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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1answer
29 views

Help finding the inverse of an exponential function

$$f(x)=6^{3x+9}-2.$$ I got to one step, but I became lost. I understand that I'm converting it into logarithmic form, but I don't understand what the next steps are. \begin{align*} ...
2
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1answer
36 views

If for every k the interval $[a,ak]$ contains $n$ specials numbers how many special numbers $[az,akz]$ must contain?

The purpose of my question is to determine if a specific kind of reasoning is true or false. Let's say that for every positive natural number $a$, there is a at least $n$ "special numbers" in the ...
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0answers
26 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
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0answers
16 views

What is the fastest way to calculate the character of a fundamental representation?

Suppose we know the Cartan matrix $C$ of a Kac-Moody algebra $\mathbb g$: $$ C=\left(\begin{array}{cccc} 2 & -1 & 0 & 0\\ -1 & 2 & -1 & -1\\ 0 & -1 & 2 & 0\\ 0 ...
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1answer
107 views

A “Theorem Style” Problem Book in Differential Geometry

I am trying to teach myself differential geometry using Lee's Introduction to Smooth Manifolds. To test my understanding, and learn the subject better, I am looking for a good problem book in ...
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1answer
72 views

Coin Flipping Game - Expected Number of Tails

Can someone help me with this problem? In this game, let $S_{t}$ denote your earnings at time $t$. Your initial earnings is one dollar ($S_{0} = 1$). For each subsequent time, $t = 1, 2, ..$, flip a ...
3
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1answer
60 views

Trying to solve $A^TA = B$ where $B$ is known

Have a matrix A that's 4x3 and a matrix B that is known. They relate by the following equation: $$ A^T A = B $$ Trying to find A, given B, such that $A^TA$ is as close to B as possible. I've tried ...
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1answer
20 views

How to stabilize cyclic tridiagonal matrix algorithm?

I've received a task which is: Solve equation by cyclic tridiagonal matrix algorithm: $$ \frac{\partial{f}}{\partial{t}} = \lambda*\frac{\partial{f}}{\partial{x}}, \\ x\in[0,1]\ t\in[0,1] \\ ...
2
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1answer
62 views

Problem Books with Problems less “intense” than Putnam Problems

As the title indicates, I'm looking for a few suggestions on problem books. The problems should be a bit less demanding than Putnam problems. Like the Putnam, however, the prerequisites should be ...
3
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2answers
25 views

Calculating the value of numbers with different operations

Calculate the value of: $$-14 + 49 \times 21 - 63 + 56 \div 35 \div 28 \times 70 - 42 \div 7$$ I noticed the numbers are a factor of $7$, so I took out $7$ as a common factor: $$7[-2 + (7 \times 3) ...
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0answers
16 views

Scheduling problem

Consider the following setting: $N$ jobs, each has a starting time, which is assumed to be a natural number and all N numbers are distinct, e.g., the 1st job has starting time at 5, the 2nd is 6, the ...
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0answers
37 views

Good problem books at a relatively advanced level?

I have been searching for problem books on advanced topics. By advanced I am referring to the undergraduate level and above. I am looking for something analogous to the olympiad type problem books ...
1
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1answer
41 views

Series solution

Given the differential equation $2(1-x)y''-3y'+\frac{y}{x}=0$ and in standard form: $y''-\frac{3}{2(1-x)}y'+\frac{1}{2x(1-x)}y=0$ I want to find the series solution for the larger root $σ = 1$ of ...
2
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1answer
69 views

Rank of a symmetric matrix. (ISI Sample Paper)

I have rephrased the question as follows : Here, $\langle v,w\rangle=v^tw$ is the usual dot product. Let $A$ be an $n \times n$ symmetric matrix. Let $l_1, l_2, \ldots , l_{r+s}$ be $(r + s)$ ...
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0answers
43 views

How to find a list of summands and factors adding up to a total?

I am neither a mathematician nor do I have an idea on how to write down my problem in accurate mathematic formulas. Please feel free to edit my question into shape and remove this paragraph. Also I am ...
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2answers
39 views

Homework Problem About Finding a Value of $k$ for Which the Given System of Equations Has No Solutions

While working through the third of three packets I'm going through to review for a pre-test for an independent-study calculus class, I came across the following problem: For what value of $k$ ...
2
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1answer
72 views

mod operation proof

Prove: $ ab\,\bmod\,d = ((a\,\bmod\,d)\,(b\,\bmod\,d))\,\bmod\,d $ where $a$, $b$ and $d$ are non-negative integers. Reference : http://en.wikipedia.org/wiki/Modulo_operation#Equivalencies Context ...
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1answer
54 views

Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}$

Find all continuous $f:\mathbb{R}\to\mathbb{R}$ satisfying $$\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}.$$ I believe the original question was $$\frac{f(x)}{3+f(x)}=\frac{4+x^2}{x^2},$$ which has a ...
1
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1answer
48 views

Polygon center which always lies inside the polygon (with no hole)

Is there is any type of centre (of polygon) which always lies inside the polygon (with no hole)? Note: Here our polygon may be any type of polygon (convex or concave) but ...
1
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1answer
63 views

Can you generate math problems that are solveable?

