Let a(n) be the expansion of n in base -2. Examples:

2 = 1(-2)^2 + 1(-2)^1 + 0(-2)^0 = 4 - 2 + 0 = 110_(-2)

3 = 1(-2)^2 + 1(-2)^1 + 1(-2)^0 = 4 - 2 + 1 = 111_(-2)

6 = 1(-2)^4 + 1(-2)^8 + 0(-2)^2 + 1(-2)^1 + 0(-2)^0 = 16 - 8 + 0 - 2 + 0 = 11010_(-2)

For which n are the digits of a(n) all 1's?

Let S be the set of integers that are the sum of 4, but no fewer, squares of positive integers: (7, 15, 23, 28, ...). Show that S contains infinitely many consecutive pairs: (n, n+1), but no consecutive triples: (n, n+1, n+2).

More general variation of this problem. What is the probability that the mean of n random numbers (independent and uniform in [0,1]) is lower than the smallest number multiplied by a factor f > 1?

Generate n random numbers, independent and uniform in [0,1]. What’s the probability that all but one of them is greater than their average?

N >= 2 players play a game - they are each given independently and uniformly a number from [0, 1]. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations.

Without any analysis, we know that the optimal strategy is to guess "higher" if one's number exceeds a certain value depending on the information available to the player so far.

Question: What is the optimal strategy?

easier variant of this recently unsolved* problem (*as of the time writing this).

Let A be a set of n points randomly placed on a circle. In terms of n, determine the probability that the convex hull of A contains the center of the circle.

note: this might give some insight to the original problem, or not... i had yet to make it work on 3D.

Let k_1, ..., k_n be uniformly chosen points in (0,1) and let A_i be the interval (k_i, k_i + 1/n). In the limit as n approaches infinity, what is expected value of the total length of the union of the A_i?

Find a combination of four tetrahedral dice with the following special conditions.

As described in Efron's Dice, a set of four tetrahedral (four-sided) dice satisfying the criteria for nontransitivity under the specified conditions must meet the following requirements:

1. Cyclic Winning Probabilities:

There is a cyclic pattern of winning probabilities where each die has a 9/16 (56.25%) chance of beating another in a specific sequence. For dice ( A ), ( B ), ( C ), and ( D ), the relationships are as follows:
Die ( A ) has a 9/16 chance of winning against die ( B ).
Die ( B ) has a 9/16 chance of winning against die ( C ).
Die ( C ) has a 9/16 chance of winning against die ( D ).
Die ( D ) has a 9/16 chance of winning against die ( A ).

This structure forms a closed loop of dominance, where each die is stronger than another in a cyclic manner rather than following a linear order.

2. Equal Expected Values:

The expected value of each die is 60, ensuring that the average outcome of rolling any of the dice is identical. Despite these uniform expected values, the dice still exhibit nontransitive relationships.

3. Prime Number Faces:

Each face of the dice is labeled with a prime number, making all four numbers on each die distinct prime numbers.

4. Distinct Primes Across All Dice:

There are exactly 16 distinct prime numbers used across the four dice, ensuring that no prime number is repeated among the dice.

5. Equal Win Probabilities for Specific Pairs:

The winning probability between dice ( A ) and ( C ) is exactly 50%, indicating that neither die has an advantage over the other. Similarly, the winning probability between dice ( B ) and ( D ) is also 50%, ensuring an even matchup.

These conditions define a set of nontransitive tetrahedral dice that exhibit cyclic dominance with 9/16 winning probabilities. The dice share equal expected values and are labeled with 16 unique prime numbers, demonstrating the complex and non-intuitive nature of nontransitive probability relationships.

easier variant of this recent problem

An adventurer is doing a quest: slay the blob of size N>=1. when a blob size n is slain, it splits into (more accurately, creates) multiple blobs of smaller positive integer size. the probability that size n blob creating size k blob is k/n independent of other values of k. The quest is completed iff all blobs are slain and no new blob is created.

The game designer wants to gauge the difficulty of blob size N.

Find the expected number of blob created/slain for each blob size to complete the quest.

edit to clarify: find the expected number of blob size k, created by one blob size n.

Define the n-hedron to be a three dimensional shape that has n vertices. Assume this n-hedron to be contained within a sphere, with each of the n vertices randomly placed on the surface of the sphere. Determine a function P(n), in terms of n, that calculates the probability that the n-hedron contains the spheres center.

A man is playing a magical pipe organ - every chord is an integer number of decibals (dB) loud. The softest chord is 0 dB. Every chord of N > 0 dB creates a random number of echoes - for every 0 <= n <= N-1, an echo of volume n dB is created with probability (N-n)/N independently of other values of n. These echoes then independently produce their own echoes.

