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Questions tagged [problem-solving]

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.

-2
votes
2answers
18 views

How many ways to choose two scoops of ice cream from five available flavors? [on hold]

Let's say you had 5 different flavors of ice cream. How many different ways can you put 2 scoops into a bowl if you can't use a flavor twice, and if the order of it can only be used once (ex: vanilla +...
-1
votes
0answers
23 views

Pigeon Hole Principle Question [duplicate]

Prove that from any set of one hundred whole numbers, one can choose either one number which is divisible by 100, or several numbers whose sum is divisible by 100. How to go about this? I tried ...
0
votes
2answers
21 views

Ice Cream Combinations [on hold]

Lets say you had $5$ different flavors of ice cream. How many different ways can you put $3$ scoops into a bowl if you can't use a flavor twice and if the order of it can only be used once. ex: ...
3
votes
2answers
87 views

Solving $8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$

Solve the equation $$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$ I have this idea: set $$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$ So $$x-1=b^2 , 2a^2+6b^2 =8b-4$$ and $$x^2+x-2 =a^2b^2$$ and ...
1
vote
1answer
48 views

Power tower last digits

Can anyone solve the following problem of finding the 6th last digit from the right of the decimal representation of the following number: $6^{6^{6^{6^{6^{6}}}}}$ Essentially it means reducing this ...
0
votes
2answers
47 views

How would you define a superproblem?

Assuming that we have a problem, how would one classify its superproblem? There could be multiple ways to generalize a problem and for example two superproblems (although including the original as a ...
-1
votes
1answer
28 views

Speed, Distance and Time [closed]

Josh cycles 2 km at y km/h and then runs 4 km at (y-4) km/h. The whole journey takes 40 min. Write an equation in y and show that it simplifies to $y^2$ – 13y + 12 = 0
0
votes
1answer
18 views

How do we find the integer corresponding to this probability problem?

There are $n$ socks, $3$ of which are red, in a drawer. What is the value of $n$ if, when $2$ of the socks are chosen randomly, the probability that they are both red is $50\%$? MY ATTEMPT ...
1
vote
2answers
93 views

Calculate the probability with a finite arithmetic progression

We have a finite arithmetic progression $ a_n $, where $ n \geq 3 $ and its $r\neq 0 $. We draw three different numbers. We have to calculate the probability, that these three numbers in the order ...
5
votes
3answers
2k views

What is the mathematical meaning of this question?

$a,b,c \in\mathbb{Z}$ and $x\in\mathbb{R}$, then the following expression is always true: $$(x-a)(x-6)+3=(x+b)(x+c)$$ Find the sum of all possible values of $b$. A) $-8$ B) $-12$ ...
0
votes
1answer
52 views

How many $10-$digit numbers are divided by $11.111$ and all the digits are different?

The Problem: How many $10-$digit numbers are divided by $11.111$ and all the digits are different? A) $3250$ B) $3456$ C) $3624$ D) $3842$ E) $4020$ The Problematic point ...
0
votes
1answer
51 views

Is there something wrong with brackets? $f(2x+(f(y)+f(f(y))=4x+8y$ [closed]

$ x,y \in\mathbb{R}$ and $f:\mathbb{R} \rightarrow \mathbb{R}$, find a function that, $$f(2x+(f(y)+f(f(y))=4x+8y$$ A) $f(x)=2^x$ B) $f(x)=2x$ C) $f(x)=2^x-3$ D) $f(x)=2x^2-3$ ...
2
votes
1answer
98 views

Prove that $3^{30} \equiv 1 + 17 \cdot 31 \pmod{31^{2}}$.

I am reading "An introduction to algebraic systems" by Kazuo Matsuzaka. There is the following problen without a solution in the book. I guess this problem is easy, but I cannot solve it. Prove ...
0
votes
2answers
39 views

two points and an equation

So I think this is the question, I forget how its worded exactly, so bear with me. Suppose we have an equation $5x^3 + bx^2 + cx + d$ cirve that passes through the points $(0,0)$ and $(2,0)$ and the ...
0
votes
3answers
42 views
0
votes
1answer
17 views

Equation/Algebra

Ethan, Mcdonald and Willie earns $\$51700$ a month. Ethan earns $\$400$ less than Mcdonald and Willie earns $\$3000$ more than Ethan. How much do they all earn each? Mcdonald's earnings is $X$. ...
0
votes
2answers
38 views

Equation to locate a square in a square

Good evening, I have been experimenting with different Sudoku checker and have come across a problem: For a nxn Sudoku where n is a square number (4,6,19,25 etcc.), there would be an n number of ...
3
votes
2answers
126 views

Problem of rooms

A rectangle is divided into some smaller rectangles.Each two adjacent rectangles share a door which connects them.Prove that we can start from one of the small rectangles and pass them all without ...
0
votes
1answer
26 views

Solving intercept from an equation

I am confused to solve (a) in this equation. $$y=x^2/(a+bx)^2$$ What I got is: $$a=(x-bx)/(sqrt(y)).$$ Is that right or not because when I use this equation by substituting numbers of the ...
3
votes
0answers
44 views

About a problem of field extension in an algebra book by Fumiyuki Terada.

