The problem-solving tag has no wiki summary.
3
votes
2answers
115 views
How to solve system of equations?
I want to sove the system of equations $$\begin{cases}
x^3 y-y^4=7,\\
x^2 y+2 xy^2+y^3=9.
\end{cases}
$$
I tried divide these two equations we obtain
$$\dfrac{x^3 - y^3}{(x+y)^2 } = \dfrac{7}{9}$$
...
16
votes
3answers
192 views
Find all polynomials that fix $\mathbb Q$ and the irrationals
Problem: Describe all polynomials $\mathbb{R}\rightarrow\mathbb{R}$ with coefficients in $\mathbb C$ which send rational numbers to rational numbers and irrational numbers to irrational numbers.
2
votes
1answer
62 views
What is the greatest amount of postage you would not be able to pay…
What is the greatest amount of postage you would not be able to pay using only a combination of seven cent and seventeen cent stamps?
I have done a similar problem and got it correct but I am just ...
2
votes
2answers
134 views
What area of mathematics is this problem asking about? [closed]
A colleague posted this on a whiteboard (as a brain-teaser I guess):
A $\rightarrow$ B;
B $\rightarrow$ C;
AD $\rightarrow$ E;
BE $\rightarrow$ C;
BF $\rightarrow$ D;
AC $\rightarrow$ F
What is ...
1
vote
1answer
35 views
Ratio isn't answering correct for this problem
Assume there are 2 products A and B made by different companies. Product A costs 1.2006 USD and Product B costs 2.8298 USD. They decided to exchange their products equally without using Money as a ...
7
votes
2answers
98 views
Show $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$
If $\lim_{n \to \infty} a_{n} = a$ and $\lim_{n \to \infty} b_{n} = b$, how can we show that $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$?
I say $\min\{a_{n},b_{n}\} $ has two cases: ...
0
votes
1answer
39 views
neighborhood - simple but i need help
I have this set of complex numbers: $\{ 1-i , 2-i , 3-i \}$, and another set $$B:= \{ w \in \Bbb{C} \mid 0 \leq \mathrm{Re}(w)\leq 4 \land -2 < \mathrm{Im}(w)\leq0 \} \setminus \{ a+bi \in \Bbb{C} ...
1
vote
1answer
46 views
elementary neighborhood problem
I am to find the proper number from $x \in \{2,3,4\}$ for which this following set is a neighborhood in $\mathbb{R}$ or in $\mathbb{C}$, $$A:= \left] 1,4 \right[ \cap \left[ 2,5 \right]$$
Firstly, I ...
1
vote
3answers
57 views
need help - $\lim_{n \to \infty} i^{3n} $
it seems first easy to me, but now i am tossing my head against wall not being able to solve the problem. i need to check for convergence of this sequence below. i dont know how to start although it ...
2
votes
2answers
40 views
simple convergence test $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$
i am pulling my hair out in solving this problem. i know, it is a stupid question but i am not that good at maths, and many thanks for any help
$\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$
...
1
vote
2answers
46 views
solve $\lim_{n \to \infty} \frac{(-2)^n}{3^{2n}} $
i am eating myself not being able to solve this problem. i somehow feel that the sequence converges to $0$, but once i calculate, it is not coming to that result. or am i making stupid mistake on the ...
0
votes
1answer
40 views
solve $\lim_{n \to \infty}\left( \frac{3+2n}{\sqrt{3}+\sqrt{2}n} + i\,n\right)$
I am checking this sequence for convergence but i am not sure whether i am on the right path in calculations, these steps are what i am doing now. $\infty + n$ will go to infinity, right?
$$\lim_{n ...
8
votes
1answer
97 views
Proving a number defined by a sequence is a square number
I found this problem in a math magazine:
Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by:
$$
x_0 = 0\\
x_1 = 1\\
x_{n+2}+x_{n+1}+2x_{n}=0
$$
Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n ...
-1
votes
1answer
196 views
Commercial Mathematics. A question on payment. Need help.
Please help me out with this problem.
Charlie buys a car for \$120,000. He pays half of the amount in cash and agrees to pay the balance in 12 annual instalments of \$5000 each. If the rate of ...
1
vote
0answers
28 views
Strategies for deriving properties of an expression
For a given $c^*$, suppose that the following system of non-linear equations in $x$ and $y$,
$f(x,y;c)=0\\
g(x,y;c)=0$
possesses a unique solution $(x^*,y^*)$. The equations are such that I do not ...
