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
92 views

What is the optimal strategy when playing `head or tail` per team

Introduction Once a week, we are playing head or tail in my favorite bar. There are $N$ people in the room and each person is guessing whether ...
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0answers
36 views

Solving a polynomial with modular arithmetic.

$x^5+ax^4+bx^3+cx^2+dx+e=0$ Assume that $a,b,c,d,e$ are not arbitrary and that they are known. I was wondering if it were possible to reduce or 'simplify' this using some modular arithmetic. It ...
0
votes
1answer
51 views

is A an even number?

Let $a,b,c,d$ be positive integers such that $(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001}$ hence, prove that $a$ is even. I tried to approach this problem reducting it modulo 6. From which we ...
2
votes
4answers
521 views

How long would it take Mustafa to do the job alone? [closed]

Murat and Mustafa can do a job together in fifteen days. After they have worked together for five days, Mustafa leaves the job. Murat completes the job in sixteen days. How long would it take ...
2
votes
2answers
39 views

Solving for $x$ in $x + \frac{s}{s^2+4} = \frac{2x}{s^2+4}$

How would I go about solving for $x$ in this equation? $$x + \frac{s}{s^2+4} = \frac{2x}{s^2+4}$$
1
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1answer
18 views

Is my intuition about this statistics problem sensible?

I'm trying to improve my knowledge of statistics and develop my intuition for solving statistical problems. While doing so I've worked on the following exercise: There are 20 players in a checkers ...
0
votes
1answer
74 views

Related Rate of Cylindrical Cone (Filling + Leaking)

So, we are only told of how to solve related rates with one underlying problem. Either a cone is leaking or Being filled up at some point $x$ but I never encountered both working at the same time. ...
1
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4answers
61 views

Explain the thought process to a given solution for $ \frac{1+n}{2^n} =\frac{3}{16} $ please

Part of the given solution to a question leaves me baffled: "...solve the equation $$ \frac{1+n}{2^n} =\frac{3}{16} $$ We can check (simply by plugging in values of $n$) that if $ \leq n \leq 5$, ...
0
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3answers
89 views

How can I prove this equation has no solution?

Solve the equation $$-x^3 + x + 2 =\sqrt{3x^2 + 4x + 5.}$$ I tried. The equation equavalent to $$\sqrt{3x^2 + 4x + 5} - 2 + x^3 - x=0.$$ $$\dfrac{3x^2+4x+1}{\sqrt{3x^2 + 4x + 5} + 2}+x^3 - x=0.$$ $$\...
0
votes
1answer
37 views

How this type of equation is solved? [closed]

I'm solving a relative and ends when the function (x²+y²)²-4x² derive out this equation, but not that I have to do to get resolve it $$f'x = 4x³+4y²x-8x$$ $$f'y = (4x²+4y⁴)y$$ How solve this ...
1
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0answers
30 views

Simple exercise in differential geometry

Problem: Prove the identity $V=\sum V[x_i]U_i$, where $x_1, x_2, x_3$ are the natural coordinate functions. (Hint: evaluate $V=\sum v_i U_i $ on $x_j$) Elementary differential geometry written by ...
1
vote
4answers
80 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
0
votes
1answer
41 views

There exist three consecutive vertices A, B, C in every convex n-gon with n≥3, such that the circumcircle of triangle ABC covers the whole n-gon

From Problem Solving Strategies by Arthur Engel: Problem to prove: There exist three consecutive vertices $A$, $B$, $C$ in every convex $n$-gon with $n \ge 3$, such that the circumcircle of triangle ...
-1
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2answers
40 views

$n = a^2 + b^2, \; p^3 = a^3+b^3, \; m^3+2p^3 = 3mn$ then prove that $m = a+b$

If $a$ and $b$ both are positive integers then $m = a+b, \;n = a^2 + b^2, \; p^3 = a^3+b^3$ Then show that $ m^3+2p^3 = 3mn$. This is a easy problem to solve. Just substitute those values and ...
0
votes
1answer
16 views

Mixture & Alligation problem

In what ratio must a person mix three kinds of wheat costing him 1.20,1.44 and 1.74 dollars per kg., so that the mixture formed is worth 1.41 dollars per kg? a)11:77:7 b)12:7:7 c)ratio other than a ...
0
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0answers
18 views

Simple ratio problem

I am currently working through some tests for graduate schemes and I am finding the maths tests relatively okay despite the time restrictions. However, I have come across one basic ratio question that ...
4
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5answers
125 views

Explanations for why someone cannot divide by $x-4$ for $x(x-4)=x(x-4)(x-5)$

A student divides both sides by $x-4$ and lost a solution $x=4$. How could you explain to the student that they are not allowed to divide by $x-4$ Here is the problem: $x(x-4)=x(x-4)(x-5)$ I am ...
3
votes
2answers
41 views

What is the speed of the car given the time taken to receive an echo?

