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
33 views

Applying invariance principle on a problem on sequence of positive integers

The problem statement: Start with the positive integers 1,...,4n-1. In one move you may replace any two integers by their difference. Prove that an even integer ...
2
votes
3answers
110 views

How to solve an irrational equation?

I want to solve this equation $$2 (x-2) \sqrt{5-x^2}+(x+1)\sqrt{5+x^2} = 7 x-5.$$ I tried The given equation equavalent to $$2 (x-2) (\sqrt{5-x^2}-2)+(x+1)(\sqrt{5+x^2}- 3)=0$$ or $$(x-2)(x+1)\left ...
2
votes
1answer
26 views

Find all sets of N addends equal to a given total W

How many distinct combinations of N natural numbers sum to a given natural number W? For example; for $W=16, N=4$ two of the combinations are $(4,4,4,4)$ and $(5,4,4,3)$ Note: Combination not ...
1
vote
1answer
40 views

Sum of $n$ positive real numbers is 1. Estimate subsums of k elements.

Sum of $n$ positive real numbers $a_1, ...,a_n$ is $1$. Let $S_k$ be maximal sum of k distinct elements of $a_n$. (they can be equal but must have different indexes). What is $\sup S_k$ and $\inf S_k$ ...
2
votes
3answers
88 views

Applying trigonometry in solving quintic polynomials?

So I came across the unsolvable quintic polynomial noticing that solutions can be found by connections with ellipses and such here. But more importantly, I was considering methods we use (or at least ...
0
votes
1answer
60 views

Clock Problem, Number of Chimes

An old fashioned clock chimes as many times as the number of hours it is when it hits a new hour. For example, the clock ticks two times when the clock reads two or the clock ticks 12 times when the ...
8
votes
2answers
95 views

Solution to $e^{e^x}=x$ and other applications of iterated functions?

While trying to solve $e^{e^x}=x$, I ran into the simple solution $x=-W(-1)$. I found it by using the equation $$e^x=x$$Then powering both sides with a base $e$.$$e^{e^x}=e^x$$Now note that the left ...
0
votes
1answer
40 views

Problem solving rolling dice

You are rolling two fair dice, and you are blindfolded, after a certain roll, your partner tells you that you have rolled at least 9. What is the probability that you have rolled at least 11? ...
2
votes
2answers
21 views

Problem involving counting about marbles

Five red cards and four blue cards are blaced in a bag, five cards are selected blind from the bag, what is the probablity that they are all red?
4
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2answers
57 views

The chart-problem; problem solving

In how many ways can we construct a $6\times 6$ chart with only $1$ and $-1$ such that in every row and column, the product is always positive?
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votes
1answer
20 views

Showing properties of the kernal and range of a linear transformation.

Let $T:\mathbb{R}^3\to V$. Show directly that the Ker($T$) is a subspace of $\mathbb{R}^3$ and that dim(Ker($T$)) $\leq 3$. Show that $R(T)$ is a subspace of $V$ and that dim$(R(T)) ...
2
votes
1answer
36 views

Finding the formula for a linear transformation given the transformation of the basis vectors.

Consider the basis $\{\vec{p},\vec{q}\}$ where $\vec{p}=(1,1)$ and $\vec{q}=(-1,0)$. Let $T:\mathbb{R}^2\to\mathbb{R}^2$ be the linear operator such that $T(\vec{p})=(1,-2)$ and ...
2
votes
3answers
70 views

Finding roots of Equation involving trig. functions.

In a problem of classical mechanics, I encounter the following equation: $$\mu \sin^4 \theta + \cos \theta = 0 \qquad \mu > 0 \qquad \frac{\pi}{2} < \theta < \pi,$$ where $\mu$ is some ...
2
votes
2answers
47 views

Determine the angle of 3 drawn lines from each corner of 3 congruent squares

Three squares are drawn next to each other. Three lines are drawn from a corner as illustrated. Determine the sum of the three angles exposed (the exact number of degrees or radians):
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vote
1answer
34 views

Problem solving: How far is the maximum distance?

The tires located on the front of the car wears out after $25000$ km, while the tires on the back wears out after $15000$ km. How far can you maximum ride with new tires if you can swap the tires ...
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votes
5answers
63 views

Christmas problem, the salesman with the nuts [closed]

At the Christmas market, a man was selling nuts in a market stall. The first person bought one nut, the next customer bought two nuts, the next bought four, and so on. That is, every new ...
3
votes
0answers
91 views

Chess tournament problem

$12$ chess players took part in a tournament. Each played against each other exactly once. After the tournament every chess player did $12$ lists of names. On the first list, the player only wrote ...
0
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2answers
83 views

A coin is tossed if a dice is rolled

I was given this question yesterday. A dice is rolled. If the number is even, a coin is tossed. If it is odd, the dice is rolled exactly once again and results are recorded. Find the probability ...
1
vote
1answer
95 views

Prove that 012345678910111213 etc is not a periodic sequence.

