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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0
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1answer
207 views

If we draw infinitely many lines on a table, can we find a triangle somewhere? [closed]

If we draw infinitely many lines on a table, can we find a triangle somewhere? We prove that there is a subgraph $C_3$ in $C_n$, which will be called a triangle. Suppose we have an infinite ...
1
vote
2answers
94 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
0
votes
2answers
92 views

How to show the existence of a number with certain divisibility conditions between two multiples?

How can we show that between two even natural numbers they're exists a natural number that isn't even? How can we show that they're exists a natural number that is odd and not divisible by 3, between ...
6
votes
5answers
228 views

How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$?

I want to sum up the partials of a harmonic series, how do I do it? If I was using the 'Lagrange trigonometric identity to solve this problem', how would I plot it on Wolfram mathematica (using which ...
1
vote
1answer
64 views

English question regarding pigeonhole principle classic question.

Mr. and Mrs. Smith invited four couples to their home. Some guests were friends of Mr. Smith, and some others were friends of Mrs. Smith. When the guests arrived, people who knew each other ...
0
votes
1answer
46 views

conditional probability Pc(B)

I am looking for the probability of Pc(B) where the event of B={no two people are born in the same month} and event C= {exactly three people were born in the summer of june, july august} and there are ...
1
vote
0answers
33 views

probability of an event question

Find the probability of the event B= { NO two people are born in the same month}. There are 9 people involved by the way. Is it 1/12 * 1/11 * 1/10 * 1/9 * 1/8 * 1/7 * 1/6 * 1/5 * 1/4 * 1/3 * 1/2 * 1/1 ...
0
votes
1answer
45 views

How to best solve a system of equations like this one

I need some help trying to solve a system of two equations with two unknowns. Background: This is not homework, just hobby. I have some LEDs that I needed to identify, but the manufacturer datasheet ...
1
vote
1answer
81 views

Question about the matrix representation of the differentiation map on the subspace generated by $\{1, t, e^{t}, e^{2t}\}$

As mentioned in a previous post (I think), I've been trying to learn some linear algebra, and so I've begun to post little questions whose answers I'm sure are obvious to most here; this is just a way ...
0
votes
1answer
57 views

Pigeonhole Principle to solve question straightforward

A store wants to celebrate its anniversary and will give a $200 shopping certi cate to the first customer to enter the store whose birthday is the same as that of two other previously admitted ...
0
votes
0answers
29 views

The product of $i$ consecutive natural numbers is divisible for $i!$ [duplicate]

There is a theorem that I've used it a few times, and never saw a demo of it, and when I tried, I could not, commenting with a teacher, it would not give me much attention and said it would use the ...
1
vote
3answers
61 views

Find $k$ ($\sqrt{x^2-3 x+3}-\sqrt{x^2-3x+2}=k$) [closed]

I would appreciate if somebody could help me with the following problem: Q:Find $k=?(k>0)$ $$\sqrt{x^2-3 x+3}-\sqrt{x^2-3 x+2}=k$$ Have two real root $a,b$ and $ab=23/16$
34
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13answers
3k views

Examples of famous problems resolved easily

Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example ...
0
votes
0answers
10 views

Ornstein-Ulenbeck and greens function problem

In the Ornstein-Ulenbeck process with a Langevin eqn such as: $$\dot{x}_{i}+\gamma_{ij} x_{i} = \phi_{i}(t)$$ then the homogenous version i.e $$ \dot{x}_{i}+\gamma_{ij} x_{i} =0$$ is clearly solvable ...
0
votes
0answers
97 views

Approximations other than taylor series and pade approximation

I have a function which has the following form: $$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$ and I want to find $x$ for $f(x)=1$. I'm ...
3
votes
0answers
63 views

A challenging non homogenous fractional inequality.

The following problem is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a ...
1
vote
3answers
199 views

Find all the integral solutions to $2x+3y=200$

What's the best way of going about this? $$2x+3y=200.$$
0
votes
0answers
68 views

Solving systems of linear congruential equations

What is the best way to solve systems of the form $$ x_1 a_{1,1} + \cdots + x_n a_{1,n} = y_1 \mod m \\ \vdots \\ x_1 a_{N,1} + \cdots + x_n a_{N,n} = y_N \mod m $$ with $n \le N$, the parameters ...
3
votes
2answers
82 views

Find $\frac{a+b+c}{x+y+z}$ given $a^2+b^2+c^2$, $x^2+y^2+z^2$ and $ax+by+cz$.

