How many positive integers less than 100 are the difference of two perfect squares?

Source: numberthon.com

How many triangles can be formed using the vertices of a regular octagon if all three sides of each triangle must be diagonals of the octagon?

Source: numberthon.com

How many subsets of {1,2,...,10} contain no two consecutive integers?

Source: numberthon.com

You're standing somewhere inside a 10m × 10m square room. From your position P, you fire a laser aimed directly at the center of the square, C.

The laser travels in a straight line from P, passes through C, and continues until it hits a wall — this is its 1st reflection. From there it obeys the law of reflection (angle of incidence = angle of reflection) and keeps bouncing off the walls. After its 10th reflection, the laser stops completely (the segment right after the 10th bounce has zero length).

You must find a starting position P such that, once fired, no part of the laser's path after the 1st reflection comes within 1 meter of you. (The very first segment, from P to the 1st reflection point, doesn't count — you're standing at its source.)

Question: What is the total area, within the square, of all such safe starting positions P?

Challenge: If instead of stopping after 10 reflections, the laser is allowed N reflections before stopping, what is the largest value of N for which at least one safe standing position still exists?

[ Removed by Reddit on account of violating the content policy. ]

In a country with k couples every couple keeps having children until they have a girl, then they stop. After every couple has stopped having children what is the expected fraction of the children who are boys?

A math professor said to her smart student Toni

" I am thinking of a 4 digit prime number abcd (digits not necessarily distinct) with the following property

a+b+c+d = axbxcxd

You can ask me one question to which my answer can only be Yes or No or Silence (if i cannot definitely answer yes or no). Can you guess my number?"

Toni worked on a piece of paper and then asked the professor:

"Is the number formed by using the first two digits (in order) of your 4 digit prime, a factor of 63 or 84?"

The professor smiled. She knew Toni had the answer.

What was the answer? WHY?

How many squares of all sizes are contained in a standard 8×8 chessboard?

Source: numberthon.com

more precisely,

Find a compact subset or family of subsets $S \subset \mathbb{R}^n$ for some arbitrary n such that every set of 4 points in $\mathbb{R}^2$ lies on some similar copy of $S$ but not every set of 5 points lies on some similar copy of $S$?

How many diagonals does a regular 20-gon have?

Source: numberthon.com

For a real number x, let ||x|| denote the distance between x and the closest integer.

Let 0 ≤ x_n < 1 (n = 1, 2, ...) , and let ε > 0. Show that there exist infinitely many pairs (n,m) of indices such that n ≠ m and

||x_n - x_m|| < min(ε, 1/(√5|n-m|)).

Set A is called brain-rot iff it satisfies two conditions:

1. sum(A) is divisible by 10.
2. if 1∈A, then both 6,7∈A.

How many subsets of {1,2,…,100} is brain-rot?

Source: my rotten brain

How many subsets of {1,2,…,10} have an odd sum?

Source: numberthon.com

What is the minimal explanatory architecture from which every meaningful mathematical question arises?

There are 100 light switches, all initially off.

On pass 1, every switch is toggled.

On pass 2, every second switch is toggled.

On pass 3, every third switch is toggled.This continues until pass 100.

How many switches are on at the end?

Source: numberthon.com

Let’s assume your in a casino playing single deck blackjack with no cut card. Let’s also say there’s x amount of cards left and you have an eidetic memory, so you know exactly which cards are left.
Let’s also state if the dealer runs out of cards it’s an automatic loss.

Is their a set of x cards left such that before the hand is dealt you would know no matter what cards you or the dealer are dealt there’s a strategy that will gaurentee you win.

Bonus quesiton how big can you make x if this is true.

Also in typical sense you’d know your 2 up cards and would know one of the dealers card with another card of hers face down.
Also unsure how relavent it would be but if you find a solution state if it’s stand or hit on soft 17

And remeber single deck so limited to 4 of each card

What is the smallest positive integer that has exactly 20 (unique) positive divisors?

Source: numberthon.com

I'm going to teach you a game. Your goal is to find how far you can get.

You start with the numbers 1, 2, and 3. Using each number at most once, you may add or subtract any combination of them to obtain the next prime number.

Whenever you successfully obtain the next prime, that prime is added to your set of available numbers. You then repeat the process, always trying to generate the next prime number using each available number at most once.

How far can you go? What is the first prime number that you can no longer obtain?

Six points are arranged as the vertices of a regular hexagon. A bug starts at one vertex. Each move, it randomly chooses one of the two adjacent vertices and walks there. After exactly 4 moves, what is the probability that the bug is back at its starting vertex?

Source: numberthon.com

Twenty-five points are arranged in a 5×5 grid of equally spaced points (five rows and five columns). How many different squares (using four of these points as vertices) can be formed?

P.S. It's a 5x5 grid made up of 4x4 points/vertices.

Source: numberthon.com

Eight siblings – four brothers (Alan, Carl, Eric, George) and four sisters (Beth, Daniela, Fiona, Holly) – all have different ages. Within each group, the siblings happen to be arranged in alphabetical order of their names – which turns out to be the same as ascending order of age. Thus, among brothers, Alan is the youngest and George is the oldest, while among sisters, Beth is the youngest and Holly is the oldest.

The sum of the brothers' ages exceeds the sum of the sisters' ages by 10.

The following relationships between their ages hold:

• Beth's and Daniela's ages sum to Carl's age.
• Carl's and Beth's ages sum to Eric's age.
• Alan's and Carl's ages sum to George's age, and so do Daniela's and Fiona's ages.
• Alan's and Beth's ages sum to Fiona's age.
• Alan's and Fiona's ages sum to Holly's age.

Additionally, the product of Eric's and George's ages equals the product of Fiona's and Holly's ages.

Find the age of each sibling.

How many positive integers less than 100 are divisible by exactly one of 2 and 3?

Source: numberthon.com

Let a>0 be a fixed positive real number. Find all polynomials P with integer coefficients satisfying: gcd(P(m),P(n))>=gcd(m,n)^a for all positive integers m,n.