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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2
votes
1answer
44 views

Simplify $y^\top x -\log(\sum_i e^{x_i})$

Simplify $\sup_x y^\top x -\log(\sum_i e^{x_i})$ The first order conditions yield $y_i=\frac{e^{x_i}}{\sum_i e^{x_i}}$. How do I eliminate $x_i$ from the equation? I know the answer to be $\sum ...
2
votes
3answers
85 views

Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...
1
vote
1answer
42 views

General solution of a second order PDE

I have the second order PDE $t^5u_{xx}-tu_{tt}+2u_t=0$ and need to find it's general solution. My problem is that since it's a second order PDE the method for first order quasi-linear PDE doesn't seem ...
5
votes
1answer
78 views

An integration question.

An help in the following problem: Let $f:[-1,1] \longrightarrow \mathbb{R}$ a $C^1$ function, i.e., continuously differentiable. Suppose that we have $$\int_{-1}^{1} f(x)\;dx = \pi ...
10
votes
1answer
174 views

Inequality $\frac{a + \sqrt{ab} + \sqrt[3]{abc}}{3} \leq \sqrt[3]{a \cdot \frac{a+b}{2} \cdot \frac{a+b+c}{3}}.$

Someone can to help me with a hint in the following problem: Show that for any $a,b,c>0$, $$\frac{a + \sqrt{ab} + \sqrt[3]{abc}}{3} \leq \sqrt[3]{a \cdot \frac{a+b}{2} \cdot ...
4
votes
0answers
99 views

Different ways of operating an infinite continued fraction

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I ...
3
votes
1answer
47 views

How many (linear) order types are there on a set of n elements?

Given number $n$ variables $a_1, a_2, \dots, a_n$. How many way can we place $>$, $=$ between them ? For example, for $n = 3$ (Let's call $a_1 = x, a_2=y, a_3=z$ for convenient). There are 13 way: ...
0
votes
1answer
36 views

At Great Value Expression

Hi, How can I find the greatest value of $2k\sqrt{(r-k)(r+k))}$ with parameter $r$ such that $(r>k)$?
0
votes
1answer
67 views

Distance Question

Two trains set out of two cities, A and B, simultaneously; One from A and one from B. Until the meeting one train has passed $108$ km more than the other. Later, one of the trains arrived at it's ...
0
votes
1answer
63 views

Stuck solving an equation using the floor operator.

I am not entirely familiar with the equation ninja'ing involving the floor operators. Here is my problem. I need to solve for $x$. Everything is an integer, including $x$: $$ a - 1 = \lfloor {\frac{x ...
0
votes
4answers
49 views

Solve for $x$ when a maximum over $x$ and a constant is involved

This may be a simple question, but I'm not sure how to find the algebraic solution for a problem like: $ax=b+\max(cx,d)$ where $a,b,c$, and $d$ are known. Wolfram-Alpha is not able to give me a ...
1
vote
3answers
89 views

There exists an integer with alternating digits $1$ and $2$ which is divisible by $2013$

Could someone give me hints in how to solve the following (rather interesting) problem? Prove that there exists an integer consisting of an alternance of $1$s and $2$s with as many $1$s as $2$s ...
5
votes
3answers
116 views

Prove that any two numbers of the form $2^{2^n}+1$ are coprime to one another.

Full problem statement: Prove that any two numbers of the follwing sequence are relatively prime: $2 + 1, 2^2+1, 2^4 + 1, 2^8+1, ... 2^{2^n} + 1 $ So far I have tried to use Euclid's algorithm with ...
1
vote
2answers
91 views

Solving quadratic system

If $a,b,c\in \mathbb{R}$ satisfy the system $a^2+ab+b^2=9$; $b^2+bc+c^2=16$;. $c^2+ac+a^2=25$. Find $ab+ac+bc$
-1
votes
3answers
227 views

simplification of an complex exponential equation

There are these steps in a solutions manual I do not follow. I struggle to find any good and problem specific information about this kind of math wizardry on my own. I don't really know what to google ...
1
vote
1answer
116 views

$\arccos$ of an imaginary number

How can I solve a $\arccos$ of an imaginary number? like: $$\cos x = 0.9i$$ Because I can't make the $\arccos$ of a imaginary number
0
votes
2answers
77 views

How to solve an equation involving the exponential and the logarithm

The equation $\log_3(\sqrt{x+1}+1)=(3^{x+1}-1)^2\,$ has two solutions, but I can't solve the equation.
4
votes
2answers
171 views

How to solve for $x$ in $\sqrt[4]{x+27}+\sqrt[4]{55-x}=4$?