We are given $a^2+b^2+c^2=m$, $x^2+y^2+z^2=n$ and $ax+by+cz=p$ where $m,n$ and $p$ are known constants. Also, $a,b,c,x,y,z$ are non-negative numbers. The question asks to find the value of ...
0
votes
1answer
79 views

General solution to a recursive equation

What is the general solution of the following recursive equation? $$N(t)=(1+f)\cdot\left(N(t-1)-N(t-T)+N(t-T-1)\right)$$ By "general solution" I mean an equation where $N(t)$ stands alone on the ...
20
votes
1answer
476 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
1
vote
1answer
188 views

Probability of two dice rolls

If two standard 6 sided dice are tossed what is the probability that a 3 is rolled on at least one of the two? How do I solve this without listing out the possibilities?
0
votes
1answer
41 views

Sequent calculus - where should I start?

I am given this formulae. $A \land B \implies C \lor D \lor E$ I want to deduce this formulae with sequent calculus. But my problem is that I dont know where to start, or which rule to take first. ...
2
votes
2answers
159 views

how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$

I am given this equation: $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$ I want to prove it: what i did is I take any $a \in f^{-1}(B_1 \cap B_2)$, then there is $b \in (B_1 \cap B_2)$ so ...
0
votes
2answers
63 views

how to find the elements of additive Group - $\mathbb{Z_7^+}$

I am given this additive Group G=$\mathbb{Z_7^+}$ I tried to find all its elements and I did: $$gcd(1,7) = 1 \\ gcd(2,7) = 1 \\ gcd(3,7) = 1 \\ gcd(4,7) = 1 \\ gcd(5,7) = 1 \\ gcd(6,7) = 1 \\ ...
-3
votes
1answer
309 views

The problem that needs some critical thinking. [closed]

The question is: If it takes 5 pipes 2 hours to fill 2 pools and they output water at the same rate, how many hours will it take 1 pipe to fill 1 pool?
0
votes
1answer
131 views

Solve 4 A (L^(3/4)) - wL (((24 - L) w)^(-3/4)) = -(((24 - L) w)^(1/4)) for L? Using Mathematica

I'm trying to solve $$4 A (L^{3/4}) - wL (((24 - L) w)^{-3/4}) = -(((24 - L) w)^{1/4})$$ for $L$ using Mathematica, and it spits the following out: ...
0
votes
2answers
123 views

Soda can contest problem

The store's rule is that you can use 3 empty cans to exchange for 1 can of soda. Alex's class has 17 students and each bought a can of soda at the beginning of the party. At most, how many more cans ...
0
votes
2answers
58 views

Distance rate time problem of two mice

Mouse A and Mouse B are separated by a distance of 1.62 meters underground. They decide to meet by digging all the way through. Mouse A will double his speed every day, that is, he starts to dig 2 cm ...
1
vote
0answers
94 views

Derive 3D point array from multiple 2D projections of same point array

Let's assume that we have an array of $n$ 3D points, we don't know their coordinates (thus we have $3n$ indeterminate scalar values). We also have $m$ 2D projections with known coordinates (thus we ...
0
votes
1answer
468 views

Dividing a rectangle into maximum number of regions using lines

At most how many regions can you divide a rectangle in using 6 lines? I got 16.
0
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0answers
192 views

Smallest positive integer divisible by and having digit sum equal to some 3-digit number.

Let $p,q,r$ be distinct digits among $1,2,4,6,8$, and consider the integer $pqr = 100p + 10q + r$. Let $N$ be the smallest positive integer that is divisible by $pqr$ and has digit sum equal to ...
1
vote
0answers
71 views

Analytical methods to solve a cubic equation

I'm having trouble solving this cubic equation: $x^3 - 7x^2 -27x+64=0$ I was able to prove that it has no rational roots, and by graphing the cubic function, I found that its 3 roots are all real, ...
1
vote
1answer
25 views

