G
Topic
First Poster
Last Poster
Number of sets S
Jackson0423 2
N
by Jackson0423
Let 
be a set consisting of non-negative integers such that:
1.
,
2. For any
, both 
and 
.
Find the number of such sets.
be a set consisting of non-negative integers such that:1.
,2. For any
, both 
and 
.Find the number of such sets
.
2 replies
F.E....can you solve it?
Jackson0423 16
N
by jasperE3
Find all functions 
such that
![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](//latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
16 replies
Find all positive a,b
shobber 14
N
by reni_wee
Source: APMO 2002
Find all positive integers 
and 
such that
![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](//latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
are both integers.
and 
such that![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](http://latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
are both integers.
14 replies
Geo metry
TUAN2k8 2
N
by TUAN2k8
Help me plss!
Given an acute triangle
. Points 
and 
lie on segments 
and 
, respectively. Lines 
and 
intersect at point 
. The circumcircles of triangles 
and 
intersect at a second point 
. The circumcircles of triangles and 
intersect at a second point 
. Point 
lies on segment 
such that 
. Prove that triangles 
and 
are similar.
Given an acute triangle
. Points 
and 
lie on segments 
and 
, respectively. Lines 
and 
intersect at point 
. The circumcircles of triangles 
and 
intersect at a second point 
. The circumcircles of triangles and 
intersect at a second point 
. Point 
lies on segment 
such that 
. Prove that triangles 
and 
are similar.
2 replies
(not so) small set of residues generates all of F_p upon applying Q many times
62861 14
N
by john0512
Source: RMM 2019 Problem 6
Find all pairs of integers 
, both greater than 1, such that the following holds:
For any monic polynomial of degree 
with integer coefficients and for any prime 
, there exists a set 
of at most 
integers, such that
![\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]](//latex.artofproblemsolving.com/f/9/1/f919f2391121c2b0551d29d4f09ea6cde631a8ba.png)
contains a complete residue system modulo 
(i.e., intersects with every residue class modulo ).
, both greater than 1, such that the following holds:For any monic polynomial
of degree 
with integer coefficients and for any prime 
, there exists a set 
of at most 
integers, such that![\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]](http://latex.artofproblemsolving.com/f/9/1/f919f2391121c2b0551d29d4f09ea6cde631a8ba.png)
contains a complete residue system modulo 
(i.e., intersects with every residue class modulo ).
14 replies
find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3
parmenides51 13
N
by SimplisticFormulas
Source: JBMO Shortlist 2017 NT5
Find all positive integers 
such that there exists a prime number , such that 
is a power of 
.
Note. A power of is a number of the form 
where is a positive integer.
such that there exists a prime number , such that 
is a power of 
.Note. A power of
is a number of the form 
where is a positive integer.
13 replies
Functional equation of nonzero reals
proglote 5
N
by TheHimMan
Source: Brazil MO 2013, problem #3
Find all injective functions 
from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](//latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](http://latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
5 replies
5-th powers is a no-go - JBMO Shortlist
WakeUp 8
N
by sansgankrsngupta
Prove that there are are no positive integers 
and 
such that 
.
and 
such that 
.
are positive isn't necessary.8 replies
1 viewing
Chess game challenge
adihaya 20
N
by cursed_tangent1434
Source: 2014 BAMO-12 #5
A chess tournament took place between 
players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly 
games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than 
and no more than 
games.
players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly 
games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than 
and no more than 
games.
20 replies
Permutations inequality
OronSH 13
N
by sansgankrsngupta
Source: ISL 2023 A5
Let 
be positive integers such that
[list=disc]
[*] is a permutation of 
, and
[*]
is a permutation of 
.
[/list]
Prove that
.
be positive integers such that[list=disc]
[*]
is a permutation of 
, and[*]
is a permutation of 
.[/list]
Prove that
.
13 replies
Cool Integral, Cooler Solution
Existing_Human1 2
N
by ysharifi
Source: https://youtu.be/YO38MCdj-GM?si=DCn6DaQTeX8RXhl0
Bonus points if you can do it without Feynman
2 replies
