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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4
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3answers
127 views

Is it possible for extremely ingenious but elementary proofs for famous problems to exist?

As Erdős put it, "Mathematics is not ready for such problems." when faced with the great conjecture of Collatz. So is it impossible altogether for simple but ingenious proofs for famous problems ...
0
votes
0answers
32 views

Optimality criterion for unconstrained convex optimization problems

Consider a general convex optimization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f_0(x) \\ & \text{subject to} & & f_i(x) \leq 0, \; i = 1, ...
0
votes
3answers
36 views

Odd and even coins

Two friends are playing a coin game. You need to give to your friend, one coin that has an even value and another coin that has an odd value. For example, a dime would be the even value ...
0
votes
1answer
22 views

Invariance problem dealing with the sums of units digits

We may write all the digits from 1 to 9 in a row in any order we like, and then we write plus signs between some digits (as many plus signs as we like). Finally, we evaluate the obtained expression. ...
0
votes
1answer
23 views

Into how many regions do they divide the plane? [duplicate]

Suppose that n lines in general position are given in a plane. (General position means that no two lines are parallel, and no three lines have a common point.) Into how many regions do they divide the ...
0
votes
1answer
17 views

Annuities-calculating interest

Janet receives a $ 10,000 life insurance benefit. If she uses the proceeds to buy an n-year annuity immediate, the annual payout will be 1613.36. If a 2n-year annuity due is purchased, the annual ...
1
vote
1answer
61 views

28 soldiers puzzle

A leader ordered his 28 soldiers to protect the castle , the castle has 4 walls or sides. He wants 9 soldiers to guard each wall. How can the be possible ?
0
votes
0answers
15 views

Calculate time needed to transmit 10 packets over 3 links

So very easy problem, but it really bugs my head and I am not sure if I am solving it in a right way. Imagine we have Sender and Destination endpoints and two routers in between: ...
1
vote
3answers
69 views

Is there an exact answer to this equation?

$$\frac{1}{x} +\ln(x)\ln(\ln x)=1$$ The solution to this equation is approximately $x \sim 5.13425\ldots$ but is there an exact answer?
0
votes
2answers
52 views

Finding the general term of a sequence (if there's any)

I would like to know if it's possible to find an expression for the following sequence $\;\{a_n\},\;n=0,\,1,\,2,\,3,\,\dots\;$ $1, 3 , 7, 13, 21...$ Someone would like to explain me what I have to ...
0
votes
1answer
29 views

Let F0 = 0, F1 = 1, F2 = 1, . . ., F99 be the first 100 Fibonacci numbers (recall that Fn = Fn−1 + Fn−2 for n ≥ 2).

Let F0 = 0, F1 = 1, F2 = 1, . . ., F99 be the first 100 Fibonacci numbers (recall that Fn = Fn−1 + Fn−2 for n ≥ 2). how many of them are divisible by 3
1
vote
1answer
24 views

How does this textbook compute the Nash Equilibrium of the two person zero sum game?

In Tamer Basar Noncooperative Game theory pg $33$ there is a $2 \times 3$ game (zero sum game) $A = \begin{bmatrix} 1 & 3 & 0 \\ 6 & 2 & 7 \end{bmatrix}$ (each element is a cost, ...
0
votes
1answer
83 views

Prove inequality $\sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3$ [duplicate]

How to prove the following inequality : $$ \sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3 $$ with $a>0,\ b>0$ and $c>0$.
0
votes
2answers
37 views

Mathematical Induction Problem solving [duplicate]

$1^3$ + $2^3$ + $2^3$ + ... + $n^3$ = ($1 + 2 + 3 + ... + n)^2$ I start with $P(1)$ and get $1 = 1$. Then I do it with $P(n+1)$ and I get stuck. $1^3$ + $2^3$ + $2^3$ + ... + $n^3$ + $(n+1)^3$ = ...
0
votes
1answer
64 views

Solve differential equation relating to air cannons

I made my own air cannon at home and would like to determine its launch velocity. I worked out the following differential equation: $$\frac{d^2}{dt^2} x(t) = \frac{A}{m} \left(\frac{P_0V_0}{V_0 + ...
2
votes
0answers
63 views

Vakil's FOAG Exercise 11.3.C

I have been thinking about the exercise 11.3.C of Vakil's FOAG. The answers posted here Trouble with Vakil's FOAG exercise 11.3.C are not clarifying enough for me, since I have though of the ...
0
votes
3answers
40 views

Solving exponential equation in one variable.

