2x^{2n-1}}{1-x}dx = 2\ln2\]

- holds for any natural number $n$.]]>

\begin{array}{rcl}

x+y+z &=& 1\\

x^2+y^2+z^2 &=& 2\\

x^3+y^3+z^3 &=& 3 \\

x^5+y^5+z^5 &=& ?

\end{array}\right.$

How to find out $x^5+y^5+z^5=?$]]>

$\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^2}$

where a,b,c are positives]]>

A) primitive symbols : (1, *) and

B) The axioms:

1) \(\displaystyle \forall x\forall y[x*=y*\Longrightarrow x=y]\)

2) \(\displaystyle \forall x[x*\neq 1]\)

3) \(\displaystyle [P(1)\wedge\forall x(P(x)\Longrightarrow P(x*))]\Longrightarrow\forall xP(x)\)

Then prove:

\(\displaystyle \forall x[x=1\vee \exists y(y*=x)]\)]]>

\(\displaystyle \frac{AB}{MZ}+\frac{AC}{ME}+\frac{BC}{MD}\geq\frac{2t}{r}\)

Where t is half the perimeter of the triangle and r is the radius of the inscribed circle]]>

So if I go to town how much money do i have to spend

Whatever is your answer prove it]]>

for all ,x : if 0<x<π/2 and |x-a|<c ,then \(\displaystyle |(\sqrt sinx +1)^2-(\sqrt sin a +1)^2|<b\)]]>

for all x,y,a,b : if 0<x<1,0<y<1,0<a<1,0<b<1,and |x-a|<B,|y-b|<B,then \(\displaystyle |x^2y-a^2b|<A\)]]>

For instance 12 has the multiple 7776.]]>

For instance 13 has the multiple 111111.]]>

\[\tan^{-1}(k) = \sum_{n=0}^{k-1}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ),\;\;\;\;\; k \geq 1,\]

- and deduce that

\[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]]]>

Find the volume of the solid obtained by rotating $R$ around the line $y = x$.]]>

Maximize the sum of squared distances:

\[\sum_{1\leq i < j \leq 2n}\left | P_i-P_j \right |^2\]

- over alle possible choices of $2n$ points (centroid of the points is the origin)

Please

The winner(s) are the penny(s) which were flipped the most times. Prove that

the probability there is only one winner is at least $\frac{2}{3}$.]]>

\[x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x} = k\]

Find the possible values of $k$.

Source: Nordic Math. Contest]]>

\[S = \frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}\]

when $a,b,c$ and $d$ are arbitrary positive real numbers.]]>

\[\lim_{n\rightarrow \infty}\int_{0}^{\infty}\frac{e^{-x}\cos x}{\frac{1}{n}+nx^2}dx.\]]]>

Draw 2 Free body diagrams, one for each joints C and E.

Define what direction represents a positive force, and write equations for all forces acting on that joint.

For example, for joint D it's:

("SUM" is used for summation symbol)

SUMF

Static Analysis of Truss]]>

Constractive dilemma being the following propositional law:

From PvQ and P=>S and Q=>T we can infer SvT]]>

$$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$]]>

1. For each $\pi\in S_n$,

$$\sum_{i=1}^n\,(\pi(i)-i)\ =\ 0.$$

2. If

$$\sigma_\pi\ =\ \sum_{i=1}^n\,\left|\pi(i)-i\right|$$

for each $\pi\in S_n$, then $\sigma_\pi$ is an even number.

Bonus challenge: Find $\displaystyle\max_{\pi\in S_n}\,\sigma_\pi$.]]>

$$3\left(\frac1{\sqrt{a^3+1}}+\frac1{\sqrt{b^3+1}}+\frac1{\sqrt{c^3+1}}\right)\ \geqslant\ 2\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right).$$]]>

the area of the base. Find the ratio of the height to the length of the side of the base

of the pyramid.]]>

has square DEFG with sides = 60 inscribed in it.

Side DE of the square lies on the hypotenuse.

Find the triangle's side lengths.

Code:

```
B
E
D
F
C G A
```

12-7*6+19-14=11

Add brackets only (no limit) so above is correct.

I can't do it...]]>

$$f_n(x)=\sum_{k=0}^{2n}x^k.$$

Show, that $m_n\to\frac{1}{2}$ as $n \to \infty$.

Source: Nordic Math. Contest]]>

31 32 33 34, Baum Baxter Daly Neary) each decide to shoot a sticky

spitball at the wall across the beverage room.

To permit this (else this puzzle could not be devised and you'd all

be disappointed) 4 spitballs magically appear on their table:

a Blue SpitBall (BSB), a green one (GSB), a red one (RSB) and

a white one (WSB).

OK: the shooting's done; believe it or not, here's how the SB's stuck:

-Don's SB directly right of...

Sticky spitballs]]>

Find the probability that throughout this process the numbers on the balls which have been drawn is an interval of integers.

