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.

learn more… | top users | synonyms

28
votes
3answers
684 views

How does one cut onions in a mathematically efficient way?

Perhaps a math degree and cooking don't go hand in hand, but hopefully they do. I have been thinking about this problem for some time when in the kitchen without making any real progress: How does ...
0
votes
1answer
185 views

how to find point where two exponential type functions intersect

I have two functions who intersect each other and i want to find time at which they intersect. The two functions are, $\left(1-\frac{1}{\text{X2}}\right)-\frac{(\text{X1}-1) (\text{X2}-1)}{e^{4 t ...
1
vote
2answers
30 views

how to express this problem as integral

I am given this word problem: Find a straight line which goes through the center of x, y coordinates so that the area between this straight line and graph of $f(x)=x^2$ is exactly $\frac{1}{6}$ I ...
4
votes
2answers
322 views

Motivation and Derivation of the Riccati Equation Transformation

Given a Riccati Equation which is differential equation of the form: $$ \frac{dy}{dx} = a_0 (x) + a_1 (x)y + a_2 (x)y^2 $$ It is well known that the transformation: $$ y = -\frac{1}{a_2(x)} ...
3
votes
4answers
100 views

How to easily prove $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$ [duplicate]

When I tried to solve some certain math problem (an inequation) for pivate exercise purposes, I had to prove that $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$, I solved it with tools from differential ...
3
votes
3answers
320 views

Is this question of sequence a Mathematical one, i.e. does it have objectively only one answer for each subpart.

This question is taken from 11th class Math book. Look at this question: At the very first glance one can tell that all the three sequences are G.P But! by using interpolation(as this answer ...
3
votes
3answers
76 views

Confusing -Probabilities.!!

Ok so far what i understand is this lets say...Having to draw a card from 52card-deck its probability is of course 1/52.Now the probability to say that i will keep drawing this same card 10 times of ...
4
votes
7answers
97 views

How to find $x^2 - x$?

I'm quite a novice when it comes to maths. I'm on a problem in which I have had to isolate $x$ , through factorials which I completed without problem. However, now I am stuck on a seemingly more minor ...
1
vote
0answers
61 views

How many lines needed to not lose in tetris game?

Suppose we play a tetris game with tetris be given randomly. Is there exists a number of lines that we can play infinitely, i.e. do not lose the game?
45
votes
9answers
5k views

When to give up on a hard math problem?

I practice olympiad problems from books like Putnam and Beyond. Often I come across a problem that I simply can't solve. After $\sim30$ minutes of deep thinking it feels like I'm ramming my head into ...
12
votes
4answers
2k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
-6
votes
5answers
665 views

How to solve the sequence: $87, 89, 95, 107, ?, 157$

This question appeared in a competitive exam. The question is: Q. Find the unknown term in $87,89,95,107,?,157$ 1)127 $\ \ \ \ \ \ \ \ $ 2)122 3)139 $\ \ \ \ \ \ \ \ $ ...
3
votes
2answers
335 views

Basic combinatorics question [closed]

In a tennis tournament there are $2n$ participants. In the first round of the tournament, each player plays exactly once, so there are $n$ games. Show that the pairings for the first round can be ...
2
votes
1answer
3k views

How to solve 2 ÷ 2 ÷ 2 ? ${}{}{}{}$

$$2 ÷ 2 ÷ 2 = (2 ÷ 2) ÷ 2 \ \ \text{OR}\ \ 2 ÷ (2 ÷ 2) ?$$ Is there any standard rule which is world wide accepted for solving this type of expressions? If I process the expression from left to ...
1
vote
1answer
70 views

Residue of this function for $z_0=0$

I have this function $$\frac{\sin (2z)-2z}{(1-\cos z)^2}$$ I want to find its residue around $z_0=0$, however I've been battling it for hours but I get nowhere. I've tried finding its Laurent series, ...
1
vote
1answer
67 views

A 20 × 20 × 20 cube is built of 1 × 2 × 2 bricks. Prove that one can pierce it by a needle without piercing a brick.

