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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2
votes
5answers
154 views

Find $x$ such that $\arctan(3/2)+\dots=\arctan x$

Find $x$ such that $$\arctan(3/2) + \arctan(5/4) + \arctan(-5/2) + \arctan(-8/3) = \arctan x.$$
5
votes
2answers
276 views

Problem understanding math

sorry for my english but I am a foreigner. I'm writing to ask you for help with a problem I have with mathematics. I'm going to go to university physics but I have serious problems as regards ...
2
votes
3answers
501 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
1
vote
2answers
56 views

elementary prove thru induction - dumb stumbling

i am trying to prove this statement for all $n \in \mathbb{N}$ with the help of induction: $4 \sum_{k=1}^{n} (-1)^kk=(-1)^n(2n+1)-1$ base case: n=1 $4 \sum_{k=1}^{1} (-1)^11=-4=(-1)^1(2*1+1)-1$ .. ...
0
votes
1answer
244 views

Uniqueness of minimum and maximum of set

i am stuck in this simple but foggy problem. i need to prove or show that the min and sup are unique if they exist. $A$ is a nonempty set and $B$ is nonempty subset of $A$. i am trying to show that ...
0
votes
1answer
43 views

monotoneness - prove the elementary statement

Let $A, B, C$ be nonempty sets with total order. and let $f\colon A \rightarrow B$ and $g\colon B \rightarrow C$ be maps. Prove these statements: a) If $f, g$ are antitone, then $g \circ f$ is ...
0
votes
1answer
106 views

Question on probability

This is a Math question I read and I badly want to know the answer. It says; There are twelve people who own 12 phones.One of them(who owns the most valuable phone) suggested to write down their names ...
22
votes
7answers
1k views

Sum of the sum of the sum of the first $n$ natural numbers

I have here another problem of mine, which I couldn't manage to solve. Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n \\ z_n = y_1 + y_2 + \dots + y_n $$ Find ...
9
votes
2answers
476 views

Evaluating $\int_{0}^{x} e^t \sqrt{2 + \sin(2t)} \, dt$

I was recently asked to evaluate the following integral: $$\int_0^x e^t \sqrt{2 + \sin(2t)} \, dt$$ It was beyond the ken of WolframAlpha, which I find quite discouraging. Does anyone have an idea ...
2
votes
2answers
223 views

How to solve generic algebraic problem using solver/library programmatically? Matlab, Mathematica, Wolfram etc?

I'm trying to build an algebra trainer for students. I want to construct a representative problem, define constraints and relationships on the parameters, and then generate a bunch of Latex formatted ...
1
vote
2answers
739 views

Factorial Word Problem

Here is the problem: "The traditional French greeting is to kiss the other person on each cheek. So, when two traditional French friends meet, one kisses the second on both cheeks and the second ...
178
votes
16answers
7k views

Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
1
vote
0answers
65 views

expiring rewards points calculation

Customers can receive (reward) points and use them for something. Each package of rewards points has a different value/size (number of points) and an expiration date sometime in the future which can ...
7
votes
2answers
276 views

Mean and Median in a Classic River Crossing Problem

Consider the following classic problem: Four people on the west side of a river wish to use their single boat to get to the east side of a river. Each boat ride can hold at most two people, and the ...
2
votes
1answer
50 views

Is such a conference possible for $n\ge 1$?

There is a conference with $2n+1$ meetings between $6n+1$ professors. The tables in the conference room are round such that one has $4$ seats and the remaining $n$ tables have $6$ seats each. We know ...
4
votes
4answers
331 views

Mathematical proof for long-term behavior of a sequence of integer vectors

There are some children sitting around a round table. Each child is given an even amount of $1$-cent coins ($0$ is even) by their teacher, all the children at once. A child will give half his money to ...
17
votes
5answers
5k views

How to Improve Mathematical Thinking and General Problem Solving Skills?

I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving ...
1
vote
1answer
389 views

Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
0
votes
4answers
118 views

How to solve the equation?

