What is the 1000th digit to the right of the decimal point in the decimal representation of (1 + √2)^{3000}?

There is a bus with 100 labelled seats (labelled from 1 to 100). There are 100 persons standing in a queue. Persons are also labelled from 1 to 100.People board on the bus in sequence from 1 to n. The rule is, if person ‘i’ boards the bus, he checks if seat ‘i’ is empty. If it is empty, he sits there, else he randomly picks an empty seat and sit there. Given that 1st person picks seat randomly, and the probability that 100th person sits on his place i.e. 100th seat is p% then what will be the value of the smallest integer nearest to p?

How many times a day do the minute and hour hands of a clock overlap?

A 12 by 25 by 36 cm cereal box is lying on the floor on one of its 25 by 36 cm faces. An ant, located at one of the bottom corners of the box, must crawl along the outside of the box to reach the opposite bottom corner. What is the integer closest to and smaller than the length of the shortest such path? The ant can walk on any of the five faces of the box, except for the bottom face, which is flush in contact with the floor. It can crawl along any of the edges. It cannot crawl under the box.

For how many integers n > 1 is x^{49} ≡ x(mod n) true for all integers x?

By Fermat's Little Theorem, the number x = (2^{p-1} − 1)/p is always an integer if p is an odd prime. Find the sum of all the values of p for which x a perfect square?

Let a, b, n, and m be positive integers, with n > 1. Show that a^{n} + b^{n} = 2^{m}. What is the minimum value of a/b?

A random number generator generates integers in the range 1...n, where n is a parameter passed into the generator. The output from the generator is repeatedly passed back in as the input. If the initial input parameter is one googol (10^{100}), find, to the nearest integer, the expected value of the number of iterations by which the generator first outputs the number 1. That is, what is the expected value of x, after running the pseudo-code given here?

Find the number of ordered pairs (a,b) of positive integers such that |3^{a} − 2^{b}| = 1.

This problem was taken from Facebook Hackercup Qualification Round 2018. Visit here to solve the problem.

Find all solutions to, i.e., number of triplets (a, b, c) which satisfies c^{2} + 1 = (a^{2} − 1)(b^{2} − 1), in integers a, b, and c.

This problem was taken from Round 2, 2016 of codejam powered by Google. View here.

An absentminded professor buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the professor opens a matchbox and finds that it is empty. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained n matches, and the prbability that the other box currently contains k matches, then find the value of 2^{15}p if n = 10 and k = 5.

For how many ordered pairs (m,n) where m and n are integers, mn(m^{420} − n^{420}) is not divisible by 446617991732222310?

Let (w, x, y, z) denote the tuple such that w,x,y, and z are positive integers which satisfy w!=x!+y!+z!. Let's also suppose that number of such tuples be n. Then find n+ max(w+x+y+z) or in other words find a tuple which has the maximum sum of its elements and add that maximum sum to the total number of possible tuples that satisfy the above equation and submit the answer.

Consider a function f:Q→Q which satisfies the following equations:

i) 3f(w+x+y+z)+f(w+x)+f(x+y)+f(y+z)+f(z+w)+f(w+y)+f(x+z) =2f(w+x+y)+2f(x+y+z)+2f(y+z+w)+2f(z+w+x)

ii) f(1) = 1

iii) f(2) = 9

Find f(3).

The terms of a sequence of positive integers satisfy a_{n+3} = a_{n+2}(a_{n+1} + a_{n}), for n = 1, 2, 3, ... . If a_{6} = 8820, what is a_{7}?

Find the smallest positive integer such that when its last digit is moved to the start of the number (example: 1234 becomes 4123) the resulting number is larger than and is an integral multiple of the original number. Numbers are written in standard decimal notation, with no leading zeroes.

A sequence of integers is defined by

1. a[0] = p, where p > 0 is a prime number,

2. a[n]+1 = 2a[n] + 1, for n = 0, 1, 2, ....

Find the number values of p such that the sequence consists entirely of prime numbers?

Alice and Bob are preparing for a holiday party, and each has pie to slice up into pieces. They decide to have a little contest to make things fun. Each person is allowed to make 20 cuts of the pie with a knife, and whoever ends up with more pieces is the winner. They agree stacking is not allowed, but that "center" pieces without the crust are permissible. What is the maximum number of pieces beyond which neither Alice not Bob will be able to cut the pie?