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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.

3
votes
2answers
37 views

Solve the equation: $x-a^2x-\frac{b^2}{b^2-x^2}+a=\frac{x^2}{x^2-b^2}$

Here is the equation: $$x-a^2x-\frac{b^2}{b^2-x^2}+a=\frac{x^2}{x^2-b^2}$$ The equation looks very simple. But, it's a little misleading (for me). $$x-a^2x-\frac{b^2}{b^2-x^2}+a=\frac{x^2}{x^2-b^...
0
votes
1answer
26 views

How to allocate these fractions? [on hold]

I'm building a game that uses a virtual rock-paper-scissor system. The rock-paper-scissor game appears in random for both players simultaneously. I want to add a leveling system for the players. 20 ...
3
votes
1answer
38 views

Optimise allocation to minimise variance

Background I am trying to allocate customers $C_i$ to financial advisers $P_j$. Each customer has a policy value $x_i$. I'm assuming that the number of customers ($n$) allocated to each adviser is ...
0
votes
2answers
41 views

A simple but painful calculation.

$$S=(x_3-x_1)^2+(x_4-x_2)^2+(x_5-x_3)^2+(x_1-x_4)^2+(x_2-x_5)^2$$. $$S'=(x_3-x_1)^2+(x_4+x_2)^2+(x_5-x_2-x_3)^2+(x_1+x_2-x_4)^2+(-x_2-x_5)^2$$ Find $S$–$S'$? I thought that, I should manipulate the ...
0
votes
0answers
38 views

How to create a function model and equation [on hold]

Hey guys hope you all are having a great day!!!. so I have one question I have been struggling on for now hours on end trying to solve it is a question on function models and equations and I just need ...
2
votes
1answer
22 views

Solve a trigonometric equation where one constant can be negative

I previously posted a code-related questions on StackOverflow here but after doing some more investigation work into the code, I had an epiphany when looking at the actual Newton-Raphson section ...
0
votes
2answers
24 views

Absolute Value Problem Using Only Variables

I recently encountered this problem, and it does not make sense to me. It looks like Given $|a(b-cx)|=d$ , find the value of $|x-\frac{b}{c}|$ This was on a multiple choice test awhile ago, and I ...
1
vote
0answers
49 views

Finding roots for $(1+c) \cos(ax)+(1-c) \cos(bx)=0$

Looking to find the analytic roots of $$(1+c) \cos(ax)+(1-c) \cos(bx)=0$$ where $a$,$b$, and $c>0$. Any ideas? For $c=0$ the answer is straight forward. P.S. The constants $a=A-B$ and $b=A+B$ if ...
0
votes
0answers
21 views

Compounding Formula (and its reverse)

I've written a program in which I'm using a compounding calculator to determine an output. But without a formula I'm having trouble reversing the calculation. Given a compounding problem whose ...
0
votes
3answers
35 views

Word Problem - what percent of the women who apply are hired?

At a certain factory, twice as many men as women apply for work. If 5% of the people who apply are hired and 3% of the men who apply are hired, what percent of the women who apply are hired?
0
votes
1answer
21 views

Find solution to exact ODE: $(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$

So I have this exact ODE $(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$ I already managed to show that $F=2bxy^2+3xy+5y+x^3+4ax^2$. Now, how can I find the solution ...
2
votes
3answers
58 views

How to derive this relation that I found intuitively?

By intuition, I found that the result of evaluating the following expression $$ \frac{1}{M} \frac{\sum_{N=0}^M \frac{M!}{(M-N)!N!} N e^{cN}}{\sum_{N=0}^M \frac{M!}{(M-N)!N!} e^{cN}} $$ does not ...
0
votes
1answer
27 views

Word problem - How many times was the player at bat during the entire season?

Its the first time I am asking a question on a forum. I have been self-studying math for a month and now am stuck in word problem chapter. I searched a lot for a solution manual, but I didn't find any....
0
votes
0answers
35 views

Puzzles and exercises to improve mathematical intelligence and spatial thinking

In your childhood or adolescence, or maybe as an adult, have there been types of exercises or puzzles that you think have improved your mathematical intelligence and in particular the spatial thinking?...
1
vote
1answer
29 views

Word problem - Problem Solving

My kid is practicing word problem. I have an issue with the question which i m unable to produce any solution. Would be appreciated if any solution is provided. QUESTION: In the village there were ...
2
votes
1answer
72 views

Solving a set of multivariate polynomials; four equations, four unknowns, maximum degree 2

I am currently trying to solve a set of four polynomial expressions for a geometric purpose. in short they take on this form: $A_0=\frac{w_0^2+w_1 w_0+w_2 w_0+w_3 w_0+27 w_0+6 w_1+6 w_2+6 w_3+117}{\...
1
vote
1answer
28 views

