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degree of f=2^k
Sayan   15
N an hour ago by Gejabsk
Source: ISI 2012 #8
Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$.

i) Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$.

ii) Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.
15 replies
1 viewing
Sayan
May 13, 2012
Gejabsk
an hour ago
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Local-global with Fibonacci numbers
MarkBcc168   26
N an hour ago by pi271828
Source: ELMO 2020 P2
Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number?

Proposed by Fedir Yudin.
26 replies
MarkBcc168
Jul 28, 2020
pi271828
an hour ago
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Cauchy functional equations
syk0526   10
N an hour ago by imagien_bad
Source: FKMO 2013 #2
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions.
(a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $.
(b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.
10 replies
syk0526
Mar 24, 2013
imagien_bad
an hour ago
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Three circles are concurrent
Twoisaprime   21
N an hour ago by L13832
Source: RMM 2025 P5
Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent.
Proposed by Romania, Radu-Andrew Lecoiu
21 replies
Twoisaprime
Feb 13, 2025
L13832
an hour ago
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IMO Shortlist 2011, Algebra 3
orl   45
N an hour ago by Ilikeminecraft
Source: IMO Shortlist 2011, Algebra 3
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.

Proposed by Japan
45 replies
orl
Jul 11, 2012
Ilikeminecraft
an hour ago
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Hard FE with positive reals
egxa   8
N 2 hours ago by megarnie
Source: Turkey Olympic Revenge 2023 Shortlist A4
Find all functions $f:\mathbb{R^+}\to \mathbb{R^+}$ such that for all $x,y\in \mathbb{R^+}$
$f(xf(y)+y)=f(f(y))+yf(x)$
Proposed by Şevket Onur Yılmaz
8 replies
egxa
Jan 22, 2024
megarnie
2 hours ago
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Like Father Like Son... (or Like Grandson?)
AlperenINAN   1
N 2 hours ago by hakN
Source: Turkey TST 2025 P4
Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.
1 reply
AlperenINAN
Today at 6:09 AM
hakN
2 hours ago
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Crazy number theory
MTA_2024   5
N 2 hours ago by bjump
Find all couple $(p;q)$ of primes (greater than 5) such that : $$pq \mid (5^q-3^q)(5^p-3^p)$$
5 replies
MTA_2024
4 hours ago
bjump
2 hours ago
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hard number theory problem
Zavyk09   0
2 hours ago
Source: forgotten
Find all couple $(x, y)$ of positive integers such that:
$$2^n + 3^n \mid x^n + y^n, \forall n \in \mathbb{N}^*$$
0 replies
Zavyk09
2 hours ago
0 replies
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Slightly weird points which are not so weird
Pranav1056   9
N 2 hours ago by Retemoeg
Source: India TST 2023 Day 4 P1
Suppose an acute scalene triangle $ABC$ has incentre $I$ and incircle touching $BC$ at $D$. Let $Z$ be the antipode of $A$ in the circumcircle of $ABC$. Point $L$ is chosen on the internal angle bisector of $\angle BZC$ such that $AL = LI$. Let $M$ be the midpoint of arc $BZC$, and let $V$ be the midpoint of $ID$. Prove that $\angle IML = \angle DVM$
9 replies
Pranav1056
Jul 9, 2023
Retemoeg
2 hours ago
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The return of an inequality
giangtruong13   2
N 3 hours ago by sqing
Let $a,b,c$ be real positive number satisfy that: $a+b+c=1$. Prove that: $$\sum_{cyc} \frac{a}{b^2+c^2} \geq \frac{3}{2}$$
2 replies
giangtruong13
4 hours ago
sqing
3 hours ago
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a+b+c=3 ine
jokehim   1
N 4 hours ago by Cool12345678
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
1 reply
jokehim
Today at 9:48 AM
Cool12345678
4 hours ago
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a+b+c=3 ine
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jokehim
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jokehim
#1 h  Today at 9:48 AM
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Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
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Cool12345678
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Cool12345678
#2 h  4 hours ago
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Since $(2,0,0)$ majorizes $(1,1,0)$ and given the condition $a+b+c=3$, by Muirhead, we rewrite $$\sum_{cyc} \frac{a\left(b+c\right)}{bc+3}\leq\sum_{cyc} \frac{a\left(3-a\right)}{a^2+3}$$
Now, let $f(x)=\frac{x\left(3-x\right)}{x^2+3}$. Then, setting $y=\frac{x\left(3-x\right)}{x^2+3}$, we get $$ x^2y + 3y = 3x-x^2 \implies x^2(y + 1) + 3y-3x = 0.$$Using the discriminant of a quadratic, we get $$9-4(y+1)(3y)\geq0 \implies y\leq \frac{1}{2}$$Therefore, the maximum value of $f(x)$ is $\frac{1}{2}$ so for all $a,b,c$ we have $$\frac{a\left(3-a\right)}{a^2+3}\leq \frac{1}{2}$$Hence, $$\sum_{cyc} \frac{a\left(3-a\right)}{a^2+3}\leq \frac{3}{2}$$as desired.
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