Algebra
How many numbers have the property that when you square them and subtract one, you get a prime number?
1
How many digits are in 2^41?
13
Three 8-sided dice are rolled. What is the probability that they all return the same value?
1/64
A vector space has basis vectors {1,i,j,k} such that i^2 = j^2 = k ^2 = ijk = -1. What is the group {1,i,j,k,-1,-i,-j,-k} under multiplication called?
The quaternion group
Perfect numbers are numbers whose proper factors (factors excluding themselves) add up to themselves. For example, 6 and 28 are perfect numbers, since 6 = 1+2+3 and 28 = 1+2+4+7+14. The third smallest perfect number can be expressed as 16n. Determine n.
31
How many numbers less than 2025 divide either 3 or 5?
944
An analog clock shows 12:34 with no seconds hand. After 10 seconds, it is still 12:34. What is the expected # of seconds until 12:35? (Assume that the starting time was uniformly randomly generated)
25
What is the name of the finite simple group with order 808,017,424,794,512,875,886,459,904, 961,710,757,005,754,368,000,000,000?
The Monster Group
What is the sum of the reciprocals of all positive integers whose prime factorization consists of no primes greater than 11?
77/16
Given that each letter represents a unique digit, and that FLOAT = LOAF + EEEE, determine T.
0
On average, how many times do you have to flip a coin until you flip three heads in a row?
14
The maximum number of steps an n state Turing Machine runs before halting is called ...
BB(n), the nth busy Beaver Number
What is the maximum number of disjoint unit circles that can be externally tangent to a circle with radius 99?
314
Arthur evaluates the sum of all positive divisors of 10000 that are multiples of 100 and writes the result on a blackboard. Underneath, they evaluate the sum of all positive divisors of 100 and write down that result. Compute the ratio of the top number to the bottom number
100
Up to reflections and rotations, how many unique cyclic quadrilaterals with perimeter 12 and integer sides are there?
8
The constant 1/1^3 + 1/2^3 + 1/3^3 ... is irrational. Who proved this? (Hint: the constant is also named after this person)
Roger Apery
Compute:
5 / (1*2*3*4) + 7 / (2*3*4*5) ... 99 / (48*49*50*51)
[SMT Alg Round]
832/2499
Let S(1) = 1 and S(n) be the smallest number greater than S(n-1) with n factors. What is S(19)?
262144
Starting with the number 0, Arthur performs an infinite sequence of moves - they choose a number from {1, 2} at random and add it to the current number. Let p_m be the probability that Arthur reaches the number m at some point. Find p_20 − p_15.
What is 11/2^20?
How many regular polyhedra are there? (trick question)
48