I am trying to solve this question- The driver of an engine produced a whistle sound from a distance $800m$ away a hill to which the engine was approaching.The driver heard the echo after $4.5s$....
1
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1answer
47 views

Is there another way to solve this equation?

My problem is Solve the equation $$2 \left(\sqrt{x^3-7 x^2+17x-14}+\sqrt{x^4-7 x^3+23x^2-37 x+28}\right)=4x^2-17 x+25.$$ And my solve. We have $$\sqrt{x^3-7 x^2+17x-14}= \sqrt{(x-2) \left(x^2-5 x+7\...
0
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1answer
62 views

Resources for solving Euclidean geometry problems using symmetries

I know a number of books that treat geometry from the viewpoint of transformations/symmetries. However, very few of them actually teach someone to solve Euclidean geometry problems using said ...
0
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1answer
22 views

Does this System of Complex Variables Has Solutions?

Find all the complex vectors $\mathbf{x}=[x_1,\ldots,x_n]^\top$ and $\mathbf{y}=[y_1,\ldots,y_n]^\top$ in $\mathbb{C}^n$ such that $$ \sum_{i\in S}x_i\bar{y_i}=1,\text{ for all } S\subset\{1,\ldots,n\}...
2
votes
1answer
59 views

Solving diophantine equations

So the equation I am trying to solve is $x^2=y^4-77$ So far I have rearranged and factorised the equation to get: $$(y^2-x)(y^2+x)=77$$ But I am really unsure of how to solve it from here. Thanks in ...
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0answers
28 views

Understanding foundational terms: notions, objects and meta-objects

I am trying to take my problem solving skill to next level. It looks like It takes a lot of mathematical discipline. Here, This post buys me to get better at proof writing. So, I think is useful to ...
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0answers
14 views

How to split out tax components

I am working on a problem where I want to split out different tax rate components. Example: Income $= \$1,230,903.00$ Dividends $= -\$2,456.00$ $1231 -\$5,116.00$ Credits $-\$2,161.00$ Long ...
0
votes
1answer
48 views

Transformation of $\log(X)+\log(1-X)$, where $X$ is uniform.

I am trying to calculate the variance of: $$\log(X)+\log(1-X)$$ where $X \sim \mathrm{Unif}(0,1)$. So far, I have tried to use a random variable transformation, i.e. define $Y=\log(X)+\log(1-X)$, ...
0
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0answers
28 views

Weird behaviour of CLT's application to binomial.

I am carrying out the simulations of the following experiment for all $n$ in the set $\{1,2,3,...,100\}$. (0) Set $k=0$. (1) Generate $n$ $Bernoulli(0.9)$ trials. (2) Construct estimate $\hat\theta=...
1
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1answer
50 views

Solving $x^e =c$ in $\mathbb{F}_{p}$

Find all solutions to the equation $x^3=7$ in $\mathbb{F}_{13},\mathbb{F}_{19}$ and $\mathbb{F}_{35}$. In An Introduction to Mathematical Cryptography (Hoffstein et al), we have that proposition 3.2....
0
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1answer
60 views

Show that one equation equals another (trigonometry)

I'm studying for a test and when going through old exams I find this one which I'm not able to solve. Show that $$a^2 = (b−c)^2 + 4bc \sin^2 \left(\frac A2\right)$$ equals $$a^2 = b^2 + c^2 ...
0
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2answers
81 views

Complex proof - Not sure where to go from here. (homework)

Knowing $2\pi r =\dfrac{h}{m \left(\sqrt{\frac{e^2}{mr}}\right)}$, How do I prove $r = \dfrac{h^2}{((2\pi)^2m e^2)}$? I started by dividing both sides by $2\pi$ to get $r = \dfrac{h}{m\left(\sqrt{\...
0
votes
1answer
16 views

Given an integer $i\in\{1,\ldots,NM\}$, find its place in a matrix of size $N\times M$?