A 20 × 20 × 20 cube is built of 1 × 2 × 2 bricks. Prove that one can pierce it by a needle without piercing a brick. Taken from Engel's book, but no solution was given. Here's my solution: Look ...
0
votes
1answer
35 views

Find Laurent's series of these two functions around $z_o$

Find the Laurent series of $f(z)=\frac{z}{(z+1)^2}$ around $z_o=-1$, and $g(z)=z\exp(\frac1{z+i})$ around $z_o=-i$. For $f$, what they're asking is to find the series in $0<|z+1|$. On the ...
0
votes
0answers
132 views

Prove solution does not exist for inequalities system

I have an inequalities sytem like the following: Example > x+y+z <= A > x+y <= B > x+z > C > y+z > D > x >= E Let A,B,C,D,E be any ...
0
votes
1answer
39 views

How do I prove this statement?

I have to prove that if $$u=t^{\lambda}y(z)$$ and $$z=\frac{x}{\sqrt{t}} \,\,,$$ then $$\frac{\partial{u}}{\partial{t}}=\frac{\partial ^{2}{u}}{\partial{x}^{2}} \Rightarrow ...
7
votes
2answers
179 views

Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$

Determine all $a\in[0,1]$ such that for ${\it every}$ continuous function $f:[0,1]\to \mathbb{R}$ with $f(0)=f(1)$ there exists at least one $x$ where $f(x) = f(x+a)$. First of all, $a=0,1/2,1$ ...
1
vote
1answer
1k views

Find the Laurent series of $\sin z/z^2$ using Laurent's theorem

I have the function $f(z)=\frac{\sin z}{z^2}$, wich is analytic over $\Bbb C\setminus\{0\}$, I want to find the Laurent series of $f$ valid for $0<|z|<R\le\infty$. Using Laurent's theorem we ...
0
votes
1answer
86 views

More problems like Engel's Problem Solving

I've been working through Arthur Engel's Problem Solving and I've been enjoying it very much. I especially liked Chapters 1, 3, and 4, which cover the Invariant, Extremal, and Pigeonhole Principles ...
1
vote
1answer
83 views

Using Poisson's integral formula

The problem asks to prove the following equality using Poisson's integral formula (or Poisson kernel, if I understood correctly from Wikipedia): $$\int_0^{2\pi} \frac{e^{\cos ...
1
vote
0answers
46 views

Trying to use the deformation theorem to solve integral

I have this integral: $$\int_{|z|=2}\frac{\cosh z}{(z+1)^3(z-1)}dz$$ Both singularities $z=1,z=-1$ are inside the circle. I have already solve this using partial fractions, and I don't have much ...
0
votes
1answer
42 views

How does $2N_{h-2}$ become $2^{h/2}$?

I'm reading the Lecture 6 notes from MIT OCW Introduction to Algorithms, which discusses AVL trees, and I'm confused about one of the relations below: Balance: Worst when every node differs ...
0
votes
0answers
48 views

Proof Verification: If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3.

If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3. The area of any such ring is $5\pi$ and the area covered by the union of all rings ...
2
votes
2answers
84 views

Solve this equation. Can anybody do it?

$$x / y = a + b \log(y)$$ Above is a relation between x and y. I have x. How do I find ...
1
vote
1answer
1k views

How to calculate per unit costs for multiple items

I had a supplier give me a quote last week that seems very strange, can someone help me out? The quote is for IT hardware, but for simplicity (and anonymity) I'll use apples and oranges: ...
5
votes
2answers
261 views

I still forget concepts even after answering numerous math problems

Note: this is particularly aimed at high-school/entry level college problems When I'm learning a new topic: 1) I read the theory given in the textbook at the start of each topic 2) proceed to read ...
2
votes
2answers
8k views

Solving the exact differential equation y'=(x-y)/(x+y)

I need to solve the following exact equation: $y' = \frac{(x-y)}{(x+y)}$ I've been taught to put those in the form $M(x, y)dx + N(x, y)dy = 0$ and to make sure ${dM}\over{dy}$ = ${dN}\over{dx}$ So ...
1
vote
1answer
46 views

Solving $Ae^x=Bx$ analytically, where $A$ and $B$ are constants?