9
votes
5answers
462 views
Prove $|a+b|+|a-b| \geq |a|+|b|$
I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$
I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$
How can I use this ...
1
vote
2answers
77 views
how to prove this simple statement
i am trying to prove this statement.
for any $a,b \in \mathbb{R}$, $$\max\{a,b\}=\frac{1}{2}\big(a+b+|a-b|\big)$$ and $$\min\{a,b\}=\frac{1}{2}\big(a+b-|a-b|\big)$$
i am eating myself not knowing ...
0
votes
2answers
134 views
Help solving probability problem!
Can you help solve this problem?
Place contains $N$ coupons $(N>3)$. $3$ tickets have prizes $ \$10000, \$1000, \$500$ respectively. $X$ is the number of coupons drawn before a prize-containing ...
3
votes
1answer
65 views
greatest common divisor is 7 and the least common multiple is 16940
How many such number-pairs are there for which the greatest common divisor is 7 and the least common multiple is 16940?
0
votes
2answers
61 views
Finding roots of product of two polynomials
Let $P$ and $Q$ be polynomials of degree $2$ and $3$ respectively. If we know the roots of both $P$ and $Q$, is there an easier way of finding the roots of the product $PQ$? Do we really have to ...
-1
votes
1answer
78 views
simple arithmetic problem but..
i am having problem in this proof. i need to find the certain coefficients of this statement on the right side.
given:
$P: \mathbb{C} \Longrightarrow \mathbb{C}, \quad P(x) := 12 − 7x + x^2$
$Q : ...
1
vote
3answers
65 views
simple question about min and max functions
i am asked to draw the graph of this min and max functions. these functions are given.
$f_1(x) := 2, \quad f_2(x) := 3x, \quad f_3(x) := x^2$
now i need to draw the graph of these functions:
...
1
vote
1answer
174 views
Maximum area for fixed perimeter of a triangle
I'm trying to prove that the triangle of largest area for a given perimeter is equilateral, but I'm having some difficulties.
I've done 2 different proofs for a similar problem but for rectangles - ...
15
votes
4answers
325 views
Maximize $x_1x_2+x_2x_3+\cdots+x_nx_1$
Let $x_1,x_2,\ldots,x_n$ be $n$ non-negative numbers ($n>2$) with a fixed sum $S$.
What is the maximum of $x_1x_2+x_2x_3+\cdots+x_nx_1$?
5
votes
1answer
163 views
How did Euler solve the 4-whole-numbers-adding-up-to-a-perfect-square problem?
So I was watching a video on Leonhard Euler about how he amazingly solved so many difficult problems and one of the many problems that he solved was this:
...
2
votes
5answers
142 views
How to show $A-B \subseteq C \Rightarrow A\cup B \subseteq B\cup C$?
I really need help with this logical proof.
Show that $A-B \subseteq C \Rightarrow A\cup B \subseteq B\cup C$.
Please show the steps to the solution. Thank you!
3
votes
1answer
1k views
Putnam 2012 B3 - Tournament combinatorics
A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ ...
0
votes
1answer
119 views
Interview Question: Best method to gather groceries [closed]
I went to an interview the other day and the interviewer asked me this question:
"You are in the grocery store shopping for 10 people with 1 shopping cart. You have a different shopping list for each ...
1
vote
1answer
66 views
simple statement proof
i am proving this statement about strict isotoneness. i will try on my own and you will tell me whether i am okay or not :)
$A$ is subset of $\mathbb{R}$
$f$ is strict isotone $ \Longleftrightarrow ...
2
votes
2answers
64 views
complex numbers - proof of this statement
i am trying to prove this statement, i dont but how to start.
$$\forall z,w \in \mathbb{C}\quad |z|^2+|w|^2=\frac{1}{2}(|z+w|^2+|z-w|^2)$$
can someone please show me how start?
0
votes
1answer
38 views
what is the difference - sorry for over-simplicity
i am asking too simple question, sorry for that. what is the difference between these two imaginär numbers?
$\operatorname{Im}(| \sqrt2+3i|^2)$ vs. $\operatorname{Im}((\sqrt2+3i)^2)$
$| ...
1
vote
3answers
54 views
minimum of this simple set
i need again some help here. i am defining the minimum and max and inf and sup of this set
$A:=(]1,2[ \cup ]2,3]) \cup \{2\}$ which is equal to the interval $(1,3]$
i say, max is 3, and sup is also ...