I want to solve this equation $$\sin\dfrac{(x+1)\pi}{4x^2 -4x + 2} = \cos\dfrac{(x-2)\pi}{4x^2 -4x + 2}.$$ The equation has solutions $x = 1$, $x = \dfrac{1 \pm \sqrt{5}}{2}$ and $x = \dfrac{5\pm ...
3
votes
0answers
272 views

How to deal with a mathematical problem if I don't know the answer?

The problem is I'm looking on shortest path between points problem and the intuition tells me that the shortest path between points happens when paths don't cross. It's a step one. Then for all ...
1
vote
1answer
113 views

solve for $f(x)$ in $4\lfloor\frac{x}{3}\rfloor - f(x) - 3\lfloor\frac{x-3f(x)}{9}\rfloor=x$

Given that $x$ is an integer in the interval $[0,8]$ and $f(x)$ is also an integer in the interval $[0,2]$, I have the following equality: $$4\lfloor\frac{x}{3}\rfloor - f(x) - ...
4
votes
2answers
379 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
4
votes
0answers
466 views

Conditional expectation of the product of two independent random variables

Suppose that $a$ and $b$ are independently distributed random variables, with means; $\mu_a$, $\mu_b$ and variances; $\sigma_{a}^2$, $\sigma_b^2$, respectively. Further, let $c=ab + e$, where $e$ is ...
2
votes
2answers
172 views

Find expectation of conditional random variables

Let $X$ and $Y$ be independent exponential random variables with means 1 and 2 respectively. Let $Z = 2X + Y$. How can I find $E(X|Z)$?
0
votes
1answer
286 views

Find normal random variable from uniform

Suppose $U$ and $V$ are independent random variables with density $f(u)$ and $g(v)$ respectively. The domain of $U$ is the interval $(0, 1)$ and the domain of $V$ is $v > 0$. After the ...
1
vote
1answer
245 views

Find joint density from transformation

I'm trying to learn some tricks for how to find joint densities from transformations, but this problem looks really hard. Any help? Suppose the joint density of continuous random variables $X,Y,Z$ is ...
3
votes
1answer
90 views

Expected number of cards fixed in a deck after a certain operation

Problem: Given a standard deck of $52$ cards, extract 26 of the cards at random in one of the $52 \choose{26}$ possible ways and place them on the top of the deck is the same relative order as they ...
1
vote
1answer
160 views

Related Rates calculus problem

For some reason I keep getting this question wrong. Suppose a 6 feet tall man is walking away from a 15 foot tall lamp post at 5ft/s. What is the rate at which the man's shadow is moving when he is ...
1
vote
2answers
269 views

How to solve this trigonometric equation?

I want to solve the equation $$(3-\cos 4x)(\sin x - \cos x)=2.$$ I solve by putting $t = \sin x - \cos x$, but I can not find all solutions.
1
vote
1answer
98 views

uncertainty in solving constraint problem

There is a class of problems that I need to investigate the constraints of the variables in the problem and build a linear system to solve it. I find that I have no "feeling" about whether the ...
1
vote
1answer
332 views

Mathematics of Tetris 2.0

Based on the question The Mathematics of Tetris, I was wondering if it is possible to have a series of tetris blocks that is impossible to clear. For example, getting the string TTTSS.. forces the ...
1
vote
2answers
74 views

simple Inequality.

Prove: if $x \in (0,1)$ then: $$ \sqrt{\frac{1-x}{1+x}} < \frac{\ln{(1+x)}}{\sin^{-1}x} < 1$$ I have been thinking on using maximization to solve this problem, but the function is not nice! Is ...
1
vote
2answers
107 views

Why $e=g \pm \sqrt{g}$ when $ \left( g-e \right)^2=g$?

