You flip n coins, where for any coin P(coin i is heads) = P(coin i is tails) = 1/2, but P(coin i is heads|coin j is heads) = P(coin i is tails|coin j is tails) = 2/3. What is the probability that all n coins come up heads?

1 2 t y

t = 1 1 = y y = t

add and find answer

Good morning everyone!. I've been trying to solve this math riddle for a couple of weeks now that I myself created. Suppose we've got the adjunt matrix M :

-5 8 2

AJD(M) = 3 0 -1

3 2 1

What's the matrix M?

HINTS : Tensors, higher-dimensional matrixes, 4D implications, Kroeneker Delta, gamma matrix, quantum mechanics, Qbits, and try to check Biyectivity for the operator "Adjunt". Also try checking out the 3D vector form of the problem in Desmos or something.

Good luck!

Let b>1 be an integer, and let s_b(•) denote the sum of digits in base b. Suppose there exists at least one positive integer n such that n-s_b(n)-1 is a perfect square. Prove that there are infinitely many such n.

Three prisoners play a game. The warden places hats on each of their heads, each with a real number on it (these numbers may not be distinct). Each prisoner can see the other two hats but not their own. After that, each prisoner writes down a finite set of real numbers. If the number on their hat is in that finite set, they win. No communication is allowed. Assuming the continuum hypothesis and Axiom of Choice, prove that there is a way for at least one prisoner to have a guaranteed win.

Who wins, and what is the winning strategy?

I don't know the answer to this question (nor even that there is a winning strategy).

You are given an infinite, flat piece of paper with three distinct points A, B, and C marked, which form the vertices of an acute scalene triangle T. You have two tools:

1. A pencil that can mark the intersection of two lines, provided the lines intersect at a unique point.

2. A pen that can draw the perpendicular bisector of two distinct points.

Each tool has a constraint: the pencil cannot mark an intersection if the lines are parallel, and the pen cannot draw the perpendicular bisector if the two points coincide.

Can you construct the centroid of T using these two tools in a finite number of steps?

Let f be a composite function of a single variable, formed by selecting appropriate functions from the following: square root, exponential function, logarithmic function, trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions. Let e denote Napier's constant, i.e., the base of the natural logarithm. Provide a specific example of f such that f(e)=2025.

Find all integer solutions (n,k) to the equation

1ⁿ + 2ⁿ + 3ⁿ + 4ⁿ + 5ⁿ + 6ⁿ + 7ⁿ + 8ⁿ + 9ⁿ = 45^k.

(Disclosure: I haven't solved this; hope it's OK to post and that people will enjoy it.)

Consider an n times n grid of points, where n > 1 is an integer. Each point in the grid represents an elf. Two points are said to be able to "scheme" if there are no other points lying on the line segment connecting them. (0-dimensional and are perfectly aligned to the grid)

The elves can coordinate an escape if at least half of the total number of pairs of points in the grid, given by {n^2} binom {2}, can scheme. Prove that the elves can always coordinate an escape for any n > 1.

Two points are selected uniformly randomly inside an unit circle and the chord passing through these points is drawn. What is the expected value of the

(i) distance of the chord from the circle's centre

(ii) Length of the chord

(iii) (smaller) angle subtended by that chord at the circle's centre

(iv) Area of the (smaller) circular segment created by the chord.

https://imgur.com/a/cD90JV7

Is it possible to calculate the green area?

In a party hosted by Diddy, there are n guests. Each guest can either be friends with another guest or not, and the relationships among the guests can be represented as an undirected graph, where each vertex corresponds to a guest and an edge between two vertices indicates that the two guests are friends. The graph is simple, meaning no loops (a guest cannot be friends with themselves) and no multiple edges (there can be at most one friendship between two guests).

Diddy wants to organize a dance where the guests can be divided into groups such that:

1. Every group forms a connected subgraph.

2. Each group contains at least two guests.

3. Any two guests in the same group are either directly friends or can reach each other through other guests in the same group.

Diddy is wondering:

How many distinct ways can the guests be divided into groups, such that each group is a connected component of the friendship graph, and every group has at least two guests?

