Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.
13656
Prove existence of a real root.
126445
Proving the second root of a quadratic equation
41364
How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
76175
Does the equation have a real solution...please help!
106011
Proving it has all real roots
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$
33469
Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$
152293
Solution of $y''+xy=0$
90261
Find all solutions of the equation $n^m=x^2+py^2$ which satisfy the following properties
114872
If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$
111847
For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
42713
Solutions to the equation $ze^{a-z}=1$
18903
Finding solutions to equation of the form $1+x+x^{2} + \cdots + x^{m} = y^{n}$
63505
Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $
111696
Show $\sin x = bx$ has at least 3 real solutions.
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$