query: IMO 2019: Find all functions $f: \mathbb Z \to \mathbb Z$ such that $f(2a) + 2\, f(b) = f \big( f (a+b)\big)$.
positive_passages.docid: 155806
positive_passages.text: (Olympiad Question) $f(2x) + 2f(y) = f(f(x+y))$
negative_passages.docid: 89257
negative_passages.text: Give an example of a function $f:\mathbb{R}^2 \mapsto \mathbb{R}^2$ such that
negative_passages.docid: 138026
negative_passages.text: To show that $n!f(x) \in \mathbb{Z}[x]$
negative_passages.docid: 73519
negative_passages.text: Determine all functions satisfying $f(x + f(x + y)) + f(xy) = x + f(x + y) +yf(x)$
negative_passages.docid: 157924
negative_passages.text: $\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
negative_passages.docid: 133427
negative_passages.text: analytic function on $\mathbb{C}$ such that $f\Bigl(\frac{1}{2n}\Bigr)=f\Bigl(\frac{1}{2n+1}\Bigr)=\frac{1}{2n}$
negative_passages.docid: 47856
negative_passages.text: $f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
negative_passages.docid: 135611
negative_passages.text: Extensions of $\mathbb{Z}$ by $\mathbb{Z}$
negative_passages.docid: 82675
negative_passages.text: What is an example of function $f: \Bbb{N} \to \Bbb{Z}$ that is a bijection?
negative_passages.docid: 66867
negative_passages.text: Question regarding the function $z=a+bi \mapsto f(z)=a+i(a+b)$.
negative_passages.docid: 124847
negative_passages.text: Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
negative_passages.docid: 37976
negative_passages.text: Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$
negative_passages.docid: 17677
negative_passages.text: Prove or show a counter example for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$
negative_passages.docid: 10253
negative_passages.text: Proving that all entire & injective functions take the form $f = ax + b$?
negative_passages.docid: 24810
negative_passages.text: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
negative_passages.docid: 119360
negative_passages.text: Find all subrings of $\mathbb{Z}^2$
negative_passages.docid: 44717
negative_passages.text: Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges?