cotpi

Two sequences of natural numbers

Susam Pal — Sun, 06 May 2012 00:00:00 +0000

There are two sequences of natural numbers. The first sequence consists of all natural numbers that do not contain a particular digit, in ascending order. The second sequence consists of all the remaining natural numbers in ascending order. Every natural number occurs exactly once in exactly one of the sequences.

How many terms are there in the first sequence such that each of those terms is greater than the corresponding term at the same position in the second sequence?

Powers of ten

Susam Pal — Sun, 29 Apr 2012 00:00:00 +0000

How many positive integers n are there such that \(10^{10^{10^{n}}} + 10^{10^{n}} + 10^{n} - 1\) is prime?

Product of divisors

Susam Pal — Sun, 22 Apr 2012 00:00:00 +0000

Given the number of all positive divisors of an unknown positive integer and the product of those divisors, how can one find the unknown integer?

Extra coin

Susam Pal — Sun, 15 Apr 2012 00:00:00 +0000

Akio has one more coin than Bansi. They throw all of their coins and count the number of heads. If all the coins are fair, what is the probability that Akio obtains more heads than Bansi?

Raised to the same power

Susam Pal — Sun, 08 Apr 2012 00:00:00 +0000

Two given positive integers are raised to the same power and added. The power to which they are raised is an integer greater than 1. The sum is a power of 2. What is the maximum possible difference between the two given integers?

Coins heads up

Susam Pal — Sun, 01 Apr 2012 00:00:00 +0000

You are blindfolded and taken into a room with two tables. There are coins scattered on one table. You are told the number of coins which are heads up on this table. The second table is empty. You are allowed to move coins from one table to another or flip them. Before you leave the room, there must be an equal number of coins heads up on each table. How can you do it?

Five-card trick

Susam Pal — Sun, 18 Mar 2012 00:00:00 +0000

Two mathematicians perform a trick with a shuffled deck of distinct cards. One mathematician asks a member of the audience to select five cards at random from the deck while the other mathematician is blindfolded. The audience member hands the five cards to the first mathematician who examines the cards, hands one of them back to the audience member, arranges the remaining four cards and places them face down into a neatly stacked pile on a table. The audience member is then allowed to move the pile on the table or change its orientation without disturbing the order of the cards in the pile. The second mathematician now removes his blindfold, examines the four cards on the table and determines the card held by the audience member from these four cards and their order in the pile.

If there were one more distinct card in the deck, the mathematicians cannot perform this trick. How is this trick done?

Non-decimal square

Susam Pal — Sun, 11 Mar 2012 00:00:00 +0000

What are the possible positive integer values for the base of a numeral system in which 11111 is a perfect square?

Denominators of reduced fractions

Susam Pal — Sun, 04 Mar 2012 00:00:00 +0000

The sum of two given fractions reduced to their lowest terms is an integer. Their denominators are positive. What is the maximum possible difference between their denominators?

Counting locally prime numbers

Susam Pal — Sun, 26 Feb 2012 00:00:00 +0000

In a set of ten consecutive integers, what are the possible values for the number of integers in the set that are coprime to all other integers in the set?

Odd product of tens digits

Susam Pal — Sun, 19 Feb 2012 00:00:00 +0000

The product of the tens digits of a few given perfect squares is odd. What is the units digit of the product of these perfect squares?

Equal sums in rows and columns

Susam Pal — Sun, 12 Feb 2012 00:00:00 +0000

Each square of a chessboard contains a number. The sum of the two largest numbers in each row is the same. The sum of the two largest numbers in each column is also the same. What is the maximum possible difference between the two sums?

Adding factors of factorials

Susam Pal — Sun, 05 Feb 2012 00:00:00 +0000

What is the largest integer \(n\) such that \(n!\) can not be expressed as the sum of \(n\) distinct factors of itself?

Sum of differences

Susam Pal — Sun, 29 Jan 2012 00:00:00 +0000

For a given even positive integer, we create two lists of integers such that every positive integer less than or equal to the given integer belongs to exactly one of the two lists and both lists contain an equal number of integers. There is a third list which is initially empty. Then, at every step, we remove the smallest number from the first list, the largest number from the second list and insert the positive difference between the two numbers into the third list.

What do we get if we divide the square of the given even positive integer by the sum of all the numbers in the third list when the first list becomes empty?

Splitting square into squares

Susam Pal — Sun, 22 Jan 2012 00:00:00 +0000

For a given perfect square, there is a set of five integers such that the sum of the squares of the integers in the set is ten times the given perfect square and the smallest positive difference between any pair of integers in the set is maximum. What do we get when we subtract the square of this difference from the given perfect square?

Grid of binomial coefficients

Susam Pal — Sun, 15 Jan 2012 00:00:00 +0000

There is an infinite grid of numbers. The first row contains all the natural numbers in ascending order. Any other number in this grid is the sum of those numbers from the row above it which do not exceed the number above it.

Given two positive integers n and k such that n ≥ k, where in the grid can we find the binomial coefficient C(n, k)?

Dropping marbles

Susam Pal — Sun, 08 Jan 2012 00:00:00 +0000

There is a 200-storey building. You are given 5 identical glass marbles. You are allowed to drop any marble from any floor to the ground. A marble either breaks or remains intact after a drop. If it remains intact, the marble can be reused.

In the worst case, what is the minimum number of drops needed to find the highest floor in the building from which you can drop the marbles without breaking them?

Unbiased rack of books

Susam Pal — Sun, 01 Jan 2012 00:00:00 +0000

Albert is opening a new library. He wants to keep a special rack of books facing the entrance to give the visitors a glimpse of the kinds of books it has. This special rack can hold only 10 books at a time. Every time a new book arrives at the library, he needs to decide whether to add it to the special rack or just place it in one of the regular racks in the library. A book is removed from this rack only if Albert decides to add a new book to it.

He wants to ensure that at any point in time, each book that has ever arrived at the library will be equally likely to be on the special rack. How is he going to do this?

Big binary sum

Susam Pal — Sun, 11 Dec 2011 00:00:00 +0000

If a_n represents the largest n-bit integer for a positive integer n, how many bits are 1 in a₁ + a₂ + … + a_1000000?

Growing list of nested radicals

Susam Pal — Sun, 27 Nov 2011 00:00:00 +0000

There is an empty list initially. First, 2 is inserted into the list. Then, at every step, every number in the list is replaced with (2 + √n) and (2 − √n) where n is the number being replaced.

What is the product of all numbers in the list when this step has been performed 1000 times?