I'm stuck with solving this exponential equation: $(1-v)^v = 2$ PS. I'm sorry I can show no solution that I could try myself.
2
votes
2answers
39 views

e of a propositional function such that the statement ∃!x ∃!y p(x,y) is true but the statement ∃!y∃!x p(x,y) is false

Give an example of a propositional function such that the statement ∃!x ∃!y p(x,y) is true but the statement ∃!y∃!x p(x,y) is false (be sure to specify the domain for each variable)
2
votes
2answers
120 views

rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$

This question concerns the equation $$\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$$ and the possibility of rational number solutions with $abc \ne 0$. In comments arising from: Using ...
2
votes
0answers
36 views

Problem of $\pi$- and $\lambda$-systems

I have some trouble with a theoretic-like exercise about measure theory, and $\pi$- and $\lambda$-systems, and I would like to have some help. The problem is stated in the book Mathematical ...
2
votes
3answers
140 views

Can you recommend a Maths Book for general Problem Solving?

So, here is my intent and situation. I am OK at solving novel problems. I've been through general University calculus, and have always gotten mainly A's and B's in my maths courses. But I've mainly ...
2
votes
1answer
73 views

How do you find the angle between two vectors? [closed]

How to solve this question? Find the acute angle between the line joining $(1,3,2)$ to $(2,5,-1)$ and the line joining $(-1,4,3)$ to $(3,2,1)$.
0
votes
1answer
17 views

Water & salt liquid

If I'd have an $500$ ml bottle of water and it would have $1\%$ salt, how much I would need to add water to make it, lets say $0.3\%$? I cannot find the way to figure this problem. Could anyone ...
0
votes
1answer
61 views

For how many different ways, can you place 14 students in two rows of desks?

For how many different ways, can you place 14 students in two rows of desks, where 5 of them sit always in first row, and 4 of them sit always in second row? Eight students can sit in on row.
1
vote
3answers
49 views

Engineering Calculus Problem

If $L\frac{di}{dt}+Ri=E $, where $L, R$ and $E$ are constants, and it is known that $i=0$ at $t=0$, show that: $$ \int_{0}^{t} (Ei-Ri^2)dt = \frac{Li^2}{2} \tag1 $$ I'm very frustrated with ...
11
votes
0answers
239 views

If $a^n-1$ divides $b^n-1$ too often, then $b=a^k$

I am looking for references to the following problem, which I saw a long time ago and I think is a well-known problem (maybe from IMO or American Mathematical Monthly), I hope to remember it ...
1
vote
0answers
29 views

Blackjack Dice Probability - Odds of winning and where to stand

Hi I'm looking for help/advice in finding the solution to the following problem. Suppose I have a fair $101$ sided dice labelled $0$ to $100$, Your objective is to roll the dice any number of times ...
4
votes
3answers
367 views

How to find the numbers which satisfies this equation? [closed]

I have the following equation, $3(x\times y\times z) = z^3$; where x,y and z are positive integers How can I find the different possibilities of numbers which suits those three undefined values?
0
votes
0answers
25 views

Finding the magnitude of a vector which maps the trajectory of an object which passes through a given point.

Given the point $(0.5, 0.8)$ which lies along the curve mapped by the trajectory of a projectile $\vec{r}(t)=|\vec{R}|\langle{t\cos{\theta},t\sin{\theta}-4.9t^2}\rangle$, as well as it's angle of ...
0
votes
0answers
33 views

Is there a way to solve this problem - sum of powers

$N_1^a + N_2^a + \ldots + N_n^a$ Where a is any number. Example $40^7+ 37^7 + 35^7 + 32^7 + ... 1^n=R$ If Iknow what $R$ is and that $ N_n$ is restricted to $0 \lt N_n \le 40$, Is there a way to ...
0
votes
0answers
5 views

How to sketch this graph by considering appropriate transformation?

Sketch the curve $(y^2-2)^2+(x^2-2)^2=2$ By considering $x^2 \mapsto u, y^2 \mapsto v$, I see that I could first sketch a graph of $v$ against $u$, which is a circle centred at $(2,2)$ with ...
1
vote
1answer
57 views

Variation on the “Number of non-bald people in NYC” problem

I have two questions about the following problem, taken from Challenging Problems in Algebra by Posamentier and Salkind: (1) Why is the answer not 1 person? (2) The answer given, without solution, is ...
0
votes
2answers
19 views

Show that if $\exists \vec{x} : \vec{x} = \vec{x}_0+\vec{x}_1 \iff \exists \vec{x}_0: A\vec{x} = \vec{b}$ and $\exists \vec{x}_1 : A\vec{x}=\vec{0}$.

Matrix $A\vec{x}=\vec{b}$ represents a system of $m \times n$ unknowns with a specific solution $\vec{x}_0$. Show that any solution of this system $\vec{x}$ can be written in the form ...
3
votes
1answer
45 views

Where is this converging to on $y=x$?

Well I was playing with graphs and I started plotting equations as the following: $$\underbrace{x+y}_{degree=1}=1 \tag{1}$$ $$\underbrace{x^2+y^2+xy}_{degree=2}+\underbrace{x+y}_{degree=1}=1 ...
0
votes
1answer
24 views

Solving given homogoneous systems.

It is given that $ A = \left[ \begin{array}{rrr} 1 & -2 & 0 \\ 1 & 1 & 1 \\ -1 & 3 & -2 \\ \end{array} \right] $ and $ \vec{x} = \left[ \begin{array}{r} x_1 \\ x_2 \\ x_3 \\ ...
0
votes
1answer
25 views

Any chances this can be further reduced?

I've come with the following equation, after a lot of simplification, but can't reduce further. Any chances it can be solved by reducing the $b$ and get the value of $a$? $$a = \frac{1000(1000 - ...
1
vote
1answer
25 views

I am having a problem in understanding a notation in the book Matrix Analysis by C Meyer.

I am having a problem in understanding a notation in the book Matrix Analysis by C Meyer. The notation of $A_{*1}$ is where I am facing problem. Also I want a detail of Reflector....where will I ...
12
votes
2answers
267 views

Trying to apply mathematics into a real-life context

Earlier today, I was trying to place my storybook into my wardrobe drawer, only to find that I could only place it horizontally as the height of my book exceeds the height of my drawer. Out of ...
0
votes
2answers
61 views

How to decompose $z^8-1$?

I want to express $z^8-1$ as a product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form. I know ...
1
vote
1answer
54 views

Free group and commutator question from Artin's book

Let $F$ be the free group on $x$, $y$ and $R$ be the smallest normal subgroup containing the commutator $xyx^{-1}y^{-1}$. a) Show that $x^2y^2x^{-2}y^{-2}$ is in $R$. b) Prove that $R$ is the ...
0
votes
1answer
46 views

What is the probability that All people select the same thing

If we have 3 types of things that were given for 4 people what the probability that all people will select the same thing, if they can select what they want Or have to get more information to solve ...
0
votes
2answers
46 views

Quadratic Equations - Problem Solving

A hall charges $30 per person for a sports banquet when 120 people attend. For every 10 extra people that attend, the hall will decrease the price by 1.5 per person. What number of people will ...
1
vote
3answers
20 views

Question about substances and percentages

My friend translated this problem from his native language, and I am unsure of the accuracy of the translation. If necessary, please edit to correct any faults. A mineral substance, to be sent to a ...
0
votes
1answer
175 views

Finding a general solution for {sin(x)=sqrt(3)/2 , cos(x)=-0.5} using an algebraic method

I am trying to form a trig general solution for this system of equations: $${ \sin(x)=\frac{\sqrt{3}}{2} , \cos(x)=-\frac{1}{2}}.$$ I found out that the general solution is $$x=2\pi n ...
0
votes
0answers
40 views

Solving for intersection of line and a vector function.

How would one approach a problem of finding intersection points between a line $\vec l = \vec S + d \vec t$ and vector of the form $$\vec v = \begin{pmatrix} x \\ y \\f(x, y) \end{pmatrix}$$ I am ...
1
vote
1answer
18 views

Sets of numbers Probability - need help

Ok so say I have 40 numbers set as 4 lines of 10... (1-10,11-20,21-30 and 31-40). How do we work out every combination possible to get 8 numbers while using a maximum of 5 numbers from each of the 4 ...
1
vote
1answer
69 views

Solving the simultaneous equations $x + y - z = 12$ and $x^2 + y^2 - z^2 = 12$

I found the following question online (it was a past question on the BMO): find all positive integer solutions x,y,z solving the simultaneous equations $ x + y - z = 12$ and $x^2 + y^2 - z^2 = 12 $ ...
2
votes
1answer
23 views

A modular arithmetic exponent problem

I was learning modular arithmetic and there was a problem that $4^{62} \equiv \ ? \pmod{7}$ The solution was to find the mods of the same number's exponents starting from one until the result equals ...
2
votes
2answers
48 views

Fitting two parallel lines to a set of points

In two dimension I have a set of points X = $\{x_1,..., x_N\}$. I want to fit two parallel lines to these points like $l_1$ and $l_2$ $$l_1 = p_1 + \lambda n^\perp$$ $$l_2 = p_2 + \lambda n^\perp$$ ...
4
votes
1answer
85 views

Find all possibles values. Looks Hard. Is it hard?

I need some help, (or advise) how I can solve this problem and in which category I need to put it. The problem state: Let m be a solution of the equation $ y^{2015}-15 y+ 14=0$. Find all possible ...