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2020 EGMO P2: Sum inequality with permutations
alifenix- 29
N
by Markas
Source: 2020 EGMO P2
Find all lists 
of non-negative real numbers such that the following three conditions are all satisfied:
[list]
[*]
;
[*]
;
[*] there is a permutation
of such that 
[/list]
A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example,
is a permutation of 
, and they are both permutations of 
. Note that any list is a permutation of itself.
of non-negative real numbers such that the following three conditions are all satisfied:[list]
[*]
;[*]
;[*] there is a permutation
of such that 
[/list]A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example,
is a permutation of 
, and they are both permutations of 
. Note that any list is a permutation of itself.
29 replies
IMO 2018 Problem 2
juckter 97
N
by Markas
Find all integers 
for which there exist real numbers 
satisfying 
, 
and
for 
.
Proposed by Patrik Bak, Slovakia
for which there exist real numbers 
satisfying 
, 
and
for 
.Proposed by Patrik Bak, Slovakia
97 replies
prove that a_50 + b_50 > 20
kamatadu 8
N
by Markas
Source: Canada Training Camp
The sequences 
and 
are such that, for every positive integer 
,
![\[ a_n > 0,\qquad\ b_n>0,\qquad\ a_{n+1}=a_n+\dfrac{1}{b_n},\qquad\ b_{n+1} = b_n+\dfrac{1}{a_n}. \]](//latex.artofproblemsolving.com/4/f/c/4fc9fd1cb31a25378268f20e019e639667e52910.png)
Prove that 
.
and 
are such that, for every positive integer 
,![\[ a_n > 0,\qquad\ b_n>0,\qquad\ a_{n+1}=a_n+\dfrac{1}{b_n},\qquad\ b_{n+1} = b_n+\dfrac{1}{a_n}. \]](http://latex.artofproblemsolving.com/4/f/c/4fc9fd1cb31a25378268f20e019e639667e52910.png)
Prove that 
.
8 replies
EGMO P4 infinite sequence
aditya21 29
N
by Markas
Source: EGMO 2015, Problem 4
Determine whether there exists an infinite sequence 
of positive integers
which satisfies the equality![\[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \]](//latex.artofproblemsolving.com/4/a/9/4a9cc873fd72a001413f0bc1558025e0673ea9cd.png)
for every positive integer .
of positive integerswhich satisfies the equality
![\[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \]](http://latex.artofproblemsolving.com/4/a/9/4a9cc873fd72a001413f0bc1558025e0673ea9cd.png)
for every positive integer .
29 replies
IMO 2014 Problem 1
Amir Hossein 133
N
by Markas
Let 
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that
![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](//latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.
133 replies
Sequences and limit
lehungvietbao 16
N
by Markas
Source: Vietnam Mathematical OLympiad 2014
Let 
be two positive sequences defined by 
and
![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](//latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
Prove that they are converges and find their limits.
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.Prove that they are converges and find their limits.
16 replies
Real triples
juckter 67
N
by Markas
Source: EGMO 2019 Problem 1
Find all triples 
of real numbers such that 
and
of real numbers such that 
and
67 replies
Social Club with 2k+1 Members
v_Enhance 24
N
by mathwiz_1207
Source: USA December TST for IMO 2013, Problem 1
A social club has 
members, each of whom is fluent in the same 
languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let 
be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of .
members, each of whom is fluent in the same 
languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let 
be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of .
24 replies
A strong inequality problem
hn111009 1
N
by Tung-CHL
Source: Somewhere
Let 
be the positive number satisfied 
Find the minimum of 
be the positive number satisfied 
Find the minimum of 
1 reply
Altitude configuration with two touching circles
Tintarn 3
N
by NumberzAndStuff
Source: Austrian MO 2024, Final Round P2
Let 
be an acute triangle with 
. Let 
denote the feet of its altitudes on 
and 
, respectively. Let 
denote the intersection of lines 
and 
. Prove that the circumcircles 
and 
of the two triangles 
and 
touch in 
.
(Karl Czakler)
be an acute triangle with 
. Let 
denote the feet of its altitudes on 
and 
, respectively. Let 
denote the intersection of lines 
and 
. Prove that the circumcircles 
and 
of the two triangles 
and 
touch in 
.(Karl Czakler)
3 replies