If you take Linear Programming, it problems are formulated like this: You know that Cabinet X costs 10 cents per unit, requires 6 square feet of floor space, and holds 8 cubic feet of files. ...
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1answer
36 views

Odds of Winning Office NCAA Pool

I have 6 coworkers competition in a NCAA bracket. I'm trying to find out how to calculate who has the best chance of winning. For example currently the score card looks like: Player 1. Current Right ...
2
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1answer
74 views

Solving an equation for x, characteristics

I am trying to plot characteristics on Matlab for a hyperbolic pde. I need to compute \begin{equation} x=\frac{t}{(1+x^2)}+x_i \end{equation} for every spatial step. Any help with how to do this? ...
2
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1answer
28 views

Problem about sum of polynomials

I have this problem I don't know how to solve: Let $f(x)$ be a polynomial of degree $n$ with real coefficients and such that $f(x) \geq 0 \forall x \in \mathbb{R}.$ How do I show that $f(x) + f'(x) + ...
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1answer
32 views

Some clarifications and a question on basic probability.

I have a few questions and some clarifications. CLARIFICATIONS: 1. Assume we roll 2 four sided dice. What is P({sum of the rolls is even})? I answered the question correctly I: Odd + Odd = Even J: ...
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2answers
63 views

$\lim_{n\to\infty}a_nb_n=? $ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=\infty$

A. $\lim_{n\to\infty}a_nb_n=?$ given that $\lim_{n\to\infty}a_n=0$ and $ \lim_{n\to\infty}b_n=\infty$ B. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=0$ and ...
2
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2answers
77 views

Minimize : $\sqrt{(1+{1\over a})(1+{1\over b})}$ subject to $a+b=\lambda$.

Given positive real variables $a$ and $b$, find the minimum of $$f(a,b)=\sqrt{\left(1+{1\over a}\right)\left(1+{1\over b}\right)}$$ subject to $a+b=\lambda$ where $\lambda$ is a constant . [ISI ...
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1answer
18 views

Non-convex quadrilateral and pentagon?

Is it possible to draw a non-convex quadrilateral/pentagon and an additional straight line such that the straight line cuts through the interior of each of the quadrilateral/pentagon's edges?
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1answer
28 views

A question about subspace and finding basis

guys help me I couldn't solve this question I've been working on subspaces for sometime but still cant do this kind of questions.
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2answers
44 views

Surface Area and Volume relationship

I know that the $SA = 6s^2$ and that the volume is equal to the base $x$ the $side = s^3$. However, I'm not sure how to approach this though.
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1answer
22 views

How does one find the change?

I tried using ratios but I failed. I need to subtract one to get the correct answer. I remember finding the change before, but I've forgotten how to. Any hints?
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1answer
64 views

Prove that following polynomial has no non-zero real solution.

Prove that following equation has no non-zero real solution. $$ \sum_{ 1 \leq n \leq 120,\, 2|n \;\textrm{or}\; 3|n } x^n = 0$$ Any idea?
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1answer
66 views

Find the condition for a center of a circle with exactly one lattice point on its circumference

Statement Find the condition for a center of a circle with exactly one lattice point on its circumference (this lattice point must not be the only one lattice point of the disk) What I have ...
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3answers
99 views

Pennies, Nickels, Dimes, and Quarters Summation of Money

Peter has only pennies, Norma only Nickels, Diane only dimes, and Quincy only quarters. Peter and Norma have the same number of coins, and Diane and Quincy have the same number of coins. What is the ...
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2answers
104 views

Solve for “x” and “y” [duplicate]

What would be the easiest way to solve the following set of equations:$$ x + y^2 = 7 $$$$ x^2 + y = 11$$ I've been trying substitution method but end up in a $4$th degree bi-quadratic equation. ...
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0answers
13 views

Determinant of partition matirx

Let $X$ be $n\times p$ matrix as $X=(x_1, x_2, \ldots x_p)$. I partition the matrix as follows $X=(X_1, X_2)$ where $X_1$ is a $n\times p_1$ matrix and $X_2$ is a $n\times (p-p_1)$ matrix. Then how ...
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3answers
396 views

