what is the expected number of integer solutions for x^2+y^2=n, given distribution of n is

(a) uniform between [0,N], and then N → ∞

(b) geometric distribution, i.e. P(n+1) / P(n) = constant for all n>=0

fun fact, solution of (a) and (b) can be related in some way, how?

edit: (b) does not work the way i though it would... thanks to imoliet for pointing it out!

Show that the blue area equals the red area. These are convex quadrilaterals, where the sides are split into n equal segments. n=4 and n=3 in the examples below, but the question is for generic n. If n is odd, remove the middle square.

Source: https://www.amazon.com/Bicycle-Unicycle-Collection-Intriguing-Mathematical/dp/1470447592

Let λ be randomly selected from [0,∞) with exponential density δ(t) = e^(–t). We then select X from the Poisson distribution with mean λ. What is the unconditional distribution of X?

(Flaired as easy since it's a straightforward computation if you have some probability background. But you get style points for a tidy explanation of why the answer is what it is!)

This one is cool. A bunch of wine bottles are stacked inside a bin as shown. The bottles at the bottom are not necessarily evenly spaced, but the left-most and right-most bottles touch the walls. Show that the top bottle is centered between the sides of the bin.

Source: https://www.amazon.com/Bicycle-Unicycle-Collection-Intriguing-Mathematical/dp/1470447592

The volume of a ball of radius R can be computed by inscribing the ball in a pile of cylinders, whose volumes are known, and taking the limit as the height of each cylinder goes to 0. The total volume of the cylinders then converges to the (expected) 4/3 π R^3.

Without doing any heavy computation: What is the limit of the areas of these shapes?

Edit: I interpret the phrase "single sequence of colors" to mean that the same sequence works for every starting city. Not necessarily that there's only one such sequence.

Source: pg 19 of https://msri-app-assets.s3.us-west-1.amazonaws.com/s32odum4qqn19sgnjg8ptepujial

Can you relabel the sides of two standard four-sided dice (with not necessarily distinct positive integers) in such a way that they produce the same distribution of outcomes for their sum as rolling a regular pair of four-sided dice?

How about two six-sided ones?

Consider all strings in {0,…k}^(n) . For each string, Alice scores a point for each ’00’ substring and Bob scores a point for each ‘xy’ substring (see below). Show that the number of strings for which Alice wins with n=m equal the number of strings that end in '0' for which Bob wins with n=m+1 (alternatively, the number of strings for which Bob wins with n=m with an extra '0' appended at the end).

1. For k=1 and xy=01
2. For any k>=1 and xy=01
3. For any k>=2 and xy=12

I’ve only been able to prove (1) so far, but based on simulations (2) and (3) appear to be true as well. Source: related to this

Consider all integer geometric sequence, what is the longest possible arithmetic subsequence that is not a constant sequence?

bonus: i originally was thinking of real domain, i have a strong suspicion that the longest is >!three!< but not yet prove it. any ideas are welcomed.

Show that every integer arithmetic progression contains as a subsequence an infinite geometric progression.

Let T be a triangle with integral area and vertices at lattice points. Prove that T may be dissected into triangles with area 1 each and vertices at lattice points.

You may know that there are no equilateral lattice triangles. However, almost equilateral lattice triangles do exist. An almost equilateral lattice triangle is a triangle in the coordinate plane having vertices with integer coordinates, such that for any two sides lengths a and b, |a^2 - b^2| <= 1. Two examples are show in this picture:

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https://preview.redd.it/noinxhc9mpqc1.jpg?width=695&format=pjpg&auto=webp&s=0aa6a629698fad40ed1725725edaebdc177e1beb

The left has a side parallel to the axes, and the right has a side at a 45 degree angle to the axes. Prove this is always true. That is, prove that every almost equilateral lattice triangle has a side length either parallel or at a 45 degree angle to the axes.

There are four mathematicians having tea and crumpets.

"Let our ages be the vertices of a graph G where G has an edge between vertices if and only if the vertices share a common factor. Then G is a square graph," declares the first mathematician.

"These crumpets are delicious," says the second mathematician.

"I agree. These crumpets are exceptional. We should come here next week," answers the third mathematician.

"Let the Collatz function be applied to each of our ages (3n+1 if age is odd, n/2 if age is even) then G is transformed into a star graph," asserts the fourth mathematician.

How old are the mathematicians?

