Suppose p is a prime. Suppose n and m are integers such that:

• 1 <= n <= m <= p
• n^2 + m^2 = 0 (mod p)

For each p, how many pairs (n,m) are there?

Let a(n) be the least common of the first n integers.

• Show that the longest run of consecutive terms of a(n) with different values is 5: a(1) through a(5).
• Show that the longest run of consecutive terms of a(n) with the same value is unbounded.

On the first day of Christmas my true love sent to me
A partridge in a pear tree

On the second day of Christmas my true love sent to me
Two turtle doves,
And a partridge in a pear tree.

On the third day of Christmas my true love sent to me
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

If this continues, how many gifts will I have on the nth day of Christmas?

Let Z^n be the n-dimensional grid of integers where the distance between any two points equals the length of their shortest grid path (the taxicab metric). How many points in Z^n have a distance from the origin that is less than or equal to n?

Show that C(3n,n) is odd if and only if the binary representation of n contains no adjacent 1's.

Show that all primes that appear in the Fibonacci sequence, except 2 and 3, are congruent to 1 mod 4.

Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.

Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends, and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.

Determine the minimum value of n for which Turbo has a strategy that guarantees reaching the last row on the n-th attempt or earlier, regardless of the locations of the monsters.

We start with 1 teacher and 1 student on day 1.

• After 1 day of instruction, a student becomes a teacher.
• On their nth day of teaching, a teacher will teach n new students.

On the nth day, how many students and teachers are there?

A bagel is a loop of 2a + 2b + 4 unit squares which can be obtained by cutting a concentric a × b hole out of an (a + 2) × (b + 2) rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length a + 2 of the rectangle.)

Consider an infinite grid of unit square cells. For each even integer n ≥ 8, a bakery of order n is a finite set of cells S such that, for every n-cell bagel B in the grid, there exists a congruent copy of B all of whose cells are in S. (The copy can be translated and rotated.)

We denote by f(n) the smallest possible number of cells in a bakery of order n.

Find a real number α such that, for all sufficiently large even integers n ≥ 8, we have: 1/100 < f(n) / n^α < 100

Hi all,

I have a cup of tea in a different coloured mug every day of the week. Blue, Red, Pink, Yellow, Orange, Green and Violet. Next year I plan to change the order so that I'm drinking from a different colour of mug on every day. Trying to figure out the order of mugs for 7 years - so that across the 7 different years every colour of mug is drank from on every day of the week. The tricky part is if possible, it would be great to have it so that the new colour is not adjacent to the previous years day (aka if I had red the first year on Thursday - the second year could not have red drank on Wed or Friday and of course Thursday). It would also be great if the two mugs never were adjacent in the same order You can only have red then yellow once (yellow then red fine)

Year 1 and 2 are already set

M T W T F S S

1 G V B R Y O P

2 B Y P O V G R

3

4

5

6

7

Bonus points if it's possible to have the R O Y G B P V as year 7.

I am a very sad man

https://preview.redd.it/7cze6tyeym5e1.png?width=2048&format=png&auto=webp&s=b4bb1d6aff95eb036d8a37af589e1afdfe4ef75d

What is the sum of the reciprocals of the Catalan numbers?

Let a(n) be the sequence of perfect powers except for 1:

• 4,8,9,16,25,27,32,36,49,64,81,100, . . .

Let b(n) = a(n) - 1, the sequence of subperfect powers.

• 3,7,8,15,24,26,31,35,48,63,80,99, . . .

What is the sum of the reciprocals of b(n)?

Show that all primorials, except for 1 and 2, are integer-perfect.

Primorial numbers: the product of the first n primes.

• 1, 2, 6, 30, 210, 2310, 30030, 510510, . . .
• Example: 2*3*5*7*11 = 2310 therefore 2310 is a primorial number.

Integer-Perfect numbers: numbers whose divisors can be partitioned into two disjoint sets with equal sum.

• 6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, . . .
• Example: 1 + 3 + 4 + 6 + 8 + 16 + 24 = 2 + 12 + 48, therefore 48 is integer-perfect.

Let π be a given permutation of the set {1, 2, ..., n}. Determine the smallest possible value of

∑ (from i=1 to n) |π(i) - σ(i)|,

where σ is a permutation chosen from the set of all n-cycles. Express the result in terms of the number and lengths of the cycles in the disjoint cycle decomposition of π, including the fixed points.

Prove that for any finite bipartite planar graph, one can assign a circle to each vertex such that:

1. The circles lie in a plane,
2. Two circles touch if and only if the corresponding vertices are adjacent,
3. Two circles intersect at exactly two points if the corresponding vertices are not adjacent.

Let q > 1 be a power of 2. Let f: F_q^2 → F_q^2 be an affine map over F_2. Prove that the equation

f(x) = x^(q+1)

has at most 2q - 1 solutions.

Let A > 0 and B = (3 + 2√2)A. Prove that in the infinite sequence a_k = floor(k / √2), for k in (A, B) ∩ Z,the number of even and odd terms differs by at most 2

An urn initially contains one red ball and one blue ball. At each step, a ball is selected randomly with uniform probability. The following actions occur based on the selected ball:

If the selected ball is red, one new red ball and one new blue ball are added to the urn.

If the selected ball is blue (for the k-th time it is selected), one new blue ball and 2k + 1 new red balls are added to the urn.