Question: What is the mean, median and mode of the number of echoes produced by a chord of volume N dB?

Notes:

• In the abscene of exact values, approximations and asymptotics are welcome.

• By median, we mean the smallest n for which the number of echoes is less than n with probability at least 1/2.

• By mode, we mean that value of n that has the greatest chance of occurring.

One sphere is inside another sphere. Which sphere has the largest surface area?

Find all non-decreasing and continuous f: ℝ-> ℝ such that f(f(x))=f(x) for all x∈ ℝ

Problem is not mine

Place points on the plane independently with density 1 and draw a circle of radius r around each point (Poisson distributed -> Poisson = fish -> fish puddles).

Let L(r) be the expected value of the supremum of the lengths of line segments starting at the origin and not intersecting any circle. Is L(r) finite for r > 0?

You have 8 batteries, 4 are dead and 4 have charges, 2 charged batteries are required to use a flashlight. How many battery pairings must you test for the flashlight to turn on? The question works on worst case scenario so there's no finding a working pair on the first try. It's always explained that you need to be sure, in the worst case scenario, that you have two working pairs.

A basic strategy (explained below) gets you to 8 tests. The optimal solution is said to be 7 tests. I have worked out it can be done in 4 tests. Can anybody find out how?

8 Test Strategy:

4 good batteries and 4 bad batteries for a total of 8 batteries.

Label them 1, 2, 3, 4, 5, 6, 7, 8.

Test them in pairs: 1+2, 3+4, 5+6, 7+8. In the worst case no pairs turn on the torch.

We know from these tests there has to be done one bad battery per tested pair. Since there are four bad batteries in total we must have exactly one bad battery per pair. Thus each pair also has one good battery.

We go on to test one pair with another pair.

1+3, 1+4, 2+3, 2+4.

In the end the pair of 2+4 work be good and the torch will turn. Here we have 8 tests.

Can anybody see how we can get a working pair in only 4 tests?

For every r > 0 let C(r) be the set of circles of radius r around integer points in the plane except for the origin. Let L(r) be the supremum of the lengths of line segments starting at the origin and not intersecting any circle in C(r). Show that

lim L(r) - 1/r = 0,

where the limit is taken as r goes to 0.

We have 2 distinct sets of 2n points on 2D plane, set A and B. Can we always bisect the plane (draw an infinite line) such that we have equal number of points on both sides from both sets (n points of A and n points of B on side 1 and same on side 2)? (We have n points of A and n point of B on each side)

Edit : no 3 points are collinear and no points can lie on the line

In a cylindrical grid of offset squares, each row has 2N cell arranged in a cycle. The first row starts with alternating white and green cells. For every row after that, a cell copy the color above it if both cells above are the same, otherwise it has a 50% chance of being green or white. Is it almost surely (P=1) that the cells will converge to mono-color? Why or why not?

https://preview.redd.it/syjq2cp4ymtd1.png?width=223&format=png&auto=webp&s=d080ef4953eed2112ff788c6630de3406ef15855

In an infinite grid of offset squares, the first row starts with one green cell and the rest white. For every row after that, a cell is white if both cells above are white, green if both cells above are green, and otherwise has a 50% chance of being green or white. Is there a non-zero probability the green cells will continue forever? Why or why not?

https://preview.redd.it/2pyn2c0f2etd1.png?width=418&format=png&auto=webp&s=3c490ceef2229838096bc402a13c977440054795

Given an acute angle triangle ∆ABC, there is an ellipse (not given) inscribed in ∆ABC such that one focus is the orthocenter of ∆ABC.

By compass and straightedge, identify the 3 points of tangency between the triangle and the inellipse.

side note: >!this problem is rephrasing of someone's recently deleted post, i guess because a large portion is bloated/irrelevant text, and the real problem is buried in the last paragraph. i tried to solve it and to be fair the solution is pretty satisfying.!<

!the original post (given sides 13,14,15, find length of the major axis) seems to suggest the solution involve a lot of tedious calculation. so i rephrase to discourage that, and still keep the essence of the solution intact.)!<

You are playing a game with a standard 52 card deck. All four aces are laid out in a 1x4 line. Next to this line, 5 randomly drawn cards are laid face down to indicate "steps" 1-5. All the aces are initially at step 0. The remaining 43 cards are then flipped one by one. An ace only advances to the next step if its suit is drawn. If all 4 aces are at a specific step, you flip one of the cards that is used to indicate a step (You do not necessarily have to flip the card that has all four aces on that step --- also no matter what, when all four aces are on a specific step you flip one of the face down cards. If you have flipped all 5, you do nothing). You then advance the ace that has a suit correspondent to the card flipped. What is the expected number of total cards flipped (including the initially face down cards) to conclude the game which ends when one ace reaches step 6 (passing through the final step 5).