Distance Rate Time of 2 mice

Mouse A and Mouse B are separated by a distance of 1.62 meters underground. They decide to meet by digging all the way through. Mouse A will double his speed every day, that is, he starts to dig 2cm ...
3
votes
0answers
70 views

why does table look like this? - multiplicative table for Group

I have a Group $\mathbb{Z_2}[x]/q$ where $q(x)=x^3+x+1$ In my textbook, there a multiplication table: but I dont understand how the last $x+1$ comes. it should be $(x^2+x+1)*(x^2+x+1) \quad mod ...
1
vote
1answer
58 views

how these elements come? - finite field with polynomial $x^2+1$

I am given this finite field. $$K=\mathbb{Z_3}[x]/p$$ with $$p(x)=x^2+1$$ In my textbook, i see that the elements of this field are: $$\{0, 1, 2, x, x +1, x+2, 2x, 2x + 1, 2x + 2\}$$ I dont ...
0
votes
1answer
38 views

how do I find all Elements of a Group?

I am given a Group $\mathbb{Z_{11}^*}$. a multiplicative group. How do i find all elements of this Group?
0
votes
1answer
30 views

how comes this step ? inductive proof with binomial

I am sitting on this problem for couple of hours now but still cannot get the reason why the step which i marked is possible... how has $(n+1)$ disappeared here?
1
vote
1answer
31 views

how comes $s=4$ and $t=3$ for $4=7s-8t$

i am given these two problems: $x\equiv 1 (\bmod 7 )$ and $x \equiv 5( \bmod 18)$ I tried this way: $x\equiv 1 (\bmod 7 )$ is basically $x = 1 + 7s$ and $x\equiv 5 (\bmod 18 )$ is $x=5+18t$ then ...
0
votes
0answers
40 views

On the solution of a stacking/spending optimisation problem

Description of the problem Each week our hero, say John, can stack a minimum of $2.5$ hours and a maximum of $6.5$ hours if he works more than $8$ hours per day. Each day John cannot stack less than ...
2
votes
1answer
28 views

Question of how to write certain response

What is the domain of the function $$f(x,y)=\frac{1}{x^2+y^2-1}$$ The answer is clear, right?! $$x^2+y^2-1\neq 0\Longrightarrow x^2+y^2\neq1$$What is the point contained in the circle of ...
1
vote
2answers
153 views

Problems with the existence of limits of several variables

I started some time studying calculus of two variables, and I'm having difficulty in knowing (and prove) that a limit does not exist, how could I resolve, for example, these, whose statement asks to ...
0
votes
0answers
65 views

Solving an equation with several summations and a falling factorial

Can you help me to solve quite a complicated inequation that contains summations? I'd like to solve it for b (leaving balone on ...
22
votes
10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
2
votes
2answers
205 views

Allocation optimization problem

Imagine that I have $1$ million dollars which I want to invest. I have a set of $N$ elements in which I can put the money and obtain a revenue. Each element has a function that determines how much ...
1
vote
0answers
72 views

Recommendation for a good book on equation solving theory from the basics

I'm relearning calculus but I often find myself applying algebraic operations to equations mechanically without having a solid understanding of the side-effects of those operations. Such as extra or ...
0
votes
2answers
58 views

how to prove this: $f(A)=B$

I am given two sets: $A$ and $B$ and a function $f: A \rightarrow B$. I am asked to show and prove whether $f(A)=B$ is true or false. I am stuck not knowing how to do this. How can I do this?
1
vote
1answer
115 views

A proof about affine subsets

Let $V$ be a vector space and $S$ a nonempty subset of $V$. I want to show that $S$ is an affine subset (a translated subspace of the form $\{v\} +$ $U$, for $U$ a subspace of a vector space $V$ and ...
1
vote
1answer
68 views

equation with series

Hello to everybody I have a problem because I can't solve this equation: $$960 - \frac{84.60}{(1+x)^{\frac1{12}}} - \frac{84.60}{(1+x)^{\frac2{12}}} - \cdots - \frac{84.60}{(1+x)^{\frac{11}{12}}} - ...
4
votes
3answers
287 views

How to solve $e^x = 2$

I know that $\ln(x)$ is the inverse of the exponential function $a^x$. So I thought that $$ e^x=2 \Leftrightarrow x = \ln(2) $$ but my calculator says $x = \ln(2) + 2 i \pi n$, where $N \in ...