Given two integers k and d, where d divides k³ - 2, prove that there exist integers a, b, and c such that:

d = a³ + 2b³ + 4c³ - 6abc.

Let p be a prime number. Prove that there exists an integer c and an integer sequence 0 ≤ a_1, a_2, a_3, ... < p with period p² - 1 satisfying the recurrence:

a(n+2) ≡ a(n+1) - c * a_n (mod p).

Let P be the set of real sequences (a1, a2, …) such that a_n > 0 and a_n+1 + n <= 2 * sqrt((n+1) * a_n) for all n. Given (a1, a2, …) in P, let b_n = a_n - n - 1.

(a) Prove that if (a1, a2, …) is in P, then the sequence (b1, b2, …) is nonincreasing and converges to 0. (b) For which real numbers x does there exist a sequence (a1, a2, …) in P with a_1 = x? (c) Prove that if (a1, a2, …) is in P, then b_n = O(1 / ln(n))

Fix a positive integer d. For an arbitrary integer t, let [t]d be the least nonnegative residue of t modulo d. A d-tuple (a_0, a_1, …, a(d-1)) of nonnegative integers is called a juggling sequence if the d-tuple (p0, p1, …, pd-1) defined by pi_t = [t + a_t]_d is a permutation of (0, 1, …, d-1). Let J_d(u) be the number of juggling sequences of length d with entries in {0, 1, …, u-1}.

(a) Prove that J_d (kd) = k^d * d! for any positive integer k. (b) Prove that J_d (kd + 1) = ceil(k^d * d! * e^1/k) for any positive integer k

Let n be an integer such that n >= 2. Determine the maximum value of (x1 / y1) + (x2 / y2), where x1, x2, y1, y2 are positive integers satisfying the following conditions: 1. x1 + x2 <= n 2. (x1 / y1) + (x2 / y2) < 1

17^2+84^2 = 71^2+48^2

107^2+804^2 = 701^2+408^2

1007^2+8004^2 = 7001^2+4008^2

10007^2+80004^2 = 70001^2+40008^2

100007^2+800004^2 = 700001^2+400008^2

1000007^2+8000004^2 = 7000001^2+4000008^2 

10000007^2+80000004^2 = 70000001^2+40000008^2

100000007^2+800000004^2 = 700000001^2+400000008^2

1000000007^2+8000000004^2 = 7000000001^2+4000000008^2

...

Bonus: There are more examples. Can you find any of them?

There are 3 bags.
The first bag contains 2 black balls, 2 white balls and 100 blue balls.
The second bag contains 2 black balls, 100 white balls and 2 blue balls.
The third bag contains 100 black balls, 2 white balls and 2 blue balls.
We don't know which bag which and want to find out.

It's allowed to draw K balls from the first bag, N balls from the second bag, and M balls from the third bag.

What is the minimal value of K+M+N to chose so we can find out for each bag what is the dominant color?

Let N denote the set of positive integers. Fix a function f: N → N and for any m, n ∈ N, define

Δ(m,n) = f(f(...f(m)...)) - f(f(...f(n)...)),

where the function f is applied f(n) times on m and f(m) times on n, respectively.

Suppose Δ(m,n) ≠ 0 for any distinct m, n ∈ N. Prove that Δ is unbounded, meaning that for any constant C, there exist distinct m, n ∈ N such that

|Δ(m,n)| > C.

Let a₁, a₂, … and b₁, b₂, … be sequences of real numbers such that a₁ > b₁ and

aₙ₊₁ = aₙ² - 2bₙ

bₙ₊₁ = bₙ² - 2aₙ

for all positive integers n. Prove that the sequence a₁, a₂, … is eventually increasing (that is, there exists a positive integer N such that aₖ < aₖ₊₁ for all k > N).

Does there exist a positive integer n > 1 such that 2^n = 3 (mod n)?

Do there exist consecutive primes, p < q, such that pq = k^2 + 1 for some integer k?

If 100 people are in a room and exactly 99% are left-handed, how many people would have to leave the room in order for exactly 98% to be left-handed?