This equation mixes both exponential terms and linear terms, something which I do not know how to deal with. Any pointers?
2
votes
1answer
548 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
6
votes
3answers
160 views

Find all real solutions of $6^x+1=8^x-27^{x-1}$

Find all real solutions of $6^x+1=8^x-27^{x-1}$. Things I tried: We want solutions of $$2^x3^x+1 = (2^x)^3-\frac{(3^x)^3}{27}.$$ Write $a=2^x$ and $b=3^x$. This gives $$ab+1 = a^3-\frac{b^3}{27}$$ or ...
3
votes
1answer
76 views

Solve for “lucky” numbers

A rational number is called "lucky" if it equals both $a+\frac{b}{c}$ and $a\times\frac{b}{c}$ for some positive integers $a,b,c$. How many lucky numbers are there between $5$ and $10$? Here's what I ...
0
votes
2answers
134 views

Question involving square equality between fractions and square roots [closed]

Find the values of the constants $p$ and $q$ such that $$\frac{\sqrt{p}}{\sqrt{p}+2p} = \frac{2\sqrt{p}-q}{3p+q} \tag{$p,q\ge0$}$$ How would you solve this? I've tried everything...
1
vote
0answers
276 views

Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
0
votes
2answers
37 views

Finding the number of solutions satisfying an equation?

Given one condition $x_1+x_2+x_3=n$ where n is known number. Given a set of data X={$a_1,a_2....a_n$}. Can you help me find all possible cases satisfying the above condition $x_1+x_2+x_3=n$ ???
0
votes
2answers
43 views

Solving two systems with two unknown?

Let's say if we are giving the following two equations: $$ 1= X/(X^2 +Y^2) $$ $$ 2= Y/(X^2 +Y^2) $$ How are we going to solve for X and Y [ by HAND ] ? Why would Summing the squares of the two ...
2
votes
4answers
165 views

Challenge: “Dividing” a number above 0 and ending up with the same, or a greater number (creative task)

Here's a question/challenge for those of you who know quite a bit about math, or enjoy to be creative with what you do know (just for reference: I'm virtually illiterate when it comes to any math more ...
1
vote
5answers
72 views

$|x| + |x-1| = 3$ how come its cases?

$$|x| + |x-1| = 3$$ in my textbook, they say that for this equation, there are 3 cases: $x\geq1$, $0 \leq x < 1$ and $ x < 0$ where do these come from and why? i thought, there are 4 cases ...
3
votes
2answers
32 views

$5-3|x-6|\leq 3x -7$

I have this inequation: $$5-3|x-6|\leq 3x -7$$ i solved this this way: i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : 1) for $x\geq6$ ...
0
votes
1answer
24 views

Word problem with $p$

In the year 2000, there are $p$ penguins. After $t$ years, the number of penguins is given by $$ 2500 \times 1.02^t$$ Calculate the number of penguins in the year $2000$. I tried to substitute random ...
1
vote
1answer
39 views

Manipulating series

I have come across this in a solution for a BMO problem where you have to find $a_{2013}$ for: $a_n$ = $\frac{n+1}{n-1}$($a_1 + a_2 + ... + a_{n-1}$) where $a_1$ = 1. It says that you manipulate it ...
1
vote
1answer
76 views

Find a graph with at least two vertices and no self-loops in which all vertices have different degrees

I am an high-school senior interested in Graph Theory, on a web forum a CS teacher teased me with ("an easy but non-trivial") a terrific Graph Theory problem: Find a graph with at least two ...
2
votes
2answers
203 views

Olympiad Modulo Problem

I have begun preparing for the British Mathematical Olympiad and hope to do well. However, I have been working on the first problem in the book: A Mathematical Olympiad Primer by Geoff Smith, captain ...
13
votes
2answers
2k views

Improving concentration and stamina when solving difficult problems.

I am trying to improve my problem solving skills by solving olympiad problems (Putnam, IMO, etc). So far, I have discovered that problem solving is somewhat like panning for gold: you think of all the ...
0
votes
1answer
95 views

Vector force application problem

I'm having trouble starting off this question. Any help would be appreciated! "Lisa is trying to hold on to her toy car. Her sister Ruby is pulling with a force of 8 N on a bearing of 023° and her ...
5
votes
2answers
152 views

why is $\int_{\pi/2}^{5\pi/2}\frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}=\pi$?

I cannot make progress on the definite integral $$\int_{\pi/2}^{5\pi/2}\frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}\,dx=\pi$$ I know the result is $\pi$ from numerical ...
6
votes
1answer
85 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
2
votes
0answers
72 views

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...