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
34 views

Working Backwards to Determine Winning Strategy

There are two piles of candy. One pile contains 20 pieces, and the other 21. Two players take turns eating all the candy in one pile and separating the remaining candy into two (not necessarily equal) ...
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1answer
13 views

Approaching concepts involving graphs in analysis

At least in undergraduate algebra, we can discuss the properties of algebraic structures and their elements without losing generality with notation such as let $G$ be a group and $g\in G$. In using ...
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1answer
22 views

Proper mathematical description for outer perfect shuffling

I was given the following problem: Consider a pack of $2 n$ cards, numbered from 0 to $2 n − 1$. An outer perfect shuffle is a shuffle of the cards, in which one first splits the pack in two ...
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1answer
97 views

Clarification on the intended meaning of a probability problem [closed]

I am just wondering if anyone can help with this question: A radio station held a competition where contestants were invited to pick a number from $1$ to $50$. If a contestant picked the ‘winning’ ...
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2answers
33 views

Prove diophantine equation $S^2+R^2+(r_1-r_2)^2 = 2R(r_1+r_2)$ has at most one solution

Given this diophantine equation: $$S^2+R^2+(r_1-r_2)^2 = 2R(r_1+r_2)$$ $S,r_1,r_2$ are variables. $R$ is a given constant. all values are positive integers. How do I prove that there's at most one ...
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4answers
57 views

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$ Sorry for the really dumb question but I'd like to see the process of how this is achieved.
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3answers
35 views

How to solve for exponent when adding fractions raised to unknown exponent?

I'm sure this is probably an extremely simple problem but I'm stuck figuring this out. For example: $(\frac{1}{5})^{x} + (\frac{7}{10})^{x} = 1$ What would be the steps to solve for x?
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2answers
67 views

Number of Polynomials with Integer Coefficients that are bounded by $x^2$ and $x^4+1$

What is the number of polynomials $p(x)$ with integer coefficients, such that $x^2≤p(x)≤x^4+1$ for all real numbers $x$?
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0answers
25 views

Find a basis for the hyperplane and use the line to extend the basis for the hyperplane to a basis for $\mathbb{R}^4$

Suppose there is a hyperplane in $\mathbb{R}^4$ that is the solution set to the homogeneous equation $x+2y-3z+w=0$ and a line in $\mathbb{R}^4$ given parametrically by ...
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3answers
43 views

Finding a basis for the intersection of two vector subspaces.

Suppose: $V_1$ is the subspace of $\mathbb{R}^3$ given by $V_1 = \{(2t-s,t,t+s)|t,s\in\mathbb{R}\}$ and $V_2$ is the subspace of $\mathbb{R}^3$ given by $V_2 = ...
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2answers
23 views

Sultan's law involving outnumbering

A Sultan wanted to increase the number of women in his country, as compared to the number of men, so that men could have larger harems. (Sorry ladies!) To accomplish this, he proposed the following ...
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0answers
53 views

Finding a finite dimensional subspace of an infinite vector space.

How could one find a nontrivial example (a subspace which contains more than simply the zero vector) of an infinite dimensional vector space that contains a finite dimensional subspace and prove it's ...
3
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1answer
37 views

Determine how many paths exist from $A$ to $B$ that > travel only to the right and up.

In the picture below, you see a schematic of some of the streets in a certain town. Determine how many paths exist from $A$ to $B$ that travel only to the right and up. Two such paths are given ...
4
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2answers
61 views

What is the value of $n$ for which $n!=2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23 $

What is the value of $n$ for which $n!=2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23 $ The way I am approaching this problem is just to find the ...
0
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3answers
52 views

Let $V = \text{span}(\{\vec{v}_1,\vec{v}_2,\vec{v}_3\})$ be a $3$ dimensional subspace of $\mathbb{R}^4$. Prove that $V^{\perp}$ has dimension $1$.

Let $V = \text{span}(\{\vec{v}_1,\vec{v}_2,\vec{v}_3\})$ be a $3$ dimensional subspace of $\mathbb{R}^4$. Prove that the orthogonal complement of $V$ has dimension $1$ My approach: Set $A = ...
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1answer
50 views

Show that all the cards contain the same number.

Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible ...
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0answers
22 views

Finding positive integers with the same digits [duplicate]

Find the three smallest positive integers $K$ (two digits or greater) with the following properties: 1) $K=\frac{(n)(n+1)}{2}$ for some $n$ 2) Each digit of $K$ is the same. I was able ...
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2answers
30 views

Can we deduce anything given the equation of a curve and the fact that it has symmetry with $y=x$?

