Aditya
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1 answers
2 votes
88 views
Solving the equation of the type $g\left( x \right) = \int\limits_0^x {f\left( t \right)dt} $
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4 votes

I think all the options have at least one solution in $(0,1)$. First of all, $g'(x)=f(x)$. For option A, Let $F(x)=e^{-x}g(x)$. $\implies F(0)=F(1)=0$ $\implies \exists c \in (0,1)$ for which $F'(x)=0$...

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2 answers
3 votes
61 views
Prove that no Fermat number is a $3$ rd power of an integer.
2 votes

Just to clarify, Fermat number is a number of the type $2^{2^n}+1$. About the part that no Fermat number ($F_n$) is a perfect square, I think the reasoning provided is correct. $F_n\equiv 7 \pmod {10}$...

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2 answers
0 votes
56 views
Proofs with prime numbers.
2 votes

Actually for (1), $n=1$ is a counterexample (trivial case, but counterexample nonetheless). $n^2 \equiv 0$ or $1 \pmod 3$. $\implies n^2+2\equiv 2$ or $0 \pmod 3$ and $n^2+2 \neq 0 \pmod 3$ (and here ...

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2 answers
2 votes
94 views
How many solutions for this equation using combinatorics?
2 votes

I totally agree with the first answer. Here I'm just going to explain why it is corect. If we had $x_1 +x_2 +x_3+ \cdots +x_n =k$, then the number of integer solutions to this equation, where $x_i \...

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2 answers
-2 votes
60 views
Is there a closed form solution to $y \frac{dy}{dx} = \sqrt{x^2+y^2}+x$?
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1 votes

$\frac{dy}{dx}=\frac{x}{y}+\sqrt{\left(\frac{x}{y}\right)^2+1}$ Let $y=tx$. $\implies \frac{dy}{dx}=t+x\frac{dt}{dx}$ $\implies t+x\frac{dt}{dx}=\frac{1}{t} +\sqrt{\left(\frac{1}{t}\right)^2+1}$ $\...

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3 answers
3 votes
114 views
Integrate $\int\frac{x^4}{\sqrt{a^2-x^2}}dx$ using trigonometric substitution
1 votes

You need to substitute $\theta=\arcsin(\frac{x}{a})$, or as you said just find $\sin(4\theta) $ and $\sin (2\theta)$ using the double angle properties and put the values in the final answer.

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2 answers
0 votes
56 views
Help prove the inequality $a/\sqrt{b} + b/\sqrt{a} \ge \sqrt{a} + \sqrt{b}$
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1 votes

You have almost solved it. You need to prove that $a(\sqrt{a}-\sqrt{b})+b(\sqrt{b}-\sqrt{a})\ge 0$. which is true due to the following : $a(\sqrt{a}-\sqrt{b})-b(\sqrt{a}-\sqrt{b})\ge 0$ $\implies (a-b)...

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1 answers
1 votes
101 views
Prove that $17$ divides $xy-12x+15y$
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1 votes

Note: I'm going to introduce some integer variables like $k,n ,m $ etc. They are only for the purpose of showing the remainder of a number when divided by a particular number. For example if I say $x=...

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3 answers
1 votes
71 views
Induction with divisibility
1 votes

Q. Show that $16$ $|$ $19^{4n+1} +17^{3n+1} -4$ $\forall$ $ n \in \mathbb N$. First of all, simplify the expression. Write $19=16+3$ and $17=16+1$, then use binomial theorem to expand it. That way ...

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3 answers
-1 votes
51 views
Answer to an algebra question
1 votes

I don't think there is any value of $b$ for which it can be the vertex. Proof: $4=9-9b-5$ (Vertex lies on the parabola) $\implies b=0$ But the slope of tangent at the vertex should be zero, that is $\...

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5 answers
1 votes
72 views
A doubt while solving a question on parabola.
1 votes

Okay so you have a doubt in how $PM=y$ and $PN=x$. When we write the equation $y^2=4ax$, the $PM$ stands for axis of the parabola and $PN$ stands for the tangent at the vertex. For the parabola $y^...

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2 answers
2 votes
64 views
$k*x^2+y^2=z^2$; Elementary question
1 votes

Assuming $(x,y,z,k) \in \mathbb Z$, $z^2 \equiv 0 $ or $1\pmod 4$ and since $y^2 \equiv 0 $ or $1 \pmod 4$ $kx^2 \not\equiv 2 $$ \pmod 4$ Now, as $x^2 \equiv 0 $ or $1 \pmod 4$, $k \not\equiv ...