https://preview.redd.it/2g6utvtl5esd1.jpg?width=800&format=pjpg&auto=webp&s=3832dfe808f41d11d6d0d73d8ca2ad659c9be531

Question is just the title. I found it fun to think about, but some here may find it too straight-forward. An explanation as to how you came up with the pair of functions would be appreciated.

inspired by recent problem

there are 2048 coins and 15 robots. (because "technicians" and "Geiger counters" are such a long word lol)

exactly one of the coins is radioactive, which can only be detected by robots.

each robot scans a subset of the coins and report if one of them is radioactive. after reporting its result, it explodes (thus unusable) .

exactly zero or one of the robots is faulty, giving opposite (thus incorrect) result.

subset of coins for each robot must be decided PRIOR to any result from other robots.

the goal is to find the radioactive coin and the faulty robot if there is one.

On a n x n grid of squares, each square has one its two diagonals drawn in. There are 2*^(n)* ^(x) ^(n) grids fitting this description. For each such grid, prove that there will either be a path of diagonals joining the top of the grid to the bottom of the grid, or there will be a path of diagonals joining the left side of the grid to the right side.

The corners are of the grid are considered to be part of both neighboring sides. It is possible to have both a top-to-bottom path and a left-to-right path.

1000 guards stand in a field a unique distance away from each other, so that every pair of 2 guards are a unique distance away from each other. Each one observes the closest guard to them. Is it possible for every guard to be observed?

There are 13 gold coins, one of which is a forgery containing radioactive material. The task is to identify this forgery using a series of measurements conducted by technicians with Geiger counters.

The problem is structured as follows:

Coins: There are 13 gold coins, numbered 1 through 13. Exactly one coin is a forgery.

Forgery Characteristics: The forged coin contains radioactive material, detectable by a Geiger counter.

Technicians: There are 13 technicians available to perform measurements.

Measurement Process: Each technician selects a subset of the 13 coins for measurement. The technician uses a Geiger counter to test the selected coins simultaneously. The Geiger counter reacts if and only if the forgery is among the selected coins. Only the technician operating the device knows the result of the measurement.

Measurement Constraints: Each technician performs exactly one measurement. A total of 13 measurements are conducted.

Reporting: After each measurement, the technician reports either "positive" (radioactivity detected) or "negative" (no radioactivity detected).

Reliability Issue: Up to two technicians may provide unreliable reports, either due to intentional deception or unintentional error.

Objective: Identify the forged coin with certainty, despite the possibility of up to two unreliable reports.

♦Challenge♦ The challenge is to design a measurement strategy and analysis algorithm that can definitively identify the forged coin, given these constraints and potential inaccuracies in the technicians' reports.

Does there exist an almost surely continuous martingale that converges in probability to +∞?

Here the definition of convergence in probability is the obvious extension of the usual definition - we say a process X converges in probability to +∞ if for every M, d > 0, there exists some T > 0 such that P(X_t < M) < d for all t > T.

Consider the following game - I draw a number from [0, 1] uniformly, and show it to you. I tell you I am going to draw another 1000 numbers in sequence, independently and uniformly. Your task is to guess, before any of the 1000 numbers have been drawn, whether each number will be higher or lower than the previously drawn one in the sequence.

Thus your answer is in the form of a list of 1000 guesses, all written down in advance, only having seen the first drawn number. At the end of the game, you win a dollar for every correct guess and lose one for every wrong guess.

How do you play this game? Is it possible to ensure a positive return with overwhelming probability? If not, how does one ensure a good chance of not losing too much?

Question: For a more precise statement, under a strategy that optimises the probability of the stated goal, what is the probability of

1. A positive return?

2. A non-negative return?

Some elaboration: From the comments - the main subtlety is that the list of 1000 guesses has to be given in advance! Meaning for example, you cannot look at the 4th card and choose based on that.

An example game looks like this:

• Draw card, it is a 0.7.

• Okay, I guess HLHLHLLLLLH...

• 1000 cards are drawn and compared against your guesses.

• ???

• Payoff!

Let ℝ⁺ be the set of positive reals. Find all functions f: ℝ⁺-> ℝ such that f(x+y)=f(x²+y²) for all x,y∈ ℝ⁺

Problem is not mine