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The order of colors
Entei 0
There are 
balls, with 
red, green, and blue balls, randomly arranged in a row. Two observers, one at the front and one at the back, each record the order of the first appearance of each color. What is the probability that both observers record the same order of colors?
For example, the sequence RGGBRB would be read as RGB for the front observer and BRG for the back observer.
balls, with 
red, green, and blue balls, randomly arranged in a row. Two observers, one at the front and one at the back, each record the order of the first appearance of each color. What is the probability that both observers record the same order of colors?For example, the sequence RGGBRB would be read as RGB for the front observer and BRG for the back observer.
0 replies
Problem in probability theory
Tip_pay 2
N
by elizhang101412
Find the probability that if four numbers from 
to 
(inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by 
, or all will be divisible by 
to 
(inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by 
, or all will be divisible by 
2 replies
Find the probability
ali3985 0
Let 
be a set of Natural numbers from to 
.
Now choose
(
) distinct elements from this set.
What is the probability of these numbers to be an increasing geometric progression ?
be a set of Natural numbers from to 
.Now choose
(
) distinct elements from this set.What is the probability of these numbers to be an increasing geometric progression ?
0 replies
hard problem
Cobedangiu 1
N
by Cobedangiu
Given any natural number 
satisfying 
is a parameter; 
* and 
*)
Write each natural number from
on sheets of paper so that no sheet has the same number
Prove: The probability of drawing
satisfying of the given number is ![$\frac{[\frac{k}{m}] + 1}{k + 1}$](//latex.artofproblemsolving.com/6/a/7/6a7a281f7b6d480abd702a525e7a7a4f436bb7c5.png)
satisfying 
is a parameter; 
* and 
*)Write each natural number from
on sheets of paper so that no sheet has the same numberProve: The probability of drawing
satisfying of the given number is ![$\frac{[\frac{k}{m}] + 1}{k + 1}$](http://latex.artofproblemsolving.com/6/a/7/6a7a281f7b6d480abd702a525e7a7a4f436bb7c5.png)
1 reply
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
Student's domination
Entei 0
Given students and their test results on different subjects, we say that student dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.
Well, it seems like more clarity and formality are needed.
Given vectors in a -dimensional space, 
, 
. We say that a vector 
is dominant to another vector 
if and only if all components of are greater than those of . Assume that no components are the same, that is 
.
Find the probability that there exists a pair of vectors that one is dominant to another.
students and their test results on different subjects, we say that student dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.Well, it seems like more clarity and formality are needed.
Given
vectors in a -dimensional space, 
, 
. We say that a vector 
is dominant to another vector 
if and only if all components of are greater than those of . Assume that no components are the same, that is 
.Find the probability that there exists a pair of vectors that one is dominant to another.
0 replies
two subsets with no fewer than four common elements.
micliva 37
N
by Maximilian113
Source: All-Russian Olympiad 1996, Grade 9, First Day, Problem 4
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.
A. Skopenkov
A. Skopenkov
37 replies
Did you talk to Noga Alon?
pohoatza 35
N
by shendrew7
Source: IMO Shortlist 2006, Combinatorics 3, AIMO 2007, TST 6, P2
Let 
be a finite set of points in the plane such that no three of them are on a line. For each convex polygon 
whose vertices are in , let 
be the number of vertices of , and let 
be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of 
, 
, and 
vertices respectively. Prove that for every real number 
![\[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\]](//latex.artofproblemsolving.com/3/2/7/3279b2917f21c8f3c2b819e39a32528ad18e4b4b.png)
where the sum is taken over all convex polygons with vertices in .
Alternative formulation:
Let
be a finite point set in the plane and no three points are collinear. A subset 
of will be called round if its elements is the set of vertices of a convex 
gon 
For each round subset let 
be the number of points from which are exterior from the convex gon Subsets with 
and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial
![\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}.
\]](//latex.artofproblemsolving.com/e/b/c/ebc6f8fa2cc303e1062c5a95d4f65c9d2691d0ca.png)
Show that the sum of polynomials for all round subsets is exactly the polynomial
Proposed by Federico Ardila, Colombia
be a finite set of points in the plane such that no three of them are on a line. For each convex polygon 
whose vertices are in , let 
be the number of vertices of , and let 
be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of 
, 
, and 
vertices respectively. Prove that for every real number 
![\[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\]](http://latex.artofproblemsolving.com/3/2/7/3279b2917f21c8f3c2b819e39a32528ad18e4b4b.png)
where the sum is taken over all convex polygons with vertices in .Alternative formulation:
Let
be a finite point set in the plane and no three points are collinear. A subset 
of will be called round if its elements is the set of vertices of a convex 
gon 
For each round subset let 
be the number of points from which are exterior from the convex gon Subsets with 
and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial![\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}.
\]](http://latex.artofproblemsolving.com/e/b/c/ebc6f8fa2cc303e1062c5a95d4f65c9d2691d0ca.png)
Show that the sum of polynomials for all round subsets is exactly the polynomial
Proposed by Federico Ardila, Colombia
35 replies
Queer Dice
scls140511 2
N
by SunnyEvan
Source: 2024 China Round 1 (Gao Lian)
5 Alice has an unfair dice such that the probability of rolling a , 
, , 
, , 
forms an arithmetic sequence, in this order. Bob rolls the dice twice, getting 
and 
respectively. There is a 
probability that 
. Find the probability such that 
.
, 
, , 
, , 
forms an arithmetic sequence, in this order. Bob rolls the dice twice, getting 
and 
respectively. There is a 
probability that 
. Find the probability such that 
.
2 replies
Covering 18 points with a figure
topologicalsort 1
N
by Assassino9931
Source: Bulgarian Autumn Math Tournament 12.4
Let 
be a figure made of squares, a right isosceles triangle and a quarter circle (all unit sized) as shown below: IMAGE
Prove that any
points in the plane can be covered with copies of , which don't overlap (copies of may be rotated or flipped)
be a figure made of squares, a right isosceles triangle and a quarter circle (all unit sized) as shown below: IMAGEProve that any
points in the plane can be covered with copies of , which don't overlap (copies of may be rotated or flipped)1 reply