(That is, for $1 \leq k \leq n$, after the $k$th draw the smallest number drawn equals the largest drawn minus $k − 1$.)]]>

$$\frac{a^3-c^3}{3} \ge abc\left(\frac{a-b}{c}+\frac{b-c}{a}\right)$$

When does equality hold?

Source: Nordic Math. Contest]]>

Let $i_1,i_2, ... , i_n$ be a permutation of $1,2,...,n$.

Determine the smallest possible value of the sum:

$$\sum_{k=1}^{n}\frac{a_k}{a_{i_k}}$$]]>

$$\prod_{n=1}^{\infty}\left(1+10^{-2^n}\right)$$]]>

If the last three digits of $n$ are removed, $\sqrt[3]{n}$ remains.

Find with proof $n$.

Source: Nordic Math. Contest]]>

$$\sum_{j=0}^{n}(-1)^j{n \choose j}=0$$

- in two different ways]]>

Source: Nordic Math. Contest]]>

$$x^n+x^{-n}$$

is an integer for any integer $n$

Source: Nordic Math. Contest]]>

$$\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left(n\prod_{i=0}^{n}\frac{1}{j+i}\right)$$]]>

and Let $\displaystyle S = \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots \cdots +\frac{1}{a_{2008}}$. Then $\lfloor S \rfloor $ is]]>

$$\tan18^\circ\ =\ \sqrt{1-\dfrac2{\sqrt5}}.$$

No calculator, computer program, Excel, Google, or any other kind of cheating tool allowed.

Have fun!]]>

A so-called complicate integral has a such a simple closed form, quite amazed me, but how to prove it, is an other story.

$$\int_{0}^{1}\mathrm dt{t-3t^3+t^5\over 1+t^4+t^8}\cdot \ln(-\ln t) dt=\color{red}{{\pi\over 3\sqrt{3}}}\cdot \color{blue}{\ln 2\over 2}$$

Does anyone know to how prove this integral?]]>

\[\sqrt[3]{x}+\sqrt[3]{y} \leq \alpha \sqrt[3]{x+y}\]

- for all $x,y \in \mathbb{R}_+$]]>

|| = \/|

Move 1 match only such that both sides are equal...]]>

\[\frac{2x-1}{1-x+x^2}+\frac{4x^3-2x}{1-x^2+x^4}+\frac{8x^7-4x^3}{1-x^4+x^8}+ ...\]]]>

118=164

356=936

423=???

What does 423 equal?]]>

Let $E$ be the set of all real $4$-tuples $(a, b, c, d)$ such that if $x, y \in \mathbb{ R}$, then:

$(ax+by)^2+(cx+dy)^2 \le x^2+y^2$.

Find the volume of $E$ in $\mathbb{R}^4$.

$(ax+by)^2+(cx+dy)^2 \le x^2+y^2$.

Find the volume of $E$ in $\mathbb{R}^4$.

2ε{a,2,b} , where a,b are letters]]>

Prove the inequality:

$$\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})} \,dx \geq 1.$$]]>

\(\displaystyle (a+b)(b+c)=-1\\(a-b)^2+(a^2-b^2)^2=85\\(b-c)^2+(b^2-c^2)^2=75\)

Find \(\displaystyle (a-c)^2+(a^2-c^2)^2\).]]>

$$\int_{1}^{\infty}\left(\arcsin \left(\frac{1}{x}\right)-\frac{1}{x} \right)\,dx$$

of indeterminate form or not? Prove your statement.]]>

where $n_1,n_2$ and $n_3$ are different positive integers, that satisfy:

$\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3} < 1.$]]>

$a+b+c=ab+bc+ac=-\dfrac{1}{2}\\abc=\dfrac{1}{8}$

Evaluate $a^{\tiny\dfrac{1}{3}}+b^{\tiny\dfrac{1}{3}}+c^{\tiny\dfrac{1}{3}}$.]]>

$$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt=\color{green}{1\over \pi}\cdot\color{blue}{{\ln 2\over 2}(1-\ln 2)}$$]]>

This problem seems very tricky and you might think you need to expand one by one, but if you think carefully, you will find out that the answer is very simple!!

Solution:

[YOUTUBE]CnHBE4SbRRs[/YOUTUBE]

\(\displaystyle \log_7 10\) or \(\displaystyle \log_{11} 13\)]]>

======

The number-caller announced: under the G...n!

Gertrude, Josephine and Waltzing Mathilda all yelled "BINGO!".

Happens that all 3 filled the top line of their bingo cards.

The 15 numbers are all odd, plus numbers under each letter

are like this:

B: > 10 and none equal

I: < 20

N: > 40 and none equal

G: > 50 (and all equal!)

O: > 70 and none equal

I ask you: what is the n the number-caller announced with the G ?

You reply: can't tell; gimme a clue.

I then say...

Bingo!]]>