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Some Length Equality
SatisfiedMagma 7
N
by BlueCrate04
Source: RMO 2024/5
Let 
be a cyclic quadrilateral such that 
. Let 
be the circumcenter of and 
be the point on 
such that 
is perpendicular to . Prove that
![\[ OB\cdot(AB+CD) = OL\cdot(AC + BD).\]](//latex.artofproblemsolving.com/8/8/1/881af4ec195f404a0521582095aa065060866f27.png)
Proposed by Rijul Saini
be a cyclic quadrilateral such that 
. Let 
be the circumcenter of and 
be the point on 
such that 
is perpendicular to . Prove that![\[ OB\cdot(AB+CD) = OL\cdot(AC + BD).\]](http://latex.artofproblemsolving.com/8/8/1/881af4ec195f404a0521582095aa065060866f27.png)
Proposed by Rijul Saini
7 replies
functional equation with exponentials
produit 6
N
by produit
Find all solutions of the real valued functional equation:
f(\sqrt{x^2+y^2})=f(x)f(y).
Here we do not assume f is continuous
f(\sqrt{x^2+y^2})=f(x)f(y).
Here we do not assume f is continuous
6 replies
Geomerty
Teacher886699 2
N
by mickeymouse7133
Source: Geometry plese help Me
To each side aof a convex polygon we assign the maximum area of a triangle contained
in the polygon and having a as one of its sides. Show that the sum of the areas assigned to all
sides of the polygon is not less than twice the area of the polygon.
in the polygon and having a as one of its sides. Show that the sum of the areas assigned to all
sides of the polygon is not less than twice the area of the polygon.
2 replies
Inspired by lbh_qys
sqing 2
N
by BraveHedgehog91
Source: Own
Let 
be real numbers such that 
and 
Prove that
Where 
be real numbers such that 
and 
Prove that
Where 
2 replies
2025 KMO Inequality
Jackson0423 0
Source: 2025 KMO Round 1 Problem 20
Let
be real numbers satisfying![\[
x_1 + x_2 + \cdots + x_6 = 6,
\]](http://latex.artofproblemsolving.com/6/2/2/622a329112ce9613a5c389b4c9078cd43c1c3f05.png)
![\[
x_1^2 + x_2^2 + \cdots + x_6^2 = 18.
\]](http://latex.artofproblemsolving.com/8/f/a/8fa3f74ba7c3587576116a7bdbb4e9f88d389133.png)
Find the maximum possible value of the product![\[
x_1 x_2 x_3 x_4 x_5 x_6.
\]](http://latex.artofproblemsolving.com/3/b/4/3b48315e2775b4ac3913d7c9b02dbc81daed8d15.png)
0 replies
JBMO TST- Bosnia and Herzegovina 2022 P1
Motion 5
N
by justaguy_69
Source: JBMO TST 2022 Bosnia and Herzegovina P1
Let 
be real numbers such that 
. Find the value of 
and find at least one triplet 
that satisfy those conditions.
be real numbers such that 
. Find the value of 
and find at least one triplet 
that satisfy those conditions.
5 replies
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic 0
Source: Serbian selection contest for the IMO 2025
For an 
table filled with natural numbers, we say it is a divisor table if:
- the numbers in the
-th row are exactly all the divisors of some natural number 
,
- the numbers in the
-th column are exactly all the divisors of some natural number 
,
-
for every 
.
A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
Proposed by Pavle Martinović
table filled with natural numbers, we say it is a divisor table if:- the numbers in the
-th row are exactly all the divisors of some natural number 
,- the numbers in the
-th column are exactly all the divisors of some natural number 
,-
for every 
.A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.Proposed by Pavle Martinović
0 replies
Serbian selection contest for the IMO 2025 - P4
OgnjenTesic 0
Source: Serbian selection contest for the IMO 2025
For a permutation 
of the set 
, define its colorfulness as the greatest natural number 
such that:
- For all
, , if 
, then 
.
What is the maximum possible colorfulness of a permutation of the set
? Determine how many such permutations have maximal colorfulness.
Proposed by Pavle Martinović
of the set 
, define its colorfulness as the greatest natural number 
such that:- For all
, , if 
, then 
.What is the maximum possible colorfulness of a permutation of the set
? Determine how many such permutations have maximal colorfulness.Proposed by Pavle Martinović
0 replies
Serbian selection contest for the IMO 2025 - P3
OgnjenTesic 0
Source: Serbian selection contest for the IMO 2025
Find all functions 
such that:
-
is strictly increasing,
- there exists
such that 
for all 
,
- for every
, there exists 
such that
![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](//latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
such that:-
is strictly increasing,- there exists
such that 
for all 
,- for every
, there exists 
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
0 replies
Upper bound on products in sequence
tapir1729 10
N
by Mathandski
Source: TSTST 2024, problem 7
An infinite sequence 
, 
, 
, 
of real numbers satisfies
![\[
a_{2n-1} + a_{2n} > a_{2n+1} + a_{2n+2} \qquad \mbox{and} \qquad a_{2n} + a_{2n+1} < a_{2n+2} + a_{2n+3}
\]](//latex.artofproblemsolving.com/4/d/a/4da3b88a20c42c1798141b8db086de341cfb9a67.png)
for every positive integer . Prove that there exists a real number 
such that 
for every positive integer .
Merlijn Staps
, 
, 
, 
of real numbers satisfies![\[
a_{2n-1} + a_{2n} > a_{2n+1} + a_{2n+2} \qquad \mbox{and} \qquad a_{2n} + a_{2n+1} < a_{2n+2} + a_{2n+3}
\]](http://latex.artofproblemsolving.com/4/d/a/4da3b88a20c42c1798141b8db086de341cfb9a67.png)
for every positive integer . Prove that there exists a real number 
such that 
for every positive integer .Merlijn Staps
10 replies
