Question

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

score 167 · Answer 1 · 2026-03-25 04:43:52

167

Views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Published on 25 Mar 2026 - 4:43

score 287 · Answer 2 · 2018-11-26 22:35:45

287

Views

Showing that $a^2+b^2+c^2+d^2+e^2+65=abcde$ has integer solutions greater than $2018$?

Published on 26 Nov 2018 - 22:35

score 30 · Answer 3 · 2018-11-27 01:51:53

30

Views

What would be the another possible answer?

Published on 27 Nov 2018 - 1:51

score 569 · Answer 4 · 2018-11-27 10:22:04

569

Views

frequencies comparison

Published on 27 Nov 2018 - 10:22

score 2.3k · Answer 5 · 2026-03-27 03:49:41

2.3k

Views

Inequalities - Tangent line trick

Published on 27 Mar 2026 - 3:49

score 187 · Answer 6 · 2018-11-28 04:37:02

187

Views

Technique behind solving 4 = $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$

Published on 28 Nov 2018 - 4:37

score 286 · Answer 7 · 2026-03-28 04:53:16

286

Views

Maximising the area of a triangle when the 3 sides are at most 2, 3, 4

Published on 28 Mar 2026 - 4:53

score 169 · Answer 8 · 2018-11-29 17:55:25

Prove that if $a_k \in [0, 1] $ then $\frac {1}{1+a_1} + \frac {1}{1+a_2} + \cdots +\frac {1}{1+a_n} \le \frac{n}{1+ \sqrt[n]{a_1a_2\cdots a_n}}$.

score 327 · Answer 9 · 2018-11-30 04:03:02

327

Views

If $a+b+c=\frac 1a +\frac 1b +\frac 1c$ then prove that $ab+bc+ca \geq 3$

Published on 30 Nov 2018 - 4:03

score 499 · Answer 10 · 2026-03-28 04:52:11

499

Views

A curious geometry problem: Find the $\angle OBC$

Published on 28 Mar 2026 - 4:52

score 102 · Answer 11 · 2026-03-25 23:53:08

102

Views

Is it possible to find a closed form for $x$?

Published on 25 Mar 2026 - 23:53

score 1.1k · Answer 12 · 2026-03-25 01:17:34

1.1k

Views

Maximum number of kings on the chessboard subject to some rules

Published on 25 Mar 2026 - 1:17

score 1k · Answer 13 · 2018-12-05 01:06:39

1k

Views

Synthetic solution to this geometry problem?

Published on 05 Dec 2018 - 1:06

score 2.9k · Answer 14 · 2018-12-05 13:34:14

2.9k

Views

if such three condition find the $\sin{\angle OPA}$

Published on 05 Dec 2018 - 13:34

score 64 · Answer 15 · 2018-12-05 17:53:36

64

Views

Prove that the $AX$ bisects $BC$ in $\triangle ABC$ where $X$ is the intersection of a side $EF$ of the contact triangle $DEF$ and $ID$

Published on 05 Dec 2018 - 17:53

List Question

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Showing that $a^2+b^2+c^2+d^2+e^2+65=abcde$ has integer solutions greater than $2018$?

What would be the another possible answer?

frequencies comparison

Inequalities - Tangent line trick

Technique behind solving 4 = $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$

Maximising the area of a triangle when the 3 sides are at most 2, 3, 4

Prove that if $a_k \in [0, 1] $ then $\frac {1}{1+a_1} + \frac {1}{1+a_2} + \cdots +\frac {1}{1+a_n} \le \frac{n}{1+ \sqrt[n]{a_1a_2\cdots a_n}}$.

If $a+b+c=\frac 1a +\frac 1b +\frac 1c$ then prove that $ab+bc+ca \geq 3$

A curious geometry problem: Find the $\angle OBC$

Is it possible to find a closed form for $x$?

Maximum number of kings on the chessboard subject to some rules

Synthetic solution to this geometry problem?

if such three condition find the $\sin{\angle OPA}$

Prove that the $AX$ bisects $BC$ in $\triangle ABC$ where $X$ is the intersection of a side $EF$ of the contact triangle $DEF$ and $ID$

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