I am reading an algebra book by Fumiyuki Terada. There is the following problem in this book: $E_1, E_2, K$ are fields. $K$ is a subfield of $E_1$. $K$ is a subfield of $E_2$. $p, q$ are ...
1
vote
3answers
43 views

True or False, diagonalization problem

Let $B_c= \left\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\right\}$, $T:\mathbb{R}^4\rightarrow \mathbb{R}^4$ a linear operator such that \begin{equation} \det\left[T-I\lambda\right]_{B_{c}}=(2-\lambda)^...
-3
votes
1answer
49 views

Rock-Scissors-Paper-Tournament (Hard problem)

"Twelve students of one class have held a rock-scissors-paper tournament. Each competed against each one exactly once. There were two points for a win, one point for a draw and no points for a defeat. ...
1
vote
2answers
75 views

Probability an ace lies behind first ace

Consider a deck of 52 cards. I keep drawing until the first ace appears. I wish to find the probability that the card after is an ace. Now, the method I know leads to the correct answer is that given ...
0
votes
2answers
80 views

How to find minimum point of a hanging rope with two fixed known points $(x_1,y_1)$ and $(x_2,y_2)$ and known length?

I need to find minimum point of hanging rope with two known points $p_1, p_2$ (start and end point of the rope) and known rope length. I want to model all rope shapes with different length and start ...
2
votes
1answer
40 views

Alternate way of looking into a problem

In a book called Applied Statistics and Probability for Engineers (Fourth edition) by Douglas C. Montgomery and George C. Runger there is a problem that reads: "A lot of 100 semiconductor chips ...
0
votes
0answers
52 views

Equality of perturbed Ramanujan's sum and -1/12

Consider the series $\sum_{k=1}^n k e^{-\varepsilon k}\cos(\varepsilon k)$. If one lets $\varepsilon$ be small enough (for example 0.0001) and $n$ large enough (for example 1.000.000), one sees by ...
1
vote
3answers
38 views

Need help with this combinatorics problem i made up

We are $8$ friends and want to make group chats on a messaging app which allows you to join as many group chats as you want. Now there’s $1$ group chat with all of us $8$ in it, and then $8$ more ...
0
votes
3answers
45 views

Solving $p_k=\frac{1}{2}p_{k+1} + \frac{1}{2}p_{k-1}$

I am trying to solve the recurrence relation $$p_k=\frac{1}{2}p_{k+1} + \frac{1}{2}p_{k-1}.$$ The context of this recurrence relation is as follows: if I start with \$20, and I win \$1 for every ...
3
votes
2answers
60 views

How to solve A=2BXB-Diag(BXB)

Given $X,B,A \in \mathbb{R}^{p \times p}$ how do you solve $A=2BXB - \text{Diag}(BXB)$ for $X$? Is there a closed-form? One can assume the following: $A,B$ and $X$ are symmetric positive definite. ...
0
votes
0answers
21 views

Mixing two different essential oils to their maximum safe dilutions in one carrier oil

I want to mix cinnamon bark essential oil (safe at a maximum 0.5% dilution) with tea tree essential oil (safe at a maximum 15% dilution) in shea butter. Shea butter is safe undiluted. The above ...
2
votes
1answer
75 views

Coloured partitions

I have $2n$ elements, $n$ of which are blue and the other $n$ are orange. Other than sharing a colour, they are distinct, i.e. each of those $2n$ objects is different and recognizable from any other. ...
2
votes
5answers
164 views

The “Right” Answer in Mathematics [closed]

I apologize if this question seems too vague for this site. I have a general question regarding how we determine "correct" answers in math. I am currently a college student studying physics/...
9
votes
2answers
167 views

Solutions to $a,\ b,\ c,\ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a} + \frac{c}{b} + \frac{a}{c} \in \mathbb{Z}$

I came across a puzzle in a Maths Calendar I own. Most of them I can do fairly easily, but this one has me stumped, and I was hoping for a hint or solution. The question is: What are the solutions to ...
0
votes
2answers
31 views

