is a pair of real numbers such that
.
Y by sky.mty
Given 
and 
with sum 
. Show that
![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](//latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
and 
with sum 
. Show that![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.This post has been edited 1 time. Last edited by RainbowNeos, Mar 26, 2025, 2:25 PM
Reason: typo
Reason: typo
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possible triangle inequality
sunshine_12 0
|a| + |b| + |c| − |a + b| − |b + c| − |c + a| + |a + b + c| ≥ 0
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot
0 replies
Sequence of functions
mathlover1231 0
Source: Own
Let f:N->N be a function such that f(1) = 1, f(n+1) = f(n) + 2^f(n) for every positive integer n. Prove that all numbers f(1), f(2), …, f(3^2023) give different remainders when divided by 3^2023
0 replies
Geo challenge on finding simple ways to solve it
Assassino9931 1
N
by MathLuis
Source: Bulgaria Spring Mathematical Competition 2025 9.2
Let 
be an acute scalene triangle inscribed in a circle 
. The angle bisector of 
intersects 
at 
and at 
. The point 
is the midpoint of 
. Let 
be the altitude in 
, and the circumcircle of 
intersects again at 
. Let 
be the midpoint of , and let 
be the reflection of 
with respect to . Prove that the triangles 
and 
are similar.
be an acute scalene triangle inscribed in a circle 
. The angle bisector of 
intersects 
at 
and at 
. The point 
is the midpoint of 
. Let 
be the altitude in 
, and the circumcircle of 
intersects again at 
. Let 
be the midpoint of , and let 
be the reflection of 
with respect to . Prove that the triangles 
and 
are similar.
1 reply
Infinite integer sequence problem
mathlover1231 2
N
by mathlover1231
Let a_1, a_2, … be an infinite sequence of pairwise distinct positive integers and c be a real number such that 0 < c < 3/2. Prove that there exist infinitely many positive integers k such that lcm(a_k, a_{k+1}) > ck.
2 replies
Minimize this in any way you like
Assassino9931 3
N
by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 12.1
In terms of the real numbers 
and 
determine the minimum value of 
as well as all values of 
which attain it.
and 
determine the minimum value of 
as well as all values of 
which attain it.
3 replies
1 viewing
Very hard FE problem
steven_zhang123 0
Source: 0
Given a real number 
such that 
(where 
), and a functional equation 
that satisfies 
for all , has a finite number of solutions. Find such .
(Here,
is the function obtained by composing 
times, that is, 
)
such that 
(where 
), and a functional equation 
that satisfies 
for all , has a finite number of solutions. Find such .(Here,
is the function obtained by composing 
times, that is, 
)
0 replies
Train yourself on folklore NT FE ideas
Assassino9931 1
N
by bin_sherlo
Source: Bulgaria Spring Mathematical Competition 2025 9.4
Determine all functions 
such that 
divides 
for any positive integers and .
such that 
divides 
for any positive integers and .
1 reply
FE f(x)f(y)+1=f(x+y)+f(xy)+xy(x+y-2)
steven_zhang123 4
N
by steven_zhang123
Find all functions 
such that for all 
, we have 
.
such that for all 
, we have 
.
4 replies
Heavy config geo involving mixtilinear
Assassino9931 0
Source: Bulgaria Spring Mathematical Competition 2025 12.4
Let be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of and also tangent to lines 
and . A circle passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of and also tangent to lines 
and . A circle passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
0 replies
Put this on a new Jane Street T-shirt
Assassino9931 0
Source: Bulgaria Spring Mathematical Competition 2025 12.3
Given integers 
, the points 
are chosen independently and uniformly at random on a circle of circumference 
. That is, for each 
, for any 
, and for any arc 
of length 
on the circle, we have 
. What is the probability that there exists an arc of length 
on the circle that contains all the points ?
, the points 
are chosen independently and uniformly at random on a circle of circumference 
. That is, for each 
, for any 
, and for any arc 
of length 
on the circle, we have 
. What is the probability that there exists an arc of length 
on the circle that contains all the points ?
0 replies
cyclic ineq not tight
G
H
G
H
Source: own
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![$$\min\{x_i^2, x_{i+1}\} = \min\{x_i^2, x_i^2, x_{i+1}\} \leq \operatorname{GM}(x_i^2, x_i^2, x_{i+1}) = \sqrt[3]{x_i^2 \cdot x_i^2 \cdot x_{i+1}}.$$](http://latex.artofproblemsolving.com/a/f/5/af518af531932d605b7a08e99904ffd28987e3f5.png)
![$$\sum \min\{x_i^2, x_{i+1}\} \leq \sum \sqrt[3]{x_i^4 x_{i+1}}\leq \sum \frac{\frac{1}{2} x_i + x_i^2 + 2 x_i x_{i+1} }{3} \leq \frac{1}{6} \sum x_i + \frac{1}{3}\left(\sum x_i \right)^2 = \frac{1}{2}$$](http://latex.artofproblemsolving.com/b/6/d/b6d8c8cd6947f8f2cb1bdc97f593e529116f25ff.png)
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