The selected ball is not removed from the urn. Let G(n) represent the total number of balls in the urn after n steps. Prove that there exist constants c > 0 and α > 0 such that, with probability 1,

G(n) / n^α → c as n → ∞.

Let n be a positive integer. There are n(n+1)/2 marks, each with a black side and a white side, arranged in an equilateral triangle, where the largest row contains n marks. Initially, all marks have their black side facing up.

An operation consists of selecting a line parallel to one of the sides of the triangle and flipping all the marks on that line.

A configuration is called admissible if it can be reached from the initial configuration by performing a finite number of such operations. For each admissible configuration C, define f(C) as the minimum number of operations required to transform the initial configuration into C.

Determine the maximum possible value of f(C) over all admissible configurations C.

Imagine you are the best math-logic puzzle creator in the world. You are to make one single puzzle that will revolutionize the universe of puzzles by using math and logic. The puzzle will be unique, like no other ever existed, and it shall be the hardest puzzle ever created and almost impossible to solve, even for the best thinkers in the world and there will be only one concrete answer, without any paradoxes. https://discord.gg/wCxJ6ueC

Generalized version of my old post

There are n users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)

Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least two friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?

Prove that there exists a positive constant c such that the following statement is true: Consider an integer n > 1, and a set S of n points in the plane such that the distance between any two distinct points in S is at least 1. It follows that there is a line l separating S such that the distance from any point of S to l is at least c * n^(-1/3).

(A line l separates a set of points S if some segment joining two points in S crosses l.)

Note: Weaker results with c * n^(-1/3) replaced by c * n^(-alpha) may be awarded points depending on the value of the constant alpha > 1/3.

Let alpha ≥ 1 be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be flooded. Hephaestus is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop.

The game begins with Hephaestus moving first. On each of Hephaestus's turns, he adds one or more walls to the levee, as long as the total length of the levee is at most alpha * n after his n-th turn. On each of Poseidon's turns, every cell adjacent to an already flooded cell and with no wall between them becomes flooded.

Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop, stopping the flood and saving the world. For which values of alpha can Hephaestus guarantee victory in a finite number of turns, no matter how Poseidon chooses the initial flooded cells?

Note: Formally, the levee must consist of lattice points A0, A1, ..., Ak, which are pairwise distinct except possibly A0 = Ak, such that the set of walls is exactly {A0A1, A1A2, ..., Ak-1Ak}. Once a wall is built, it cannot be destroyed. If the levee is a closed loop (i.e., A0 = Ak), Hephaestus cannot add more walls. Since each wall has length 1, the length of the levee is k.

A snake of length k is an animal that occupies an ordered k-tuple (s1, s2, ..., sk) of cells in an n x n grid of square unit cells. These cells must be pairwise distinct, and si and si+1 must share a side for i = 1, 2, ..., k-1. If the snake is currently occupying (s1, s2, ..., sk) and s is an unoccupied cell sharing a side with s1, the snake can move to occupy (s, s1, ..., sk-1) instead.

The snake has turned around if it occupied (s1, s2, ..., sk) at the beginning, but after a finite number of moves occupies (sk, sk-1, ..., s1) instead.

Determine whether there exists an integer n > 1 such that one can place a snake of length 0.9 * n^2 in an n x n grid that can turn around.

An n times m matrix is nice if it contains every integer from 1 to mn exactly once and 1 is the only entry which is the smallest both in its row and in its column. Prove that the number of n times m nice matrices is (nm)!n!m!/(n+m-1)!.

Prove that for all sufficiently large positive integers n and a positive integer k <= n, there exists a positive integer m having exactly k divisors in the set {1,2, ....., n}

What is the minimum value of

[ |a + b + c| * (|a - b| * |b - c| + |c - a| * |b - c| + |a - b| * |c - a|) ] / [ |a - b| * |c - a| * |b - c| ]

over all triples a, b, c of distinct real numbers such that

a^2 + b^2 + c^2 = 2(ab + bc + ca)?

A Nim-style game is defined as follows: Two positive integers k and n are given, along with a finite set S of k-tuples of integers (not necessarily positive). At the start of the game, the k-tuple (n, 0, 0, ..., 0) is written on the blackboard.

A legal move consists of erasing the tuple (a1, a2, ..., ak) on the blackboard and replacing it with (a1 + b1, a2 + b2, ..., ak + bk), where (b1, b2, ..., bk) is an element of the set S. Two players take turns making legal moves. The first player to write a negative integer loses. If neither player is ever forced to write a negative integer, the game ends in a draw.

Prove that there exists a choice of k and S such that the following holds: the first player has a winning strategy if n is a power of 2, and otherwise the second player has a winning strategy.

A. Two players play a cooperative game. They can discuss a strategy prior to the game, however, they cannot communicate and have no information about the other player during the game. The game master chooses one of the players in each round. The player on turn has to guess the number of the current round. Players keep note of the number of rounds they were chosen, however, they have no information about the other player's rounds. If the player's guess is correct, the players are awarded a point. Player's are not notified whether they've scored or not. The players win the game upon collecting 100 points. Does there exist a strategy with which they can surely win the game in a finite number of rounds?

b)How does this game change, if in each round the player on turn has two guesses instead of one, and they are awarded a point if one of the guesses is correct (while keeping all the other rules of the game the same)?

It is known and not too hard to prove that any 5 points in the plane define a unique conic section.

My riddle for you is:

Given 5 points in the plane, how would you construct the tangents to the conic they define at one of the points?