(a) a cuboid is wonderful iff it has equal numerical values for its volume, surface area, and sum of edges. does a wonderful cuboid exist?

(b) a dimension n hyper-box (referred as n-box from here on) is wonderful iff it has equal numerical values for all 1<=k<=n, (sum of measure of k-box) on its boundary. for which n does a wonderful n-box exist?

for clarity, 0-box is a vertex (not used here), 1-box is a line segment/edge, 2-box is a rectangle, 3-box is a cuboid, n-box is a a1×a2×a3×...×a_n box where all a_k are positive. so no, 0x0x0 is not a solution.

A group of n people plan to paint the outside of a fence surrounding a large circular field using the following curious process. Each painter takes a bucket of paint to a random point on the circumference and, on a signal, paints towards their furthest neighbor, stopping when they reach a painted surface. What is the expected fraction of the fence that will be left unpainted at the end of this process?

Source: https://legacy.slmath.org/system/cms/files/525/files/original/Emissary-2017-Spring-Web.pdf

A function f: R -> R is called T-periodic (for some T in R) iff for all x in R: f(x) = f(x + T).

Prove or disprove: there exists a surjective function f: R -> R that is q-periodic iff q is rational (and not q-periodic iff q is irrational).

​

Note: This problem was inspired by [this one](https://www.reddit.com/r/mathriddles/comments/1bduiah/can\_this\_periodic\_function\_exist/) from u/BootyIsAsBootyDo.

Get ready to play, Name That Polynomial! Here's how it works. There is a secret polynomial, P, with positive integer coefficients. You will choose any positive integer, n, and shout it out. Then I will reveal to you the value of P(n). What is the fewest number of clues you need to Name That Polynomial? If you are wrong, your opponent will get the chance to steal.

You need to help Santa have a successful test flight so that he can deliver presents before Christmas is ruined for everyone.

In order to have enough magical power to fly with the sleigh, all nine of Santa's reindeer must be fed their favorite food. The saboteur gave one or more reindeer the wrong food before each of the three test flights, causing the reindeer to be unable to take off.

In each clue, "before test flight n" means "immediately before test flight n". Before each test flight, each reindeer was fed exactly one food, and two or more reindeer may have been fed the same food. Two or more reindeer may have the same favorite food. You must use these clues to work out what each reindeer's favorite food is, then complete a test flight by feeding each reindeer the correct food.

11: Before test flight 2, reindeer 9 was given food 5.

18: Before test flight 2, reindeer 8 was given food 2

2: Before test flight 1, reindeer 2 was given food 4.

9: Before test flight 1, 2 reindeer were given the wrong food.

10: Before test flight 1, reindeer 9 was given food 6

12: Before test flight 3, reindeer 9 was given food 1

19: Before test flight 3, reindeer 5 was not given food 7

21: Before test flight 3, reindeer 7 was given food that is a factor of 148

3: Before test flight 2, reindeer 2 was given food 4.

4: Before test flight 3, reindeer 2 was given food 6.

6: Reindeer 4's favorite food is a factor of 607

13: Before test flight 2, reindeer 4 was not given food 9

20: Before test flight 3, 3 reindeer had the food equal to their number

22: Before test flight 3, reindeer 7 was not given food 1

23: Before test flight 3, no reindeer was given food 2

5: Before test flight 3, 4 reindeer were given the wrong food.

7: Reindeer 4 was given the same food before all three test flights.

14: Before test flight 2, 2 reindeer were given the wrong food

16: Before test flight 2, all the reindeer were given different foods

17: Before test flight 1, reindeer 7 was not given food 7

24: Before test flight 1, reindeer 7 was not given food 9

1: Reindeer 2's favorite food is 4

8: Before test flight 1, reindeer 8 was given food 3.

15: Reindeer 1 was given food 1 before all three test flights

​

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Can any of you explain how to get to the answer? I have the answer, but am not sure how you get there.

had this riddle at a job interview, there has to be a more advanced solution than just pairing based on low to high price with units, but i can't figure it out

"Imagine that each fruit has its own "weight":

• Apple - 1 unit
• Pear - 6 units
• Pineapple - 3 units
• Orange - 5 units
• Pomegranate - 2 units
• Banana - 4 units

Now imagine that the hotel has different rooms with different prices:

• Business - 4011 dollars per night
• Standard - 2567 dollars per night
• Comfort - 3987 dollars per night
• Presidential - 24670 dollars per night
• Deluxe - 4096 dollars per night

You need to correlate one fruit with one room in the hotel. How would you correlate them and why?"

define function f: Z+ → Z+ that satisfy:

1. f(1) = 1
2. f(2k) = f(k) for even k; 2f(k) for odd k
3. f(2k+1) = f(k) for odd k; 2f(k)+1 for even k

find the closed form of Σf(k) for 1 ≤ k ≤ 2^(n) - 1.

alternatively, prove that the sum equals >!2·3^(n-1) - 2^(n-1)!<

There are n students in a classroom.