I have a problem that involves finding $e$ such that $\left( g-e \right)^2=g$. Maxima tells me that $e=g \pm \sqrt{g}$, but I can't work on that equation to get this result. Actually, I can't go past ...
2
votes
1answer
282 views

Expected value of randomly distributed projection

Problem: Find the mathematical expectation of the area of the projection of a cube with edge of length 1 onto a plane with an isotropically distributed random direction of projection. Source: ...
1
vote
1answer
51 views

Shorter proof for this?

if $G = T + N$ and $T = N * t$, where $t$ is a $\%$, find $T$ if the only given values are $G$ and $t$. I already has a formula but it needs to compute for $N$ first which is lengthy, is there a ...
0
votes
1answer
905 views

Can this crate have even numbers in all rows and columns?

A milk crate holds 24 bottles in four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate have an even number of bottles in it?
0
votes
2answers
41 views

How to evaluate this formula

How can I evaluate $1/e^{\ln(x)}$? I really don't have experience on this and appreciate if you can explain it to me. Thanks.
1
vote
2answers
173 views

Algebraic expression based on problem sloving statement.

A bus can only have x passenagers seated and 12 passenagers standing. How many passenagers can y buses have? Leaving your answer in an algebraic expression.
2
votes
2answers
69 views

Proving two expressions can never be simultaneously satisfied

I have a very simple problem, the technique of which I want to apply to more difficult problems. Here is the simple example version: Suppose we have four functions: $f_1(x)=\sin(x)$, $g_1(x)=0$, and ...
1
vote
1answer
397 views

Algebra: Finding rate

Here Is The Problem. Julia drove from her home to her aunt's house in 3 hours and 30 minutes. The distance between the houses is 175 miles. Knowing that distance = rate X time, find that car's ...
2
votes
1answer
537 views

problem solving with Conditional Probability

Team $A$ facing Team $B$. Team $A$ Hosts the first and Second and if needed fifth and seventh games in the best of seven contest. Team $A$ has a 65% chance of beating Team $B$ at home in the first ...
6
votes
7answers
233 views

Solving $1.1^n = n^{100}$

How do I go about solving for $n$ in the following equation: $$(1.1)^n = n^{100}$$ A hint suffices.
1
vote
1answer
463 views

Help understanding confusing present value question (Theory of Interest)

I'm currently enrolled in a theory of interest class that covers the same material that's on exam 2/FM. At the instructors recommendation, I'm working through all of the problems in the book on my own ...
1
vote
2answers
172 views

Independence of two variables

I ran into some problems while doing an exercise. The problem goes as follows: Suppose we have two random independent variables $X$ and $Y$. Both are distributed normally with parameters $(0, ...
6
votes
5answers
2k views

Show me some pigeonhole problems [closed]

I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve. I would be thankful if ...
11
votes
2answers
370 views

Literature on Mathematical Problem Posing

I know there is a fair bit of literature on mathematical problem solving (e.g., Polya, Schoenfeld). I am wondering if anyone can direct me toward good sources on mathematical problem posing. More ...
2
votes
1answer
834 views

Calculating percentage

I'm wondering if my answer for C would be correct and if I was understanding A correctly? I heard drawing venn diagram could help but I'm not sure how to convert the numbers to a diagram. A ...
3
votes
1answer
60 views

What is the method of solving an equation set in which the unknowns are in denominator?

$ \frac{a_1}{x} + \frac{b_1}{y} + \frac{c_1}{z} = d_1 \\ \frac{a_2}{x} + \frac{b_2}{y} + \frac{c_2}{z} = d_2 \\ \frac{a_3}{x} + \frac{b_3}{y} + \frac{c_3}{z} = d_3 $ $ a_i, b_i, c_i, d_i $: known ...
3
votes
2answers
2k views

Particular solution to a 3rd order ode (method of undetermined coefficients not working)

I am trying to get the particular solution to the equation - $y''' + 4y'' + 5y' + 2y = e^{-t} $ We are taught the method of undetermined coefficients to solve such equations. However, one of the ...
2
votes
0answers
182 views

Sought-Given-Solution-Answers

I am a German student attending lectures delivered in English at the KTH Stockholm. There I am supposed to solve a problem sheet, using a "Sought-Given-Solution-Answer approach". I am not really into ...