Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

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1answer
96 views

How do pupils solve 2nd degree equations in Germany? (different from Spain)

I'm from Spain and in Spain the undergraduate pupils learn to solve a 2nd degree (i.e. quadratic) equation using the formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ but years ago I had a colleague who did ...
1
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2answers
31 views

Under what conditions would the function $\prod_{i=1}^{n}{\frac{r_i}{r_i - 1}}$ be decreasing with respect to $n$?

So I know that $$\frac{r}{r - 1}$$ is a decreasing function of $r$. My question is: Under what conditions would the following function be decreasing with respect to $n$? $$\displaystyle ...
1
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3answers
39 views

Problem leading simple equations

A sum of Rs. 8.85 is made up of 124 coins which are either 10 paisa coins or 5 paisa coins ; how many coins are there each Note : Rs. 1 = 100 paisas
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2answers
23 views

Problem leading equations

The question is : "A and B begin to play with 60$ each. If they play till A's money is double B's, what does A win?" Now i tried to solve it like they both have 60\$ each, then A got his money ...
1
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2answers
65 views

Defining the $L^2$ norm of a vector valued function

I am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space. I am defining the a norm of ...
0
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2answers
24 views

System of equations that I'm having trouble with

$a/(x+y) - b/(x-y) = 1$ $b/(x+y) + a/(x-y) = (b^2-a^2)/2ab$ The answer to the values of $x$ and $y$ are given as $x=a-b, y=a+b$. How is that achieved?
1
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0answers
23 views

General and sufficient condition of independence

I'm having troubles with this proof: Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean and unit standard deviation. For $(a_0, a_1, ..., a_r)$ a sequence of $r$ real numbers ...
1
vote
1answer
76 views

$\mathbb{A}^2\setminus (0,0)$ is not affine

I want to prove that $X = \mathbb{A}^2\setminus (0,0)$ is not affine. My attempt: If $\Bbbk[X] = \Bbbk[x,y]$ then $X$ is not affine since $(x,y) \subset \Bbbk[x,y]$ is a proper ideal, but $V(x,y) ...
4
votes
1answer
51 views

Inverse Fourier transform of $\frac{1-e^{-2\pi ift}}{2\pi if}$

I would like to calculate the inverse Fourier transform of the following $$H(f) = \frac{1-e^{-2\pi i f t}}{2\pi i f}$$ Can anyone tell me and explain to me how to do that? I don't want just an ...
2
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0answers
46 views

Reference request - Problem book by subject

I'm looking for good problem textbooks for self-study. I know only of two of this sort: "Introduction to Measure Theory" by Terry Tao, and "Problems in Algebraic Number Theory" by Esmonde and Murty. ...
1
vote
1answer
68 views

Strange sum of random variables

So guys, I'm having this hard proof to solve in probability. I don't really know how to tackle it! Hope that someone can help. Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean ...
3
votes
3answers
95 views

What is the value of $a^4+b^4+c^4$?

Consider $a,b,c$ such that $a+b+c =1, a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$. Find the value of $a^4+b^4+c^4$, if possible. Trial: I observe that \begin{align} a^4+b^4+c^4 ...
1
vote
1answer
14 views

Solving a matrix for color manipulation

I'm making an application that deals with color transforms. The idea is that if you give it an RGB color and apply a color matrix transform it outputs another color. In this case I'm giving the color ...
1
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2answers
50 views

Can I use eigenvalues to find the inverse of a vector?

I have two 1D matrices (say dimension 1xn) called A and B. Multiplying these: A . B = M. Where M is a scalar. Knowing B and M, can I find A? One cannot take the inverse of a vector, but is it ...
3
votes
4answers
418 views

SAT Maths Question About Fractions

Whilst revising, a problem caught my eye and I cannot seem to find an answer. I am usually bad at these types of questions. On a certain Russian-American committee, $\frac23$ of members are men, ...
1
vote
1answer
129 views

Kill the creeps with minimum cost

Oz plays popular ARTS Dota 2. Invoker is one of the favourite Oz's heroes. Oz's skills are not perfect yet, so he uses only two spells - SunStrike and Tornado. Each of these spells takes some mana ...
9
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2answers
158 views

Proving $\sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}$

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: $$ \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2} ...
2
votes
5answers
231 views

