Prove that $x^7+x^5+x^3+1=0$ has only one solution in $\mathbb R$
134736
Prove using Rolle's Theorem that an equation has exactly one real solution.
88333
Find all integral solutions of $y^2=x^3+7$
87567
How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
106374
Show that $2x^6+12x^5+30x^4+60x^3+80x^2+30x+45=0$ has no real roots
73478
Show that the equation $a_1e^{\alpha_1x} + a_2e^{\alpha_2x} + \cdots + a_ne^{\alpha_nx} = 0$ has at most $n - 1$ real roots.
69631
Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$
147325
Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either
116363
If $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$ , then prove that $P(x)$ is divisible by $x-1$
133863
All pairs (x,y) that satisfy the equation $xy+(x^3+y^3)/3=2007$
69328
(USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$
134885
proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $
81671
Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$
53244
Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polynomials from $\mathbb{R}[x]$
139305
Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.
152451
REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
124073
solving for one solution to a system of polynomials
15730
Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution