(a) 25%

(b) 50%

(c) 50%

(d) 100%

Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?

Let G be a connected graph with n vertices such that the chromatic number of G is k. Prove that the number of edges |E(G)| is at least kC2 + n - k, where kC2 represents the number of ways to choose 2 items from k.

For $1, you can pick up any number of regular 6-face dice and roll them.

If more odd than even numbers come out, you lose the biggest odd number in dollars (eg 514 -> lose $5, net loss $6).

If more even than odd numbers come out, you win the biggest even number in dollars (eg 324 -> win $4, net win $3).

If the same number of odd and even numbers come out, you win or lose nothing (eg 1234 -> win $0, net loss $1).

What is your average win with best play ?

Let's have some fun with games with incomplete information, making the information even more incomplete in the problem that was posted earlier this week by /u/Kindness_empathy

Initial problem:

3 people are blindfolded and placed in a circle. 9 coins are distributed between them in a way that each person has at least 1 coin. As they are blindfolded, each person only knows the number of coins that they hold, but not how many coins others hold.

Each round every person must (simultaneously) pass 1 or more of their coins to the next person (clockwise). How can they all end up with 3 coins each?

Before the game they can come up with a collective strategy, but there cannot be any communication during the game. They all know that there are a total of 9 coins and everything mentioned above. The game automatically stops when they all have 3 coins each.

Now what happens to the answer if the 3 blindfolded players also wear boxing gloves, meaning that they can't easily count how many coins are in front of them? So, a player never knows how many coins are in front of them. Of course this means that a player has no way to know for sure how many coins they can pass to the next player, so the rules must be extended to handle that scenario. Let's solve the problem with the following rule extensions:

A) When a player chooses to pass n coins and they only have m < n coins, m coins are passed instead. No player is aware of how many coins were actually passed or that the number was less than what was intended.

B) When a player chooses to pass n coins and they only have m < n coins, 1 coin is passed instead (the minimum from the basic rules). No player is aware of how many coins were actually passed or that the number was less than what was intended.

C) When a player chooses to pass n coins and they only have m < n coins, 0 coins are passed instead. No player is aware of how many coins were actually passed or that the number was less than what was intended. Now the game is really different because of the ability to pass 0 coins, so we need to sanitize it a little with a few more rules:

• Let's add the additional constraint that players cannot announce that they want to give 10 or more coins and therefore guarantee that they pass 0 (though of course if they announce 9 in the first round, they are guaranteed to pass 0 because they cannot have more than 7 initially).
• Let's also say that players can still pass all their coins even though they may receive 0 coins, meaning that they might end a turn with 0 coins in front of them.

D) When a player chooses to pass n coins and they only have m < n coins, n coins are passed anyway. The player may end up with a negative amount of coins. Who cares, after all? Who said people should only ever have a positive amount of coins? Certainly not banks.

Bonus question: What happens if we lift the constraint that the game automatically ends when the players each have 3 coins, and instead the players must simultaneously announce at each round whether they think they've won. If any player thinks they've won while they haven't, they all instantly lose.

Disclaimer: I don't have a satisfying answer to C as of now, but I think it's possible to find >!a general non-constructive solution for similar problems!<, which can be another bonus question.

A class consists of 10 girls and 10 boys, who are seated randomly, forming 10 pairs. What is the probability that all pairs consist of a girl and a boy?

3 people are blindfolded and placed in a circle. 9 coins are distributed between them in a way that each person has at least 1 coin. As they are blindfolded, each person only knows the number of coins that they hold, but not how many coins others hold.

Each round every person must (simultaneously) pass 1 or more of their coins to the next person (clockwise). How can they all end up with 3 coins each?

Before the game they can come up with a collective strategy, but there cannot be any communication during the game. They all know that there are a total of 9 coins and everything mentioned above. The game automatically stops when they all have 3 coins each.

Same setup as this problem (and spoiler warning): https://www.reddit.com/r/mathriddles/comments/1i73qa8/correlated_coins/

Depending on how you modeled the coins, you could get many different answers for the probability that all the coins come up heads. Suppose you flip 3k+1 coins. Find the maximum, taken over all possible distributions that satisfy the conditions of that problem, of the probability that all the coins come up heads. Or, show that it is >!(k+1)/(4k+2)!<.

correlated coins is a fun problem, but the solution is not unique, so i add more constraints.

there are n indistinguishable coins, where H (head) and T (tail) is not necessary symmetric.

each coin is fair , P(H) = P(T) = 1/2

the condition prob of a coin being H (or T), given k other coins is H (or T), is given by (k+1)/(k+2)

P(H | 1H) = P(T | 1T) = 2/3

P(H | 2H) = P(T | 2T) = 3/4

P(H | 3H) = P(T | 3T) = 4/5 and so on (till k=n-1).

determine the distribution of these n coins.

bonus: prove that the distribution is unique.

edit: specifically what is the probability of k heads (n-k) tails.

Same setup as this problem(and spoilers for it I guess): https://www.reddit.com/r/mathriddles/comments/1i73qa8/correlated_coins/

Depending on how you modeled the coins, you could get many different answers for that problem. However, the 3 models in the comments of that post all agreed that the probability of getting 3 heads with 3 flips is 1/4. Is it true that every model of the coins that satisfies the constraints in that problem will have a 1/4 chance of flipping 3 heads in 3 flips?

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

1^n + 2^n + 3^n + 4^n + 5^n + 6^n + 7^n + 8^n + 9^n = 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 p^2 - 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?