1
vote
1answer
82 views
Getting Practice on Finding Charts for a Manifold
I just want to ask for a suggestion on the study of differential geometry. When I study it I understand the theorems, their proofs, I understand perfeclty the concepts and so forth, but I'm having ...
2
votes
3answers
236 views
archimedean Property - proof
i am stumbling across this statement. i need to show minimum, maximum, infimum and supremum, if they exist.
$$ C:= \bigcup_{n \in \mathbb{N}} [0,1/n[$$
the archimedean property says: let $e,x$ be ...
3
votes
0answers
55 views
Optimizations for Travelling Salesman Problem
I have to design a branch-and bound algorithm that solves the optimal tour of a graph on the cartesian plane every time. I have been given the hint that identifying hopeless branches earlier in the ...
1
vote
2answers
161 views
Solve $x^2(1-x) = 1$
I have a problem $x^2(1-x) = 1$
This can be simplified ( I think ) to $-x^3 + x^2 - 1 = 0$
Google shows that there is 1 solution for this in its graph.
I am not sure how to get to that solution ...
3
votes
2answers
356 views
Proof by induction for Fibonacci numbers
How can we prove by induction the following?
$
F_{n+1} = \left\{
\begin{array}{l l}
F_{n/2}^2+F_{(n+2)/2}^2 & \quad \text{if $n$ is even}\\
...
1
vote
1answer
433 views
fibonacci numbers - induction proof
i am trying to prove this statement of fibonacci numbers by induction, i am stuck though on the way:
my steps:
definition:
$F_{0}:=0, F_{1}:=1 $ and $F_{n}:=F_{n-1}+F_{n-2}$
The Hypothesis is: ...
4
votes
1answer
108 views
Showing a function is not of a bounded variation.
$V_a^b(P,f):=\sum_{i=1}^k|f(x_i)-f(x_{i-1})|$, where $P$ is a partition.
$$f(x)= \begin{cases}
x^2\sin(\frac{1}{x^2}), &\text{if } x\neq0, \\
0, &\text{if } x=0
\end{cases}
...
1
vote
3answers
166 views
Problems like the handshake problem
I am in college and my RA has been putting up little thought problems on his door for us to see as we pass by, but the ones he puts up aren't too interesting. I wrote up the handshake problem (invite ...
1
vote
5answers
134 views
Find $x$ such that $\arctan(3/2)+\dots=\arctan x$
Find $x$ such that
$$\arctan(3/2) + \arctan(5/4) + \arctan(-5/2) + \arctan(-8/3) = \arctan x.$$
6
votes
2answers
196 views
Problem understanding math
sorry for my english but I am a foreigner. I'm writing to ask you for help with a problem I have with mathematics. I'm going to go to university physics but I have serious problems as regards ...
0
votes
2answers
58 views
Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$
I am struggling to prove this map statement on sets.
The statement is:
Let $f:X \rightarrow Y$ be a map.
i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$
ii) $\forall_{A,B \subset X}: ...
1
vote
2answers
47 views
elementary prove thru induction - dumb stumbling
i am trying to prove this statement for all $n \in \mathbb{N}$ with the help of induction:
$4 \sum_{k=1}^{n} (-1)^kk=(-1)^n(2n+1)-1$
base case: n=1
$4 \sum_{k=1}^{1} (-1)^11=-4=(-1)^1(2*1+1)-1$ .. ...
0
votes
1answer
71 views
Uniqueness of minimum and maximum of set
i am stuck in this simple but foggy problem. i need to prove or show that the min and sup are unique if they exist.
$A$ is a nonempty set and $B$ is nonempty subset of $A$. i am trying to show that ...
0
votes
1answer
29 views
monotoneness - prove the elementary statement
Let $A, B, C$ be nonempty sets with total order. and let $f\colon A \rightarrow B$ and $g\colon B \rightarrow C$ be maps. Prove these statements:
a) If $f, g$ are antitone, then $g \circ f$ is ...
0
votes
1answer
81 views
Question on probability
This is a Math question I read and I badly want to know the answer.
It says;
There are twelve people who own 12 phones.One of them(who owns the most valuable phone) suggested to write down their names ...
20
votes
7answers
930 views
Sum of the sum of the sum of the first $n$ natural numbers
I have here another problem of mine, which I couldn't manage to solve.
Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n
\\ z_n = y_1 + y_2 + \dots + y_n $$
Find ...
9
votes
2answers
390 views
Evaluating $\int_{0}^{x} e^t \sqrt{2 + \sin(2t)} \, dt$
I was recently asked to evaluate the following integral:
$$\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
140 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 ...