The teacher writes a positive integer on the board and asks about its divisors.

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The 1st student says, "The number is divisible by 2."

The 2nd student says, "The number is divisible by 3."

The 3rd student says, "The number is divisible by 4."

...

The nth student says, "The number is divisible by n+1."

"Almost," the teacher replies. "You were all right except for two of you who spoke consecutively."

​

1. What are the possible pairs of students who gave wrong answers?

2. For which n is this possible?

Let S(n) be the sum of the base-10 digits of all divisors of n.

Examples:

S(12) = 1 + 2 + 3 + 4 + 6 + 1 + 2 = 19.

S(15) = 1 + 3 + 5 + 1 + 5 = 15

Let S^i(n) be i compositions of the function S.

Example:

S^4(4) = S^3(7) = S^2(8) = S(15) = 15

Is it true that for all n > 1 there exists an i such that S^i(n) = 15?

Consider a right triangle, T, with sides adjacent to the right angle having lengths a and b (just as in the Pythagorean theorem). If a^(-2) + b^(-2) = x^(-2) then what is x in relation to T?

For any point p in the plane consider the set of points with an irrational distance from p. Is it possible to cover the plane with finitely many such sets? If yes, find the minimal number needed and if no, show that at most countably many are needed.

Can a real periodic function satisfy both of these properties?

1. There does not exist any p∈(0,1] such that f(x+p) is identically equal to f(x).

2. For all ε>0 , there exists p∈(1,1+ε) such that f(x+p) is identically equal to f(x).

In other words: Can there be a function that does not have period 1 (or less than 1), but does have a period slightly greater than 1 (with "slightly" being arbitrarily small)?

Find an elementary function, f:R to R, with no discontinuities or singularities such that:

1. f(0) = 0

2. f(x) = 1 when x is a non-zero integer.

For which n does there exist an n x n matrix M such that all entries of M are in {-1,0,1} and the row and column sums are all pairwise distinct, that is, there are 2n total distinct sums?

Showing that the Cycloid is the brachistochrone curve under a uniform gravitational field is a classical problem we all enjoy.

Consider a case where the force of gravity acting on a particle (located on the upper half of the plane) is directed vertically downward with a magnitude directly proportional to its distance from there x-axis.

Unless you don't want to dunned by a foreigner, find the brachistochrone in this 'linear' gravitational field.

Assume that the mass of the particle is 'm' and is initially at rest at (0, 1). Also, the proportionality constant of the force of attraction, say 'k' is numerically equal to 'm'.

CAUTION: Am an amateur mathematician at best and Physics definitely not my strong suit. Am too old to be student and this is not a homework problem. Point am trying to make is, there is room for error in my solution but I'm sure it's correct to the best of my abilities.

EDIT: Added last line in the question about the proportionality constant.

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https://preview.redd.it/uaamrjnytrnc1.png?width=1246&format=png&auto=webp&s=ac42d958c9c1b289f02a546f6f13d99e100f4adc

Easy with the hint:

!use weierstrass product formula for sine!<

(a) Given two intersecting lines and a fixed point. construct a new line through the fixed point, such that the perimeter of the triangle formed is minimized.

insight: >! let AP, AQ be tangents of circle, where P,Q are the points of tangency, then AP=AQ.!<

(b) Given 3 fixed points P,Q,R in deep space (no gravity). A stationary rocket at P wants to reach Q for scientific observation, then to R and stay stationary there. It can maneuver by changing its velocity vector at P,Q,R at an instance, i.e. adding some Δv to its original velocity vector. ^((the distance between points are so great that the acceleration time is negligible compare to travel time between points))

If the time constraint is 1 unit, construct vectors Δv maneuvered at P,Q,R such that the |Δv| budget is minimized.

example scenario / example solution