Help With SAT Maths Problem (Percentages and Numbers)

I usually solve SAT questions easily and fast, but this one got me thinking for several minutes and I cannot seem to find an answer. Here it is: In 1995, Diana read $10$ English and $7$ French ...
16
votes
2answers
174 views

$xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ implies $\int_0^1 f(x) \,dx\leq\frac{\pi}{4}$

I want to show that if $f\colon [0,1]\to\mathbb{R}$ is continuous and $xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ then we have the following inequality: $$\int_0^1 f(x) \, dx\leq\frac{\pi}{4}.$$ The ...
0
votes
0answers
31 views

Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are $3$ teams observing the movement of the birds. Team $1$ observes that on their first flight the birds move a short distance across a ...
4
votes
3answers
164 views

Numbers with 2015

I like to build math problems; to solve the one below I should first find a certain square and use it in my solution. I would want to know if anyone can solve this problem otherwise. Thanks. ...
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0answers
32 views

A little bit more difficult problem regarding rooted plane trees

A question regarding rooted plane trees bothers me. We know that the number of rooted plane trees with $n$ nodes equals to $n-{th}$ Catalan number, that is $|Tn| = Cn$. But what is this number if we ...
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3answers
46 views

Simplifying $\Big[\dfrac{5-\sqrt{a}}{5+\sqrt a}-\dfrac{\sqrt a+5}{\sqrt a-5}+2\Big]^{-2}$

Simplifying $$\Big[\dfrac{5-\sqrt{a}}{5+\sqrt a}-\dfrac{\sqrt a+5}{\sqrt a-5}+2\Big]^{-2}$$ When I try, the numerator cancels out to $0$, yet the answer sheet says $(25-a)^2/10000$. Where am I going ...
1
vote
1answer
26 views

Fitting the closest coefficients in a system of millions of simultaneous equations?

I don't really know the correct terminology to describe this, but let's say we have many values of $(x_n, y_n, z_n)$. Also let's say that our description of 'many' means that $i$ ranges from $1$ to ...
2
votes
4answers
77 views

Solving equations with $x^x$ on any given side [duplicate]

How would you solve such an equation if it's infeasible to just start trying different $x$ values? Example: $$x^x = 6.$$
1
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4answers
44 views

Inequality for sides and height of right angle triangle

Someone recently posed the question to me for the above, is c+h or a+b greater, without originally the x and y lengths. I used this method: (mainly pythagorus) $a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$ ...
1
vote
1answer
44 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...
3
votes
2answers
124 views

Solve first order nonlinear differential equations

I want to solve this nonlinear 1-st order ODE, $$\frac{1}{1+x}=(\frac{1}{x-y}-\frac{1}{y})\frac{dy}{dx}$$ I find it non-separable, and Wolfram Alpha does not give me a closed form solution, but the ...
2
votes
0answers
105 views

Cognitive processes involved solving IMO level problems [closed]

I am currently 16 years old and, though I'm obviously not as good as most of the people on this site, I have always been considerably better than most of my classmates in mathematics. This, of course, ...
1
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2answers
46 views

Finding the missing length

How do i find the ST?? What more information do I need? I used Pythagorean theorem, but I still can't find the answer.
0
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2answers
36 views

Traveling salesman problem (TSP): what is the Relation with number of vertices and length of the found route?

I know that there are many algorithms (exact or approximate) which implement the traveling salesman problem. I would like to know the relation between the number of the vertices (i.e., the places to ...
0
votes
1answer
24 views

Compute a basic Side of two rooms, given total Area and total perimeter

My Gf's professor asked her to solve this problem: Two square rooms have an area of $52m^2$. The two rooms have a perimeter of 40 meters. Given this, we need to compute the length of the side of the ...
-2
votes
1answer
37 views

How long will it take for one of them or both of them?