I'm trying to guess a method for getting the values that work on this irrational equation: $$\sqrt[4]{x+27}+\sqrt[4]{55-x}=4, x\in\mathbb C$$ After using the formula ...
1
vote
0answers
45 views

Reputation probabilities

Is there a probabilistic model for what one's reputation can be on MSE? I can of course obtain increments of points in +2, +5, + 10, + 15, +50, +100. Are there models for what my reputation will ...
1
vote
0answers
177 views

Solving system if equations containing trigonometric functions with Ti-Nspire

In trying to solve the following system of equation: $20000\times9.81+a\cos b=0$ $a\sin b=6.17\times20000$ Find $a$ and $b$ . It gives me something containing "n2" in bold and I don't know why? ...
0
votes
3answers
181 views

GRE Question [Word Problem]

The following is a question i got wrong on the GRE practice test. There is no explanation provided. I have actually a confusion as to what is even being asked and how they get the answer $r^{2} ...
1
vote
2answers
98 views

Right angle triangle simple problem

The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both the angles in degrees. So I have found the following answers : ...
1
vote
2answers
238 views

Number of people having shaken hands an odd number of times

This is from a book called USSR Olympiad Problem book: Every living person has shaken hands with a certain number of other persons. Prove that a count of the number of people who have shaken hands ...
2
votes
2answers
92 views

solve for m by rewriting the equation (transposition)

In the following equation how would I rewrite the equation to solve for $m$? $$z=\frac{-4m-8+\sqrt{(4m+8)^2+4(4(mx+y-4m-4))}}{8}$$ when $x=66$ and $y=22$ and $z=10$
2
votes
2answers
75 views

Tricky differentials problem involving continuous functions

Suppose $f$ is a continuous function on $[0, \infty )$, differentials on $(0, \infty)$, such that $f(0)=1$ and $f'(x)> \frac{1}{2\surd (x+1)} \forall x>0$. Show that $f(x)> \surd (x+1)$. ...
2
votes
2answers
169 views

How to calculate the determinant of this matrix $A=\begin{bmatrix} \sin x & \cos^2x & 1 \\ \sin x & \cos x & 0 \\ \sin x & 1 & 1 \end{bmatrix}$

How to calculate the determinant of this matrix $A=\begin{bmatrix} \sin x & \cos^2x & 1 \\ \sin x & \cos x & 0 \\ \sin x & 1 & 1 \end{bmatrix}$ ...
1
vote
0answers
35 views

Solving a system of equations with fractional parts and a system with round parts

I have the following two systems of equations: $a = x_{11} - \{x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}d\}$ $b = x_{12} - \{x_{12} + \frac{4 - ...
6
votes
3answers
108 views

What should I do if I don't know where to start?

Sometimes getting started on a problem seems to be the hardest part. Once you find something to sink your teeth into, everything goes smoothly. What are some good things to try when you're staring at ...
1
vote
3answers
56 views

Solve inverse tangents

How do I solve the following equation: $$ \tan^{-1}\frac{x}{10^6}+\tan^{-1}\frac{x}{10^7}=90^{\circ}$$ WA Step by step solution from wolframalpha is unavailable.
1
vote
1answer
33 views

Finding a y(x) that satisfies $ y(x) = \int_0^x \! \left(\frac{t} {y(t)+1}\right)^2 \, \mathrm{d}t $

I'm having problem with finding a y(x) that satisfies $$ y(x) = \int_0^x \! \left(\frac{t} {y(t)+1}\right)^2 \, \mathrm{d}t $$ Here is what I tried to do. $$ y(x) = \int_ \! \left(\frac{x} ...
1
vote
2answers
30 views

Inequality with Arc Length

Define $L:=\int_{0}^{1}\sqrt{(x'(t))^2+(y'(t))^2} dt$. Show $L\geq|x(1)-x(0)|$. I don't know where to start.
0
votes
1answer
43 views

Power calculation problem

Three machines perform together a certain job. If only machine A works, to perform the job alone, it would require a hours more than the time required to perform the job by the 3 machines, when they ...
1
vote
1answer
57 views

A confusion(possible book mistake) about one of the proofs in Spivak's Calculus?