Prove that the sequence $012345678910111213...$ (all non-negative integers written one by one in natural order) is not periodic. I want to know the shortest and most elegant way to prove it. Can you ...
3
votes
0answers
32 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 ...
2
votes
1answer
64 views

Solve matrix vector equation

Let $A$ be a real $n\times n$ matrix and $w,x$ real $n\times 1$ vectors. For fixed $A$ and $w$ solve the following for $x$: $(x^\top A x)w - (x^\top w) (A+A^\top) x = 0$ Any hints? I do not really ...
1
vote
1answer
65 views

Optimal strategy for unlocking Cho'gall (probability intuition question)

Right now there is an event occurring in Heroes of the Storm where a special hero (Cho'gall) is unlocked if you play with another player currently playing that hero. I ran into a bit of an intuition ...
1
vote
1answer
46 views

How do we integrate $xe^{x^2}$ in this differential equation?

Yeah I did try searching how to integrate $e^{x^2}$ and mostly I stumbled upon how a similar but not this function called Gaussian function $e^{-x^2}$ is un-integrable , now I was given to solve a ...
0
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0answers
38 views

Solutions of diophantine equation: $s^2 = (ad)^2+ (bc-ad+4ac)^2$

Given diophantine equation: $$s^2 = (ad)^2 + (bc-ad+4ac)^2$$ $s,a,b,c,d$ are all variables. They are all odd. a and b are coprime. c and d are coprime. How do you parametrize all the solutions? ...
1
vote
1answer
69 views

Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$

Given this diophantine equation: $$16r^4+112r^3+200r^2-112r+16=s^2$$ Wolfram alpha says the only solutions are $(r,s)=(0,\pm4)$ How would one prove these are the only solutions? Thanks.
1
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0answers
34 views

Proof for a periodic function

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. ...
0
votes
0answers
10 views

Point set in affine euclidian planes

Let $\cal{P}$ be an affine euclidian plane, $F_1$ and $F_2$ two points of $\cal{P}$. We consider the following set: $\cal{H}$ = $\{M \in \mathcal{P} \ |\ |MF_1 - MF_2| = F_1F_2\}$ I need to ...
0
votes
1answer
44 views

How to solve the quadratic form

I am a physicist and I have a problem solving this \begin{equation} Q(x)=\frac{1}{2}(x,Ax)+(b,x)+c \end{equation} In a book it says that: "The minimum of Q lies at $\bar{x}=-A^{-1}b$ and ...
0
votes
1answer
33 views

How can I use the solve() function inside of itself?

I'm trying to use the solve function recursively on my TI-89 calculator. Minimal example to demonstrate the concept: ...
0
votes
1answer
15 views

Mandelbrot set, inequality proof

If I have the relation $z_{n+1} = z_{n}^2 + c$. How can I show that $|z_{n+1}| > k |z_n|$ for some $k>1$, if $|z_n| > |c| > 2$? I have no idea how to proof this, any help will be good.
1
vote
2answers
29 views

There exist fractal with similarity dimension between 0 an 1?

How to prove that there exist a fractal with similarity dimension D = x, where x is between 0 and 1?
4
votes
1answer
95 views

Winning Strategy with Addition to X=0

Problem: Two players play the following game. Initially, X=0. The players take turns adding any number between 1 and 10 (inclusive) to X. The game ends when X reaches 100. The player who reaches 100 ...
1
vote
1answer
25 views

Game Dealing with Multiplication and Winning Strategy

Two players play the following game. Initially X=1. The players take turns multiplying X by any whole number from 2 to 9 (inclusive). The player who first names a number greater than 1000 wins. Which, ...
1
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2answers
192 views

Determine if a 4-tuple exists

Starting with 2,0,0,3, we construct the sequence 2,0,0,3,5,8,6,..., where each new digit is the mod10 sum of the preceding four terms. Will the 4-tuple 0,4,0,7 ever occur? Any help is greatly ...
1
vote
1answer
36 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) ...
1
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1answer
17 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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
98 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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vote
2answers
34 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 ...
0
votes
4answers
58 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.
0
votes
3answers
39 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?
1
vote
2answers
69 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
74 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 ...
1
vote
3answers
47 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 = ...
2
votes
2answers
28 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 ...
0
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0answers
63 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
votes
1answer
38 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
votes
2answers
70 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
votes
3answers
53 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 = ...
5
votes
1answer
51 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 ...