Question: The line $y=x$ is a line of symmetry to the curve with equation $$y=\frac{px+q}{rx+s}$$ where $p,q,r,s \neq 0$. Which of the following must be true? $p+s=0$ $p+q=0$ ...
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1answer
10 views

One of the values of $z$ verifying

How to solve this equation of argument of complex number knowing that one of the values of $z$ verifying $\left|z+1\right|^2+\left|z-1\right|^2=2\left|z+i\right|^2$
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1answer
25 views

Finding the dimension of a vector space, and determing if it is the subspace of a parent space.

How could one determine the dimension of a some space which is the subspace of a particular vector space (or consider it a subspace of a particular space to begin with), say for instance given some ...
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1answer
22 views

Find a basis for the subspace determined by the given line.

Find a basis for the subspace of $\mathbb{R}^3$ determined by the line $x=-3t .\ y=2t ,\ z=t$. It seems to me that a basis for this subspace would be simply $\{t(-3,2,1)\}$, but could it really ...
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0answers
15 views

Determination of Bond prices

Two 1000 dollar face value bonds are both redeemable at par, with the first having a redemption date 3 years prior to the redemption date of the second. Both are bought to yield 11 percent convertible ...
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0answers
35 views

How to determine whether expression is positive or negative?

Given expressions $|x - 3 + y|$ and $|x + 3 + y|$ how can I determine, whether are those positive or negative, and determine their value in the intervals of: $y < -x - 3$ $y \in [-x - 3, 3 - x)$ ...
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1answer
19 views

Can a certain board be covered in Tetrominoes

Prove that a $15x8$ board cannot be covered by $2$ L-tetrominoes and $28$ skew tetrominoes. This is a coloring proof and I have tried a variety of colorings, from stripe colorings to other unique ...
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1answer
21 views

Diophantine equations: $x_1y_1+x_2y_2 = x_3y_3+x_4y_4$

Given 3 diophantine equations: $$x_1y_1+x_2y_2=x_3y_3+x_4y_4$$ and $$x_1+x_2 = x_3+x_4$$ and $$y_1+y_2 = y_3+y_4$$ We're interested in solutions to this system of equations when all variables ...
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2answers
36 views

A coin is tossed three times. Given that at least one head appears, what is the probability that exactly two will appear?

The "at least" confuses me. But I am assuming one head will appear. Making P(first head) = 1. Correct answer: 3/7 I start with the formula: P(A and B) = P(A) • P(B|A) Fitting the conditions into ...
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0answers
11 views

Class-participation problem modeled with game theory

I'm taking a class and the teacher has set up a system of class-participation to encourage us, the students, to, well, participate more actively. The system is as follows: each student is given 20 ...
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1answer
22 views

Creating a structure to show 2 formulas do not satisfy a 3rd using first order logic

A = (∀x∀y∀z(P xy → (P yz → P xz))) B = (∀x∀y(P xy → (P yx → x = y))) C = (∀x∃yP xy) → (∃y∀xP xy) I want to show that {A, B} does not imply C by constructing a structure. What I've done so far is ...
1
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1answer
34 views

Struggling a bit with Combinatorics Order in which to do this question?

n P 4 = 84 n C 2 Now I'm not even sure if the 84 is multiplying by the N choose 2? I don't understand. Ive done all the practice questions my teacher gave me and this came up on the homework and Ive ...
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0answers
39 views

Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as $$ p_X(x) = \left\{ ...
2
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1answer
58 views

If $x^2-3x+9=0$, can we say $(x+3)(x^2-3x+9)$ is also $0$ hence $x^3=-27$?

I was solving this question If $\dfrac x3 + \dfrac 3x = 1$ then find the value of $x^3$. I solved it as. Cube both sides and substitute $x^3$ with $t$, $$ \dfrac{t}{27} + \dfrac{27}{t}=-2$$ ...
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0answers
16 views

Uniqueness of a solution

Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties: 1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$. 2) $f_i(x_1, x_2, ..., x_n)=1$ if ...
5
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1answer
60 views

How can I improve my problem solving abilities so that I stop missing the obvious?