Minimum quadratic formula value

I have recently been researching about quadratic equations, and came across the following formula for finding the minimum $x$ and $y$ values from a "Bivariate (two equation) quadratic": $$x = -(2BC-...
2
votes
3answers
86 views

Congruence equations in the form $x^k \equiv b \bmod m$

In my textbook its described as finding $k\;$th roots moduluo $m$. How do you solve equations of this form as I can't find any examples anywhere? I am just looking for a step by step example so I can ...
0
votes
2answers
46 views

Solve for X and Y

I have the following equation: $0 = 34x^2+92xy+68y^2−250x−344y+461$ I cannot find any way to get the values of both x and y from this equation, any help would be much appreciated, especially a step ...
5
votes
1answer
65 views

Cars park randomly next to a sidewalk of length 100. Each car is length 1. How many cars will fit on average? [duplicate]

I'm looking for a mathematical solution to this problem, not an answer through a simulation. Clarification of the problem: $x$ is a random number (not only integers) uniformly distributed between $0$ ...
3
votes
1answer
51 views

Cryptogram: $XYZ\div8 = ZX$, remainder $Y$

Say we have the division algorithm Where X,Y,Z represent a non-zero digit and the remainder is Y. What is the three-digit number XYZ? From what I gather, I re-arranged the division into an equation:...
1
vote
1answer
44 views

How to prove that such measure does not exist on an infinite product space?

A previous (already submitted) exercise from my coursework was, Prove that there does not exist a probability measure $ \mathbb{P} $ on the infinite product space $ (\mathbb{R}^{[0,\infty]}, > ...
1
vote
1answer
27 views

IMC iteration series question

A question featured on a practice test reads: An infinite increasing list of numbers, has the property that the median of the first $n$ terms equals the $n$th odd positive integer. How many numbers ...
0
votes
1answer
9 views

Trying to re-write the formula for the present value of an annuity to solve for annuity value

I have the formula: $$PV = C * {{(1-(1+r)^{-n})} \over r}$$ This is the formula for the Present Value (PV) of an Annuity (C) with interest (r) (for example 5% interest is 0.05) over (n) periods. I ...
0
votes
1answer
37 views

Sort two list of jobs result in minimum execution time

Two twins, James Johnson and Jonathan Johnson, work at a factory which produces bicycles. Their job is probably the most important one: James attaches the front wheel, and Jonathan does the same with ...
0
votes
1answer
27 views

Getting probailities for two events

When producing something while having a chance of 4% for getting a bad item I can calculate the chances for having 2 bad items by just using the Bernoulli formula $$ P(x=k) = ( \begin{matrix} n \\ k ...
1
vote
1answer
31 views

How do the values of $a$ affect definiteness of this matrix $\textbf{A}$?

Let $$\textbf{A} = \begin{bmatrix} -2 & 0 & 1 \\ 0 & -2 & a \\ 1 & a & -2\end{bmatrix}$$ Given that one of its eigenvalues is equal to $-2$, how does its definiteness vary ...
6
votes
1answer
107 views

What is the correct way to solve the equation: $x^4-x^3+x^2-x+1=0$

Given the equation: $x^4-x^3+x^2-x+1=0$ we need to find both its real and complex roots. What is the easiest and correct method for solving the equation? Here is my approach, but it gives wrong ...
1
vote
3answers
61 views

puzzle on three man drinking wine

There is puzzle solution of which doesn't click for me. One person has $5$ bottles of wine, another one $3$ bottles. There is also third person. Together all three drank this $8$ bottles of wine ...
0
votes
5answers
49 views

Efficient method for solving simultaneous equation?

so I spent a surprising amount of time trying to solve something as trivial looking as that picture ^^, and while I got there in the end, I was wondering if there is a way to solve such simultaneous ...
0
votes
1answer
22 views

Problem: How many 'timetables' can be created for visiting 8 different lectures without time clashes?

Apologies for the poorly worded question but it really requires more explanation than can be done with one line. I have a statistics problem, or rather, a timetabling problem that I feel can be solved ...
-1
votes
1answer
42 views

Question on Function of function polynomial [closed]

If $f(x)=x^3-12x^2+Ax+B>0$ $f(f(f(3)))=3$, $f(f(f(f(4))))=4$ then what is the value of $f(7)$
0
votes
4answers
84 views

What is the answer to $17^{16} \pmod {10}$? Is it equal to $ 9 $ or $1$

I encounter a modular arithmetic problem. which says: Find the last Digit of $17^{16}$, by intuition the last digit of a number is the remainder of the number divided by $10$. so the statement is: $...
13
votes
2answers
2k views

Why aren't these two solutions equivalent? Combinatorics problem

I was given the following fact: there is a set $S$ of $11$ people, among which there are $4$ professors and $7$ students, $S=\{p_1, p_2, p_3,p_4, s_1, s_2,...,s_7\}$ We are requested to form from ...
-3
votes
1answer
91 views

