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Why is the old one deleted?
EeEeRUT 11
N
by Mathgloggers
Source: EGMO 2025 P1
For a positive integer 
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
Here
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
Proposed by Paulius Aleknavičius, Lithuania
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
Here
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.Proposed by Paulius Aleknavičius, Lithuania
11 replies
Congruence related perimeter
egxa 2
N
by LoloChen
Source: All Russian 2025 9.8 and 10.8
On the sides of triangle 
, points 
are chosen such that when going around the triangle, the points occur in the order 
. It is given that
![\[
AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2.
\]](//latex.artofproblemsolving.com/4/c/f/4cf9e7bc009736836bcd10c1d8ba46bd3a91cd7b.png)
Prove that the perimeters of the triangles formed by the triplets 
and 
are equal.
, points 
are chosen such that when going around the triangle, the points occur in the order 
. It is given that![\[
AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2.
\]](http://latex.artofproblemsolving.com/4/c/f/4cf9e7bc009736836bcd10c1d8ba46bd3a91cd7b.png)
Prove that the perimeters of the triangles formed by the triplets 
and 
are equal.
2 replies
+1 w
number theory
Levieee 7
N
by g0USinsane777
Idk where it went wrong, marks was deducted for this solution
Show that for a fixed pair of distinct positive integers
and 
, there cannot exist infinitely many 
such that
![\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]](//latex.artofproblemsolving.com/5/4/d/54d20546de5736770a9929ac86163e3b06553b7d.png)
Let
![\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]](//latex.artofproblemsolving.com/6/6/8/668d53723d307556536736fda0e4a19be848834f.png)
Then,
![\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]](//latex.artofproblemsolving.com/1/9/f/19f2627a432354a757d3c9e8ba42e96f83f46726.png)
So:
![\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]](//latex.artofproblemsolving.com/0/7/3/073602693604bbf3cb1f5c1c2e5051e8994a5e78.png)
Therefore,
![\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]](//latex.artofproblemsolving.com/1/1/4/11441fab3ba768fee7a0eef4dbf68124d6637aec.png)
Let
![\[
(n + a)(n + b) = k^2.
\]](//latex.artofproblemsolving.com/0/e/c/0ec5e7bac2aebe9ec507727ed21f7ca028655222.png)
Assume 
. Then we have:
![\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]](//latex.artofproblemsolving.com/7/5/f/75fc52b6662df3a4c0cb2f40f49c4b3351c90a4d.png)
or it could also be that 
.
Without loss of generality, we take the first case:
![\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]](//latex.artofproblemsolving.com/1/f/5/1f5936241f7bebcf15c68fed5267bbfad2d40c2b.png)
Thus,
![\[
k_1 k_2 = \frac{n + b}{n + a}.
\]](//latex.artofproblemsolving.com/d/4/c/d4c1958e3554b58c3def282070f849842cc10912.png)
Since
, we have:
![\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]](//latex.artofproblemsolving.com/3/0/6/306a6a00c44d4e4db1b36c7e588f684f28fdbf44.png)
For infinitely many
, 
must be an integer, which is not possible.
Therefore, there cannot be infinitely many such.
Show that for a fixed pair of distinct positive integers
and 
, there cannot exist infinitely many 
such that![\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d20546de5736770a9929ac86163e3b06553b7d.png)
Let
![\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/6/6/8/668d53723d307556536736fda0e4a19be848834f.png)
Then,
![\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/1/9/f/19f2627a432354a757d3c9e8ba42e96f83f46726.png)
So:![\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/0/7/3/073602693604bbf3cb1f5c1c2e5051e8994a5e78.png)
Therefore,
![\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/1/1/4/11441fab3ba768fee7a0eef4dbf68124d6637aec.png)
Let
![\[
(n + a)(n + b) = k^2.
\]](http://latex.artofproblemsolving.com/0/e/c/0ec5e7bac2aebe9ec507727ed21f7ca028655222.png)
Assume 
. Then we have:![\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]](http://latex.artofproblemsolving.com/7/5/f/75fc52b6662df3a4c0cb2f40f49c4b3351c90a4d.png)
or it could also be that 
.Without loss of generality, we take the first case:
![\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]](http://latex.artofproblemsolving.com/1/f/5/1f5936241f7bebcf15c68fed5267bbfad2d40c2b.png)
Thus,
![\[
k_1 k_2 = \frac{n + b}{n + a}.
\]](http://latex.artofproblemsolving.com/d/4/c/d4c1958e3554b58c3def282070f849842cc10912.png)
Since
, we have:![\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]](http://latex.artofproblemsolving.com/3/0/6/306a6a00c44d4e4db1b36c7e588f684f28fdbf44.png)
For infinitely many
, 
must be an integer, which is not possible.Therefore, there cannot be infinitely many such
.
7 replies
and 
.
3 var inquality
sqing 0
Source: Own
Let 
be reals such that 
and 
Prove that
Let be reals such that 
and Prove that
![$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$](//latex.artofproblemsolving.com/6/2/2/62211222c148c86d65526d86dde603cb6598925f.png)
![$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$](//latex.artofproblemsolving.com/3/8/3/383d38cc87c1b1ea1e17e10e4bb95bed79329786.png)
be reals such that 
and 
Prove that
Let be reals such that 
and Prove that![$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$](http://latex.artofproblemsolving.com/6/2/2/62211222c148c86d65526d86dde603cb6598925f.png)
![$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$](http://latex.artofproblemsolving.com/3/8/3/383d38cc87c1b1ea1e17e10e4bb95bed79329786.png)
0 replies
Combinatorics
TUAN2k8 0
A sequence of integers 
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
Find the smallest integer
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)
is call 
if it satisfies the following properties:
and 
for all indices 
.
.Find the smallest integer
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)
0 replies
pqr/uvw convert
Nguyenhuyen_AG 4
N
by SunnyEvan
Source: https://github.com/nguyenhuyenag/pqr_convert
Hi everyone,
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
I hope you’ll find it helpful!
Download: pqr_convert
Screenshot:
IMAGE
IMAGE
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.I hope you’ll find it helpful!
Download: pqr_convert
Screenshot:
IMAGE
IMAGE
4 replies
A nice lemma about incircle and his internal tangent
manlio 0
Have you a nice proof for this lemma?
Thnak you very much
Thnak you very much
0 replies
Nice problem about a trapezoid
manlio 0
Have you a nice solution for this problem?
Thank you very much
Thank you very much
0 replies
IHC 10 Q25: Eight countries participated in a football tournament
xytan0585 0
Source: International Hope Cup Mathematics Invitational Regional Competition IHC10
Eight countries sent teams to participate in a football tournament, with the Argentine and Brazilian teams being the strongest, while the remaining six teams are similar strength. The probability of the Argentine and Brazilian teams winning against the other six teams is both 
. The tournament adopts an elimination system, and the winner advances to the next round. What is the probability that the Argentine team will meet the Brazilian team in the entire tournament?
. 
. 
. 
. 
. 
. The tournament adopts an elimination system, and the winner advances to the next round. What is the probability that the Argentine team will meet the Brazilian team in the entire tournament?
. 
. 
. 
. 
. 
0 replies