Find the unit price of each item which was paid as whole

I am trying to help my daughter in her math and there is this question I can't quite get my head around, The sum is: Three friends go into a book shop. Salma buys a cook book and a novel, she pays \$ ...
0
votes
1answer
34 views

How to: $f(x)$ congruent to $a \pmod{b^n}$

I'm failing to understand the notes we've been given and have struggled to find something on the internet in the form of help. I'm currently stuck on a question for a class. The specific question is ...
2
votes
1answer
35 views

Differentiating an arbitrary vector function with angular momentum operator in quantum mechanics

This is part of a quantum mechanics problem, where the eigenvalues of angular momentum operator in 3D are supposed to be calculated. The 2 particle wave function is given as: $$\Psi(\textbf{r}_\...
1
vote
3answers
99 views

Find all positive integers $a$ and $b$ such that $(1 + a)(8 + b)(a + b) = 27ab$.

Here's the problem I'm having difficulties with: Find all positive integers $a$ and $b$ such that $$(1 + a)(8 + b)(a + b) = 27ab\,.$$ Does anyone have an idea how to do this? Any detailed solution ...
0
votes
0answers
46 views

lower bound of the objective function of a constrained optimization problem

Consider the following constrained optimization problem \begin{align*} &\underset{p,\theta}{\max}~R(p,\theta)\triangleq\frac{qp\theta}{1-q}+p(1-\theta)(1-p)\\ &\mbox{s.t. }~q = 1-\frac{c+\...
3
votes
1answer
55 views

Show that three point $G,H,G_1$ are collinear.

Triangle $ABC$ has centroid $ G$ and orthcenter $H$. Line (through $A$) is perpendicular to $GA$, line (through $B$) is perpendicular to $GB$, line (through $C$) is perpendicular to $GC$ cut at ...
2
votes
1answer
76 views

Right triangle geometry problem

Right triangle $\Delta ABC$ ($\angle ACB=90°$). The following is constructed: from point $C$ altitude $CD$, angle bisector $CL$ of $\angle ACB$, angle bisector $DK$ of $\angle ADC$, angle bisector $DN$...
3
votes
2answers
90 views

Prove that $BH=AH$

A triangle $ABC$ is given. There's a point $P$ inside it and also it is connected to point $H$, which lies on edge $BC$ ($H$ must not be the middle point of edge $BC$). Turns out, that bisector of ...
5
votes
1answer
109 views

Let $\Delta ABC$ be a right triangle. Prove that $\angle BEH=\angle HCI$.

Let $\Delta ABC$ is a right triangle. $D$ is chosen arbitrarily in $AB$,the segment $DH$ is perpendicular to the segment $BC$ at $H$, $E\in AC$ such that $DE=DH$. $I$ is the midpoint of $HE$. Prove ...
4
votes
2answers
78 views

Proving $ \frac{\csc x + \cot x}{\tan x + \sin x} = \cot x\csc x $

I am currently working on understanding trig identities. A question has me stumped, and no matter how I look at it, it never leads to the proof. I believe I am making a mistake when dividing multiple ...
0
votes
1answer
51 views

Petersen graph edge chromatic number

Hi I keep on getting 3 for the edge chromatic number of the Petersen graph. But the Petersen graph has edge chromatic number of 4 and I don’t know how to do that. Can someone please show this by ...
0
votes
0answers
24 views

Solving equation with euler function

Is there a way to solve the following equation for $x$: $x + \frac{1}{1+e^x} = z $ Sure, I can solve it with Newton-Raphson method but I'm looking for a closed-form solution. I'm thinking about to ...
5
votes
2answers
71 views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
1
vote
1answer
73 views

Inscribe an equilateral triangle inside a triangle

Given a triangle ΔABC, how to draw all possible inscribed equilateral triangles with given side whose vertices lie on different sides of ΔABC?
1
vote
1answer
33 views

How to make a kid understand geometry and help him solve problems?

I'm tutoring a 13 year-old boy, a middle school student. He has almost no problem with elementary algebra: he just applies the rules and everything falls into place. However he does struggle with ...
-3
votes
1answer
38 views

how to simplify and solve this equation :

Solve this equation for $x$: $$\left|x \sqrt{1-x^2} +x \right| = \sqrt{1+x^2}$$ I'm having a problem getting rid of the square root!
3
votes
1answer
70 views

Is there anything that non-newtonian calculus can do, which newtonian calculus cannot?

As well, are there problems where non-newtonian calculus leads to a more elegant or simple solution than regular calculus?