query: Find, with proof, all functions $f:\Bbb N \to \Bbb N$ such that for all $n \in \Bbb N, f(n) + f(f(n)) = 2n+3$.
positive_passages.docid: 75448
positive_passages.text: Find all function satisfying $f(f(n))+f(n)=2n+3k$
negative_passages.docid: 149328
negative_passages.text: A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020)
negative_passages.docid: 59612
negative_passages.text: Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
negative_passages.docid: 96894
negative_passages.text: Prove $f_{n+1}^2+f_n^2=f_{2n+1}$
negative_passages.docid: 148330
negative_passages.text: Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$
negative_passages.docid: 10253
negative_passages.text: Proving that all entire & injective functions take the form $f = ax + b$?
negative_passages.docid: 925
negative_passages.text: Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
negative_passages.docid: 131400
negative_passages.text: Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
negative_passages.docid: 28111
negative_passages.text: Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$
negative_passages.docid: 70128
negative_passages.text: Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
negative_passages.docid: 73277
negative_passages.text: Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
negative_passages.docid: 107659
negative_passages.text: Let $f:A \to B$ and $g:B\to A$ be arbitrary functions.
negative_passages.docid: 73454
negative_passages.text: Prove that $f$ has finite number of roots
negative_passages.docid: 3922
negative_passages.text: How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$?
negative_passages.docid: 129052
negative_passages.text: Find all functions satisfying $\frac{1}{f(a)}+ \frac{1}{f(b)} = \frac{1}{f(c)} $ whenever $\frac{1}{a}+ \frac{1}{b} = \frac{1}{c} $
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: 34343
negative_passages.text: Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$.