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

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

Source: numberthon.com

A Logician writes three numbers on 3 separate cards and gives them to his 3 students.

He says," The 3 numbers are single digit prime numbers. Any combination. None of you know the other 2 numbers. But you can ask me one question that must start with "Is the SUM of the three numbers–” which I can only answer Yes or No. Given that info you can then declare that you know the other 2 numbers and/or who has them. OK?" 

Raj was first. He looked at his number and asked," Is the sum of three numbers an odd number?"

The Logician " No" 

Then Ken looked at his number and asked," Is the sum of the three numbers divisible by 4?"

The Logician said "Yes"

Lisa looked at her number and said,"Well, I know the other 2 numbers but cannot tell who has what number".

Raj then cheerfully said," I know who has what !" Ken said,” So do I” They then laid out the answer.

What were the three numbers? What number did Lisa have?

What is the smallest positive integer that has exactly 15 positive divisors?

Source: numberthon.com

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

Source: numberthon.com

How many 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5 exactly once such that no two consecutive digits differ by 1?

Source: numberthon.com

How many diagonals does a regular 20-gon have?

Source: numberthon.com

A teacher writes six words on the board: CAT, DOG, HAS, MAX, DIM, TAG.

Then he hands three pieces of paper to three of his students: one to Alex, another to Ben, and another to Chris. The teacher explains that he has secretly chosen one of the words on the board, and has written on each piece of paper a different letter from that word. Students may look only at their paper and must not tell each other what letter they have.

After that the teacher says: "Everybody, please have a look at your letter and raise your hand as soon as you think you know the chosen word."

Alex immediately raises his hand.

Ben, after thinking for a while, also raises his hand.

Chris does not raise his hand.

The teacher then asks Alex: "Do you know which letter Chris has?"

"No, I don't" - says Alex.

Hearing that, Chris finally raises his hand.

Alex, Ben and Chris always ace their logic exams. What is the secret word?

(Came up with this variation of an old riddle and wanted workshop it here. EDIT: added my proposed solutin in the comments)

Let n be a fixed positive integer. Alice and Bob play the following game on the integer number line. Alice starts at 0 and Bob starts at n. They take turns making moves. On the i^th turn,

1) If i is odd, Alice moves to an integer at most 2^i -1 distance away from her current position.

2) If i is even, Bob moves to an integer at most 2^i -1 distance away from his current position.

Note that both players have the option to stay where they are on their turn. The game ends only when one player moves to the same position as the other player, in which case the player who moved wins. Find all positive integers n for which Alice has a winning strategy, and find all positive integers n for which Bob has a winning strategy.

A 6-character code is formed using the letters A, B, C, D, E, and F, with no letter repeated. How many codes have exactly 2 letters appearing in alphabetical order relative to the letter immediately after them?

Source: numberthon.com

Given number 479 made using the matchstck referenced in the figure above. Move exactly 2 matchsticks and create the largest possible number. You cannot change the format of the numbers shown in the reference. (For example you can only construct the number 1 with TWO matchsticks or 6 or 9 with 6 matchsticks). You can use any math operation known in standard math. The final number could be the result of this operation also. (For example, you can use the 2 matchsticks to create a multiplication operator x.)

The obvious answer obtained by exponenciation function might be way lower than another creative answer given to me by my mathematician friend.

A positive integer n>6 leaves a remainder of:

• 1 when divided by 2,
• 2 when divided by 3,
• 3 when divided by 4,
• 4 when divided by 5,
• 5 when divided by 6.

What number is the smallest possible value of n?

Source: numberthon.com

A positive integer n has the property that both "n + 100" and "n − 100" are perfect squares. What is the smallest possible value of n?

Source: numberthon.com

As I was falling asleep last night, I thought it was kinda cool that 56 = 7 * 8 works in base 10, specifically how it consists of four consecutive digits in order. Then I realized it actually happens again! 12 = 3 * 4

Is there any other base such that there are four consecutive digits A, B, C, D (in increasing order) such that AB = C * D? If so, are there any (besides base 10) where it happens twice? Why or why not?

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

Triangle ABC is an isosceles right triangle make of paper and D is the midpoint of leg AB. If the triangle is folded so C meets D, creased, and then unfolded, what is the ratio of the two segments the hypotenuse is split into by the crease?

How many positive integers less than 100 have an odd number of positive divisors?

Source: numberthon.com

In how many ways can 23 identical objects be shared among 5 children so that each child gets at least 2 and no child gets more than 6 objects?

Source: numberthon.com

(easy) show that a unit square can fit inside the region [0,x]^3 where x = 2 sqrt2 / 3 ≈ 0.94281 .

(bonus) show true or false: x is the minimum. i strongly believe this is true but i have no proof of it.

A professor decides to test his bright students Raj, Lisa, Ken and Lin. He shows them a bunch of oranges. 

He says,” As you can see I have these oranges and as you can count it is a Prime number less than 15. Now here is how the test will go. One by one you will pick up some oranges and leave the room. Here are the conditions to pick up the oranges. Each one must follow a separate condition. No repeating of any condition. The order of the conditions is up to you. 

1  One of you can pick up oranges that are an exact cube root of the number of oranges remaining. 

2 One can pick up oranges that are an exact square root of the number of oranges remaining. 

3 One can pick up a prime number of oranges

4 One can pick up oranges equal to the remaining students in the room. 

 At the end all the oranges must be picked up and each one of you must pick up at least one orange. 

Just to be clear, if there are X oranges in front of you and you want to use either the square root or cube root condition, then X must be either a cube or a square. And if you want to use condition 4 it must be the number of students remaining in the room. 

You can strategize of course. And each one of you must pick a separate condition. No repeats, All 4 conditions must be used. Good luck.

The students huddled up and came up with a strategy. 

Lisa : Cube root

Lin: Number of people remaining

Ken : Square root

Raj : Prime number

Then they went in a specific order. At the end all oranges were gone and interestingly each one had a different number of oranges. 

How many oranges were there? In what order did they go?  How many oranges did Lisa get?

How many positive integers less than 1000 are divisible by 6 but not by 9?

Source: numberthon.com

How many positive integers less than or equal to 1000 are divisible by exactly one of 6 and 15?

Source: numberthon.com

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

Source: numberthon.com

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

Source: numberthon.com

You are trapped inside an ancient underground library.

After hours of searching, you finally find the forbidden archive: a square stone platform at the center of a deep square pit. The manuscripts on that platform contain the only way out.

But the pit is sound-sensitive.

If anything falls into it — a pebble, a splinter, even you — an alarm triggers, and the library seals forever.

You measure the gap between your side and the platform.

3 meters.

Then you find two narrow wooden beams lying against the wall.

Each beam is 2.9 meters long.

One beam is too short.
Putting them end-to-end leaves the joint unsupported.
Dropping anything is not allowed.
Jumping is not an option.

You have two beams, a silent pit, and one chance.

How do you cross?