How many people are there when the word “with” is used? [closed]

How many people are there is the following statement: "Bob with his family of 4..." Is it: a total of 4 (Bob + 3 members) or a total of 5 (Bob + 4 members) ?
0
votes
1answer
84 views

A problem in single variable calculus

Given a function $a(t)$, and the condition $$\dot{a}^2 \geq 1 - \frac{a^2}{L^2} $$ and $$a(t) \leq L $$ I need to prove that if there exists a $t_0$ such that $\dot{a}(t_0)<0$ and $a(t_0)>0$, $a(...
0
votes
0answers
54 views

A question from a past BMO exam, on modulo [duplicate]

If the last three digits of the number $7^{400}$ are 001 find the last three digits of the number $7^{9999}$. Guys, I tried working out this question today, however to no success. I tried using ...
0
votes
3answers
62 views

If $a^3-b^3\equiv 0\bmod3$ prove that $a^3-b^3\equiv 0\bmod 9$ [closed]

If $a^3-b^3\equiv 0\bmod3$ prove that $a^3-b^3\equiv 0\bmod 9$. Guys, this question appeared in a preparation leaflet for a national exam, which I recently did. However, I was incapable of doing it. ...
0
votes
1answer
36 views

A question invoving combinatorics and many limitations [closed]

Three kids, Andrew, Basil and George, were listening to four different songs. None of the four songs was appreciated, by all three kids. For each of the three possible pairs of kids (Andrew and Basil ...
1
vote
1answer
12 views

Proportionally distribute a lump sum between a maximum and a minimum

I have to distribute a number, x, to a certain number of people, n. I know that I want to distribute the maximum, a, to the top person, and the minimum, b, to the lowest person. My question is, what ...
-2
votes
4answers
121 views

proving that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$ [closed]

Knowing that $a+b=c+d$ and $a^3+b^3=c^3+d^3$, prove that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$. Can you guys please help me complete this proof, as I was attempting it yesterday, without getting far. ...
0
votes
1answer
54 views

Proving the independence of an equation, from certain variables [closed]

The different and unequal to zero real numbers x, y, z satisfy the equation $$x^3+y^3+m(x+y)=y^3+z^3+m(y+z)=z^3+x^3+m(z+x).$$ Prove that $$K=\left(\frac{x-y}{z}+\frac{y-z}{x}+\frac{z-x}{y}\right)\...
1
vote
2answers
67 views

A question for the preparation of internationals [closed]

Given the real numbers $a$ and $b$, for which it is true that $$a^3+b^3+3ab=1$$, evaluate $a+b$. I tried working this question out, by factorizing, but I didn't manage to reach a conclusion, which ...
2
votes
5answers
138 views

Solving $x^2-2x-3<0$

If i have to solve $x^2-2x-3<0$ I would do $$x+1 < 0, \quad x-3<0$$ and end up getting $x<-1$ and $x<3$. Why is it wrong to use the same inequality sign? Shouldn’t both of the ...
0
votes
1answer
34 views

A question on gaming theory with variables [closed]

We have two stacks of coins, each with a and b coins respectively. On each step, the player is allowed to remove as many coins as they want(but at least one) from either of the two stacks he wants to (...
0
votes
1answer
40 views

Question for preparation for the IMC internationals [closed]

Work out the addition of: (1/(1+2))+(1/(1+2+3))+...+(1/(1+2+3+...+51)) Guys, I'm having difficulties working this one out. Using a calculator, it would have been easy, however in the internationals ...
1
vote
3answers
54 views

How to solve exponential equation of $x + e^{-x }= 3$?

I've got the following equation: $\lambda + e^{-\lambda} = 3$ $3 - \lambda = \frac{1}{e^{\lambda}}$ $\frac{1}{3-\lambda} = e^{\lambda}$ Now, I take the natural log of both sides: $ln(\frac{1}{...
0
votes
1answer
45 views

Explicit solution to a nonlinear ODE

I hope someone can help me with the following question. Find an explicit solution to the following problem $$\dot{u}(t)=\frac{k_1 u(t)}{u_* + u(t)}-k_2 u(t)$$ with initial condition $u(0)=...
1
vote
2answers
40 views

Problem-solving with ratios and similar triangles.

The following question is designed to test problem-solving and reasoning skills with ratios and similar triangles. Without using Pythagoras Theorem or Trigonometry this question is supposed to be ...
0
votes
0answers
25 views

Differential equation notation about maximal solution

I'm doing the following problem: The differential equation $$ \dot{y} = X(t,y), X(t,y) = \frac{1}{3}y^{1/4} +t^{1/3}$$ defined on $D_X = (0,\infty)\times(0,\infty)$. I already solved it with: $y(t) ...