Solving a quintic function for zero

I got this question on my homework and I cannot for the life of me figure out how to solve for $0$. $$x^5+2x-10=0$$ I have tried this every which way and this is my last resort. Thanks in advanced. ...
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1answer
421 views

Given two potatoes, prove that there is a loop of wire which fits around both

This is a classic problem in geometric continuity and I want to see if there are some solutions other than the one I'm thinking of: Two potatoes are given. Prove that there exists a closed loop of ...
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1answer
38 views

Getting rid of product of sequence sign

I am having trouble with equation containing product of sequence: $$\frac {1}{2} = 1 - \frac {\prod \limits_{i=1} ^{n} (366 - i)}{365^n} $$ How can I convert the $\prod \limits_{i=1} ^{n} (366 - i)$ ...
3
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1answer
58 views

$\prod_{i=1}^{n-1} a_i = 1 \Rightarrow \prod_{i=1}^{n-1} (1+ a_i)^{i+1} > n^n$?

Let $n>3$ be an integer number and $a_1, a_2, \dots, a_{n-1}$ positive real numbers, such that $\prod_{i=1}^{n-1} a_i = 1$. Is the following inequality true? $$ \prod_{i=1}^{n-1} (1+ a_i)^{i+1} ...
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2answers
65 views

Solve $x^4+3x+20=0$ by Ferrari's method

Comparing the equation $$x^4+3x+20=0$$ With the equation $$(x^2+\lambda)^2-(mx+n)^2=0$$ we get $m^2=2\lambda,$ $-2mn=3,$ $n^2=\lambda^2-20$ Now, $4m^2n^2=9\Rightarrow ...
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2answers
29 views

What's the probability a die irolled 4 times you will get only two kinds of faces?

Let $A$ be the event "only $2$ different faces in $4$ rolls of a die." At each roll there's $6$ possibilities, so: $$\omega = 6\cdot 6\cdot 6\cdot 6$$ Considering that it can be only two kinds of ...
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0answers
38 views

What is the probability that the fourth and fifth coins tossed are the same?

A biased coin is tossed infinitely many times and has probability $p$ of being "heads". 1) What is the probability that the fourth and fifth coins are the same? 2) And given that the first 10 tosses ...
0
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1answer
52 views

Solving a cubic equation

Solve $y=ax^3+bx^2+cx+d$ I need $x$ in terms of $y$ . I do not need the roots of the cubic equation . I need to express $x$ in terms of $y, x>0$
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2answers
27 views

What is the probability that exactly 7 of the first 10 coin tosses are heads?

A biased coin is tossed infinitely many times and has probability $p$ of being "heads". What is the probability that exactly $7$ of the first $10$ coin tosses are "heads", in terms of $p$? It's a ...
3
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4answers
57 views

Find $\min\big\{ \lfloor xy + \frac{1}{xy} \rfloor \,\Big|\, (x+1)(y+1)=2 ,\, 0<x,y \in \mathbb{R} \big\}$

I am invited to calculate the minimum of the following set: $\big\{ \lfloor xy + \frac{1}{xy} \rfloor \,\Big|\, (x+1)(y+1)=2 ,\, 0<x,y \in \mathbb{R} \big\}$. Is there any idea? (The question ...
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4answers
86 views

Solve $16x^{-3}=-2$

Solve $16x^{-3}=-2$. My working: \begin{align} 16x^{-3}&=-2\\ \frac{1}{16x^{3}}&=-2\\ \frac{16x^3}{16x^3}&=-32x^3\\ 1&=-32x^{3}\\ -32x^{3}&=1\\ -32x&=\sqrt[3]{1}\\ ...
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2answers
45 views

Frogs and switches - problem solving strategies

The question is pretty simple, consider 1000 switches and 1000 light bulbs, every time we press a switch it's light bulb changes it's state(ON to OFF and vice versa). We start with all the light bulbs ...
0
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1answer
81 views

Pigeonhole question about distinct sums

How do I show with the pigeonhole principle that no seven positive integers not exceeding $24$ can have sums of all subsets different. As observed by Ross Millikan, the simplest possible approach ...
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3answers
37 views

Solving a function for a variable, confusion

I have the function $f(t) = -4.9t^2+25t+3$, where $f(t)$ is a the height of a grapefruit after $t$ number of seconds. I need to find out how long the grapefruit is in the air, so I know i need to ...
1
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1answer
47 views

Local extension of smooth funtion to a embedded manifold

I'm trying to proof the following problem from Lee's Book: Suppose $M$ is a smooth manifold and $S\subseteq M$ is a smooth submanifold. Show that $S$ is embedded if and only if every $f\in ...