I'm a generally good math problem solver. I get decent scores on contests, top of my class in math courses, and have a pretty wide array of knowledge from which to relate concepts in order to solve ...
3
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1answer
35 views

Does this seemingly elementary question require König's theorem?

Question Let $f:\;\mathbb{R}\to\mathbb{N}$ and let $X_n$ be the set of reals mapped to the integer $n$. Show that for some $n,\;X_n$ has cardinality of the continuum. This is straightforward if we ...
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1answer
20 views

Determining outstanding balance on a loan

A loan of $17,000$ dollars is to be repaid in annual installments of $2,100$ dollars, the first due in one year, followed by a final smaller payment. If the effective rate of interest is $8.8$ ...
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0answers
33 views

Bases & Congruence's decimal problem

This is a question for a university assignment I have. I'm not after more of a "how-to". ...
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0answers
44 views

How to mathematically calculate this?

I want to know "energy price" to conduct panel data analysis. But I cannot find it on database of all web-sites. On all database, enegy price is unavailable. And according to my reseach, it's said ...
0
votes
2answers
26 views

Prove one group is the subgroup of another under a specific condition

Suppose that $H$ and $K$ are subgroups of a group $G$. Now for some $g_1,g_2 \in G$, $Hg_1 \subset Kg_2$. Prove that $H \subset K$ I tried to write the condition given as $H \subset Kg_2g_1^{-1}$ ...
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1answer
47 views

Basic probability question, choosing between two options for every stage

Suppose that two players are playing a game, players select between two choices: Scenario 1: player $1$ chooses option $1$ with probability $60\%$, option $2$ with $40\%$ player $2$ chooses option ...
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0answers
32 views

Known classic problem or not?

There is a set of positive whole numbers without null. I have to find the minimal number of subsets of the original set so, that the the sum of two numbers in a subset can't be the value of a number ...
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1answer
24 views

Solving problems involving powers

How to reach from $1+𝐸𝐴𝑅= [1+𝑇×𝐴𝑃𝑅]^1/​t $ the power is (1/T) to $$APR = \frac{\ (1+EAR)^T - 1 \ }{T}$$ enter image description here and the same goes here: $Var(aX) + Var(bY) + ...
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1answer
76 views

Solutions of diophantine eq: $x^4-2x^3y+2xy^3+y^4=2s^2$

I'm examining solutions of this diophantine equation: $$x^4-2x^3y+2xy^3+y^4=2s^2$$ It looks like all the solutions are of the form $(x,y,s) = (t,\pm t, \pm t^2)$ where $t$ is any integer. But how do ...
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0answers
22 views

Solving a problem involving powers

How to reach from 1+𝐸𝐴𝑅= (1+𝑇×𝐴𝑃𝑅)^1/𝑇 to 𝐴𝑃𝑅= (1+𝐸𝐴𝑅)^𝑇−1 /𝑇 enter image description here as simple as possible, regards.
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1answer
120 views

Who is a mathematician? [closed]

My first question in Math SE. Basically the question itself, who is a mathematician? Is it someone who solves problems on his leisure time or as a part of a job or even as a hobby? who researches ...
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1answer
62 views

Diophantine equation: $y^2=1+12x+16x^2$

The diophantine equation $$y^2=1+12x+16x^2$$ only has solutions $x=0, y=\pm1$ according to wolfram alpha. How would I go about proving these are the only solutions? Similarly the equation ...
0
votes
1answer
23 views

Showing S is a subset of A by structural induction.

I have a problem similar to: Let S defined recursively by (1) 5 ∈ S and (2) if s ∈ S and t ∈ S, then st ∈ S. Let A = {5^i| i ∈ Z+}. prove that S ⊆ A by structural induction. I've only done ...
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1answer
47 views

Series expansion at infinity

I am trying to find to generalize the limit that involves all rational functions such as $\sum_{n=0}^{l}\frac{{a}_{n}{x}^{n}}{{b}_{n}{x}^{n}}$. I believe the best way of generalizing all of them is ...
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0answers
13 views

Potential Reward Average

I'll try to get straight to the point. I'll be offering a reward system as part of a marketing scheme and it will be for a select amount of users who opt-in. What will be calculated is the overall ...
0
votes
1answer
18 views

Invariance Dealing with Infected Squares

Twelve 1x1 cells of a 10x10 square are infected. Two cells are called neighbors if they share at least one vertex (thus an inner cell has 8 neighbors). In one unit time, the cells with at least four ...
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 ...