The integers $N$ and $M$ are positive. Given the matrix $\mathbf{A}=\left[\mathrm{a}_{nm}\right]$ defined as follows: $\mathrm{a}_{nm}=m+(n-1)M$ for all $m\in\{1,\ldots,M\}$ and $n\in\{1,\ldots,N\}$. ...
0
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2answers
37 views

How can I solve this nature log equation?

$ln(x+2)=e^{(x-4)}$ Is there any way to solve this equation without graphing or using GDC ? Thank you
0
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1answer
25 views

Recover Marginal Distribution subject to a Constraint

I want to identify the marginal of a normal distribution subject to a restriction. Take two normally distributed random variables $x,y$. Their pdfs are denoted by$\phi(x)$ and $\phi(y)$. The moments ...
0
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0answers
82 views

In a sweepstakes giveaway scenario, how does having 2 chances to win the same prize affect the overall odds?

In a sweepstakes giveaway scenario where total entries are expected to result in final odds of 1:93,150.685 for/against a single entrant (after adjusting for multiple entries) and can be won by either ...
0
votes
1answer
28 views

Finding the number of multiples [closed]

I have recently been doing problem solving in math, and I came across this problem: Determine the number of positive multiples of $6$ or $9$ or both, less than $1000$. I appreciate any help. Thanks!
4
votes
1answer
82 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ \...
4
votes
2answers
46 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim t\...
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1answer
62 views

Counting and Abstract Problem Solving [closed]

Suppose that you have a bucket holds fiv-sev c, and one holds tw-one c. How could you use them to measure out thre c of water?
1
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2answers
62 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
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0answers
49 views

How do I derive the cubic formula? (without substitutions)

I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without ...
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2answers
22 views

Survival bias and probability

Imagine the following situation: A new virus is discovered that is believed to have infected 20% of the population. Anyone infected with the virus has a chance of 50% of dying in their sleep every ...
0
votes
1answer
27 views

How can we solve this system of linear inequalities?

Let $c_i$ be a given non-negative integer for all $i\in\{1,\ldots,n\}$. I would like to find the non-negative integers $a_i$ and $b_i$ for all $i\in\{1,\ldots,n\}$ such that: \begin{align} \begin{...
0
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0answers
23 views

Probabilities of each waitlist person

Coming from this, $10$ Applicants for a exclusive club membership. I found that you can use total probability to consider existing and old members leaving/returning as members in the club, ended up ...
0
votes
1answer
22 views

Mpg to l/100km conversion problem?

We have two vehicles A: old truck that does 17 mpg (13.84 l/100km) B: old car that does 47 mpg (5.00 l/100km) We are looking to replace one of these vehicles with a new one (of the same size)...
0
votes
1answer
17 views

Finding the rate at which the space diagonal of a cube is decreasing

This is the context of the question: I'm assuming by the space diagonal (although I'm not sure) to be the area of the right-angled traingle created by the diagonal. Let this space be $V$, then we ...
2
votes
1answer
73 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
3
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4answers
89 views

How to find unknowns $w_1,w_2,w_3$ that satisfy $t=w_1f_1 + w_2f_2 + w_3f_3$?

For any $i \in \{1,2,3\}$, let: $w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$. $t$ is a known number in $[0,1]$. Suppose that $t = 0.8$. $f_i$ is also a known ...
0
votes
1answer
37 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and continuity ...
3
votes
1answer
44 views

solve $54 x + 16 y = 2400$ for integer values of x,y

How to get integer values for x and y that satisfy: $$54 x + 16 y = 2400$$ Someone told me that I can do it using Euclid-Wallis algorithm, but I don't understand it so, if there isn't any else ...
0
votes
0answers
69 views

Reference request for a very particular problem solving skill

I want to start with an apology for a very verbose description of my question but if there is a way to cut it down, please let know and I will do so right away. I have been trying to get better at ...
0
votes
0answers
51 views

Books or website about solving IMO problems

Hey I want to solve IMO problems like the problem in the image below, but I cannot solve the problem or any of the problems in the IMO, so do you guys have some good website or books that teach how to ...