In Chapter 5 - Function Limits, there is a proof that: if $|x - x_0| < 1; |x - x_0| < \frac{\epsilon}{2(|y_0| + 1)}; |y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}$ then $|xy = x_0y_0| < ...
2
votes
0answers
51 views

Probability of World Series - Pascal and Fermat [closed]

In the World Series, each opponent attempts to win 4 of 7 games. In 1989, Oakland had already won the first two games as the San Francisco Giants and the Oakland Athletics were getting ready to play ...
2
votes
0answers
120 views

Can you explain the solution of this geometric problem

A year ago IBM research posted an interesting geometrical problem: A gardener plants a tree on every integer lattice point, except the origin, inside a circle with a radius of $9801$. The trees ...
3
votes
2answers
722 views

Proving Holder's inequality using Jensen's inequality

Let $p$ and $q$ be positive reals such that $\frac{1}{p}+\frac{1}{q} = 1$, so that $p,q$ in $(1,\infty)$. For $\vec a$ and $\vec b \in \mathbb{R}^2$ prove that $|\vec a \cdot \vec b | \leq ||\vec ...
2
votes
2answers
117 views

The Generalized Jug Problem - a different question

“If you have a five-liter and a three-liter bottle and plenty of water, how can you get four liters of water in the five-liter bottle?” We (all) know this problem, and its natural ...
0
votes
0answers
32 views

Bijection inequality

Suppose $a$,$b \in \mathbb{R}^n$ are such that $$0 \leq a_1 + \leq a_2 + \cdots + \leq a_n $$ and $$0 \leq b_1 + \leq b_2 + \cdots + \leq b_n $$ If $\sigma : \{1,2,3,\ldots,n\} \rightarrow ...
3
votes
1answer
102 views

Proving there are infinitely many integers having the identical set of prime factors.

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the ...
5
votes
2answers
376 views

$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}} $, what is the nearest value of $y^2 - y$?

I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question. $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the ...
1
vote
0answers
85 views

Can the following quadratic equation be solved for M without iterating over possible values

I have developed this equation for a piece of software I am writing. Not being very mathematically minded I am stumped at how I can solve for M without iterating over all possible values for M. The ...
4
votes
0answers
164 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
1
vote
1answer
124 views

Solving an equation involving floor/ceiling as a summation bound

Is it possible to solve the following equation for $\alpha$? $$ M = \lfloor \alpha \rfloor + \alpha \sum_{k=\lfloor \alpha \rfloor +1}^N \frac{1}{k}$$ where $\alpha \geq 1$. Intuitively, $M$ is a ...
0
votes
2answers
62 views

What are the ages?

A man taking the census walks up to the apartment of a mathematician and asks him if he has any children and how old they are. The mathematician says: "${\it\mbox{I have three daughters and the ...
1
vote
1answer
81 views

Show that there is a number between 1 and 1000 such that there is a perfect square

Show that there exists an integer $n \in S = \{1,2, \ldots, 1000\}$ such that $$\prod_{i\in S-\{n\}}i!$$ is a perfect square. I was thinking in trying to prove it by contradiction using the ...
3
votes
1answer
47 views

Residues computation when we need power series

I'm trying to compute residues in situation where we need to manipulate power series to get it, but I can't find a good way. Indeed for the sake of example, consider the residue of the following ...
1
vote
0answers
138 views

100th degree polynomial $P(2^k)=k$ for $k=0,1,…100$

I sometimes see this kind of question, but I completely forgot how to solve it. Could anyone solve it for me?
0
votes
1answer
272 views

Finding dy/dx as a function of x for a dog-walker dragged by a dog travelling in a straight line

Hello. I was wondering if anyone could provide some insight into how to solve the following Calculus word problem: Max is walking his dog Beau in the Cartesian plane, with the leash between them at ...
1
vote
1answer
49 views

Function with invariant area under curve

I'm trying to find a function $f$ that fulfills the following property: The area under the curve starting at some point $x_0$ with a width of $x_0$ should always be the same for all $x_0$. In other ...
1
vote
2answers
61 views

Putnam-Style Sequences Problem

Let $S_1$ denote the sequence of positive integers $1,2,3,4,5,6,\ldots,$ and define the sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to those integers in $S_n$ which are divisible by $n$. Thus, ...