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3 answers
3 votes
80 views
Is there a situation such that the 'magnitude' of discriminant is important?
0 votes

If the given quadratic is $ax^2+bx+c$ with $\Delta=b^2-4ac$, then $ax^2+bx+c=\frac{1}{4a}\left( 2ax +b \right)^2 - \frac{\Delta}{4a}$, which means the minimum or maximum value of the quadratic ($- \...

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3 answers
2 votes
33 views
Congruency application
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It's the same as proving that $n^2-1 \equiv 0 \pmod 24$. Since $n \equiv 1 \pmod 2 \implies (n-1)(n+1) \equiv 0 \pmod 8$ (as one of $n-1, n+1$ will be $2 \pmod 4 $ and the other will be $0 \pmod 4$) ...

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1 answers
0 votes
107 views
Eisenstein's criterion of irreducibility simple proof
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This has to do with Gauss's Lemma for polynomials. It states that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. So in the given ...

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5 answers
3 votes
142 views
Can you solve $\int 1/(x^2+1)^2\, dx$
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$I=\int \frac{dx}{(x^2+1)^2}=\frac{1}{2} \int \frac{1+x^2+1-x^2}{x^4+2x^2+1}.dx$ $\implies I= \frac{1}{2} \left ( \int \frac{1+x^2}{x^4+2x^2+1}.dx +\int \frac{1-x^2}{x^4+2x^2+1}.dx\right).$ $I_1=\int \...

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1 answers
0 votes
75 views
How to prove that a particular form of numbers achieve all remainders modulo $n$?
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Partial progress : Here's the proof that $n$ power numbers for any even $n>1$ (i.e. squares, $4$ th powers etc.) can't achieve all remainders modulo $k$ for $k \gt 2$. Define sets $A$ and $B$ as $...

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4 answers
2 votes
340 views
Is it a necessary condition for an even function to have a local extremum (for $f(x)=k,$ derivative${}=0$) at $x=0$
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Let $f(x)$ be a function differentiable at $x=0$. Case 1: $f$ is an even function. $$ f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{f(x)-f(x-h)}{h}.$$ Now find $f'(-x)$. $$f'(-x)= ...

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2 answers
2 votes
81 views
Proving nature of roots of polynomial.
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If you have a polynomial $p(x)$ of degree $n$, and you are given that $p(x)$ has $n$ roots, all of which are real then between any pair of consecutive roots (according to your graph, roots which are ...

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3 answers
-1 votes
460 views
How many integer solutions are there to $a + b + c + d = 15$ when $a≥-3, b≥0, c≥-2$ and $d≥-1$?
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The method suggested in the first answer is easier than what I am going to use. We can find the coefficient of $x^{15}$ in the expansion of $$(x^{-3} + x^{-2} + \cdots )(x^{0} + x^{1} + \cdots )(x^{-...

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2 answers
1 votes
69 views
I need help with induction
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0 votes

You know that $x_n \lt x_{n+1}$ $\implies x_n \lt \sqrt{3+2x_n}$ $\implies 2x_n \lt 2 \sqrt{3+2x_n}$ $\implies 3+ 2x_n \lt 3+ 2 \sqrt{3+2x_n}$ $\implies \sqrt{3+ 2x_n} \lt \sqrt{3+ 2 \sqrt{3+2x_n}}...

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3 answers
-2 votes
48 views
proof problem of combination and summation
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$(1+x)^n = \binom{n}{0} + x \binom{n}{1}+ \cdots +x^n \binom{n}{n}$ ----(1) We need $ S= 1\binom{n}{1} -2 \binom{n}{2} + \cdots + (-1)^{n-1} n \binom{n}{n} $. In equation (1), differentiate both ...

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4 answers
-2 votes
45 views
Intuition for why inverse of strictly increasing function is also strictly increasing?
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Let $f$ be a strictly increasing function. Consider two points $A$ and $B$ on the curve $y=f(x)$. Let $A \equiv (x_1 , f(x_1))$ and $B \equiv (x_1+k , f(x_1 +k))$ for some $k \gt 0$. Clearly, $x_1+...

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3 answers
1 votes
75 views
How many ways can we get a number by addition if each part of the addition has to be smaller or equal to a set value?
0 votes

If you wish to find the number of ways to express a positive integer $n$ as the sum of $m$ positive integers ($m\le n$) and each of the positive integers must be less than or equal to a certain value $...

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