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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4
votes
7answers
90 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
35 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?
31
votes
9answers
4k 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 ...
11
votes
4answers
1k 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 $\ \ \ \ \ \ \ ...
-3
votes
5answers
313 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 $\ \ \ \ \ \ \ \ $ ...
1
vote
2answers
124 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
77 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
65 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
39 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 ...
2
votes
0answers
39 views

Number of collisions of particles in a box. Application to epidemiology

I was surprised to see in this biology article a model assuming that the number of newly infected cells is a linear function of the number of (healthy) cells and of the number of viruses. I am not ...
0
votes
1answer
26 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
38 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
36 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 ...
5
votes
2answers
105 views

$\{a$ : $\forall f\in C^0$ 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 $f:[0,1]\to \Re$ with $f(0)=f(1)$ there exists at least one $x$ where $f(x) = f(x+a)$. Firstly, $a=0,1/2,1$ are obviously ...
0
votes
1answer
56 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
46 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 ...
0
votes
1answer
31 views

An organization was surveyed with regard to the number of children each member had…

I am studying for my exam and there is no solution for this question. Can anyone provide the correct answers? Thanks. An organization was surveyed with regard to the number of children each member ...
0
votes
1answer
45 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
32 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
29 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
39 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
72 views

Solve this equation. Can anybody do it?

There is this relation between x and y: $$x / y = a + b \log(y)$$ I have x. How do I ...
1
vote
1answer
40 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: ...
1
vote
2answers
62 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 ...
1
vote
2answers
369 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 ...
0
votes
0answers
21 views

Explanation for a simple comparison

Ok, Yesterday I started to learn how to solve problems with comparisons, but I couldn't understand one thing of the "solve algotithm". Here is a part from a solve from a simple example problem ...
1
vote
1answer
36 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
223 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
133 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
42 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
60 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...
0
votes
0answers
73 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 ...
1
vote
2answers
25 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
35 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
50 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
70 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
29 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
23 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
30 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
40 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
73 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
3answers
863 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
19 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
125 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
67 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 ...
1
vote
0answers
45 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 ...
1
vote
2answers
95 views

How many $a$-nary sequences of length $b$ never have $c$ consecutive occurrences of a digit?

Let $S(a,b,c): = \#\{a$-nary sequences of length $b$ without $c$ consecutive occurrences of a digit$\}$. For example, $S(2,n,3)$ would be the number of binary sequences of length $n$ without $3$ ...
2
votes
2answers
62 views

How to solve this equation or system of equations?

I want to solve the equation $$(5 x-4) \cdot\sqrt{2 x-3}-(4 x-5)\cdot \sqrt{3 x-2}=2.$$ I tried. Put $a = \sqrt{2 x-3}\geqslant 0$ and $b =\sqrt{3 x-2}\geqslant 0 $. Suppose $$5x-4=m(2x-3)+n(3x-2)$$ ...
0
votes
0answers
42 views

Find the solution of this differential equation

I want to solve $\dot{\xi}(s)=\sqrt{\frac{(n-2)^2}{4}\xi(s)^2-\frac{n-2}{n}\xi(s)^{\frac{2n}{n-2}}}$ with the condition $\xi(0)=\biggl(\frac{n(n-2)}{4}\biggl)^{\frac{n-2}{4}}$. I know that ...
9
votes
2answers
194 views

Computing $\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$

I don't know how to compute: $$\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$$ I have tried substituting $t=\tan ^{2} x$ but got nothing out of it. I know there's some trick involved, but ...