One knight can storm a castle in 15 days. He and his partner can do it in 10 days. How long does it take the partner to storm the same castle alone? Pipe A can fill a pool in 5 hours, while pipe B ...
0
votes
1answer
112 views

The Area of shaded region in a circle

I'm having trouble solving this problem. I can't solve this. I don't know where and how to start. I don't know there is any formula for finding the area for this kind of shape, and if it did, I ...
1
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0answers
28 views

Essential singularity of c.c.(z)

According to my lecture notes, $z^*$ has an essential singularity (asterix denotes complex conjugate). However, it is not explained why nor at which point. Can anyone elaborate where the singularity ...
1
vote
3answers
68 views

$f(x)$ is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2)$), given $f(2)=5.$

If f(x) is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2))$, given $f(2)=5.$ ATTEMPT:- $f(f(2))=f(5)$, We can find $f(0)$,$f(1)$ and $f(1/2)$ to be $1,2$ and $5/4$ ...
2
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0answers
14 views

Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
1
vote
1answer
45 views

Working with mathematical models, HELP.

I'm currently doing a lot of self study with mathematics. I live in The Netherlands and hope to be admitted to Leiden University somewhere in 2016. Now, I have encountered a problem in my workbook ...
1
vote
3answers
39 views

Find the interval on which $ x^{2} - \lfloor x \rfloor - 3 < 0 $ holds.

On what interval does the equation $ x^{2} - \lfloor x \rfloor - 3 < 0 $ hold? My attempt: I tried sketching the graph, but it’s a bit complicated. Is there any other approach?
3
votes
3answers
701 views

Result of solving an unsolved problem?

I was mulling over currently unsolved problems in mathematics (as I, and many others, find them quite interesting) and began to wonder what would happen if these problems were to be solved. I know ...
2
votes
2answers
402 views

Making change with prime-valued coins

Am I understanding this question correctly and how do I approach these problems? In Numberland, the unit of currency is the El (E). The value of each Numberlandian coin is a prime number of Els. So ...
2
votes
0answers
48 views

Is the matrix with these coefficients invertible?

Let $0 \leq x_{i-1} < x_i < x_{i+1} \leq 1$. Let $p, q$ be functions that depend on that such that $p$ is positive and $q$ is non-negative. Let $c_i = a_{i+1,i} = a_{i,i+1}$. Let all other ...
0
votes
1answer
80 views

How to solve problems in elementary number theory?

I have studied and solved almost all of Elementary Number Theory by David M. Burton. Yet, tough problems in NT from Olympiads seen unapproachable to me. What should I do? What should I study? I feel ...
1
vote
1answer
71 views

Is this special matrix invertible?

The symmetric, tridiagonal $n-$by$-n$ matrix with the elements $a_{ii+1} = a_{i+1i} $ and off-diagonals' absolute values equal to the diagonal (except for row 1 and row n) is invertible. The elements ...
3
votes
2answers
50 views

Find all solutions to $2x \equiv p \mod 3p$

Find all solutions to $2x \equiv p \pmod {3p}$. $p$ is prime, and $p > 3$. I found that this is equal to $2x = p(3k+ 1)$ for some $k \in \Bbb{N}$. Since $k$ can't be even, then we have $2x = ...
2
votes
0answers
44 views

On the second part of solution of a question due to Erdos

Problem. Let $a_1<a_2<\dotsb<a_n\le 2n$ be a sequence of positive integers. Then $$ \min [a_i,a_j]\le 6\left(\Big[\frac n2\Big]+1\right), $$ where $[a_i,a_j]$ denotes the least ...
2
votes
1answer
66 views

Find with proof all solutions to $2^n = a! + b!$, where $a$, $b$, $n$ are positive integers and $a \leq b$.

So far I have looked at $n=1$ with $a=1$ and $b=1$ which is $$2^1 = 1! + 1! = 2 $$ and $n=2$ with $a=2$ and $b=2$, $$2^2 = 2!+ 2! = 4$$ and finally $n=3$ with $a=3$ and $b=2$, $$2^3 = 3! + 2! = 8$$ I ...
5
votes
2answers
74 views

How to decide which moduli to check when solving a “polynomial” congruence?

Consider the following problem: Find all integer solutions to $y^2 = x^5 - 4$. The solution goes something like – check modulo 11, where $x^5 \equiv 0, \pm 1$, and then check cases to arrive at ...
1
vote
2answers
54 views

Why is this the eigenvector?

For the eigenvector how are they getting \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} when you have \begin{bmatrix} 0 & -1 & -1 \\ 0 & -1 & -3 \\ 0 & 0 & -2 \end{bmatrix} ...
2
votes
0answers
41 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...