20200614  20200715 
an easy one: Let k be an integer: For which k does 120 divides (k^5  k)? 

20200521  20200614 
an easy one: Let k be an integer: Show that 30 divides (k^5  k) 

20190925  20200521 
Find all pairs (k, n) of positive integers such that k! = (2^n − 1)(2^n − 2)(2^n − 4)*...*(2^n − 2^(n−1)) 

20190529  20190925 
http://mathworld.wolfram.com/ Let p be an odd prime. How many nonempty subsets of {1,2,...,p2,p1} have a sum which is divisible by p 

20190525  20190529 
Let p be an odd prime. How many nonempty subsets of {1,2,...,p2,p1} have a sum which is divisible by p 

20190304  20190525 
http://mathworld.wolfram.com/ https://pastebin.com/M2zDkUD3 We have to make 24 by using 1, 3, 4, and 6. We can use +,,*,/. and each number exactly once 

20180618  20190304 
https://pastebin.com/M2zDkUD3 We have to make 24 by using 1, 3, 4, and 6. We can use +,,*,/. and each number exactly once 

20180610  20180618 
We have to make 24 by using 1, 3, 4, and 6. We can use +,,*,/. and each number exactly once 

20180223  20180610 
Let S be a bounded, closed set on the real plane such that the distance between any two points < 1. Show that S lies within a circle of radius 1/sqrt(3) 

20170917  20180223 
For each rational number p/q in (0,1) set an open interval of size 1/(2q^2) centered on p/q. Show that sqrt(2)/2 is not in any of these intervals! 

20170914  20170917 
For each rational number p/q in (0,1) set an open interval of size 1/(2q^2) centered on p/q. Show that sqrt(2)/2 is not in any of these intervals! Congrats Gauss! Congrats Nick! Congrats Jcay! 

20170824  20170914 
For n a natural number, show that ceil[sqrt(n) + sqrt(n+1)] = ceil[sqrt(4n + 2)] Congrats Gauss! Congrats Nick! Congrats Jcay! 

20170615  20170824 
Let a,b,c be nonnegative real numbers, no two of which are equal. Prove that a^2/(bc)^2 + b^2/(ca)^2 + c^2/(ab)^2 > 2. Congrats Gauss! 

20170422  20170615 
Let a,b,c be nonnegative real numbers, no two of which are equal. Prove that a^2/(bc)^2 + b^2/(ca)^2 + c^2/(ab)^2 > 2. 

20170413  20170422 
Prove that n! + 1 is not a square number for all integer n>7 

20170218  20170413 
Let x_1, ..., x_n be the n nth roots of unity. Evaluate Prod_i<j [(x_i  x_j)^2] 

20170212  20170218 
Let a, b, c, d be pairwise distinct integers such that f(x) = (xa)(xb)(xc)(xd)  4 has an integer root k. Show that 4k = a + b + c + d 

20170212 (21:08:52  22:08:54) 
Let a, b, c, d be pairwise disctinct integers such that f(x) = (xa)(xb)(xc)(xd)  4 has an integer root k. Show that 4k = a + b + c + d 

20170131  20170212 
Find all integers n such that (n^2  81) is a multiple of 100 

20161121  20170131 


20161127  20161201 
Happy Birthday Jcay! may you have an exceptional year with much happiness, health and wealth. Solve: http://mathb.in/107431 

20161115  20161121 
For all natural numbers n, show that 2^(n+1) divides ceil[ (1+sqrt(3))^2n ] 

20161114  20161115 
Beauty is not always meant to reveal itself. For it knows how to continuously drive the seekers by keeping them hungry (Elizabeth Bennet) 

20161030  20161114 


20160913  20161030 
Given are 100 positive integers whose sum equals their product. Determine the minimum number of 1s that may occur among the 100 numbers. 

20160910  20160913 
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd? 

20160815  20160910 
There are 100 ladies in a club.Each lady has had tea (in private) with exactly 56 of her friends.The Board,consisting of the 50 most old ladies,have all had tea with one another.Prove that the entire club may be split into two groups in such a way that,in each group,any lady had tea with any other. 

20160207  20160815 
Let a_1,a_2,a_3,... and b_1,b_2,b_3,... be two sequences of natural numbers. Prove that there are indices i<j such that a_i <= a_j and b_i <= b_j 

20160123  20160207 
The numbers 1,2, ... ,n^2 are placed randomly in an n × n table. Prove that there are two adjacent cells (in a row or a column) such that the numbers in them differ by at least n. 

20151205  20160131 
Suppose you are given n blocks, each of which weights an integral number of pounds, but less than n pounds. Suppose that the total weight of the n blocks is less than 2n. Prove that we can divide the blocks into two groups, one of which weights exactly n pounds. 

20151103  20151205 
Prove that there is no solution to x(x+1)(x+2)(x+3) = y^2 where x,y are natural numbers. 

20151030  20151103 
Prove that any coloring of the natural numbers with a finite number of colors contains integers x,y,z of the same color such that x+y=z Bonus question: prove the same when z=x*y get an OP question: Prove that the coloring contains integers x,y,z,w of the same color such that x+y=z and x*y=w. 

20150818  20151030 
We have 2^m sheets of paper, with the number 1 written on each sheet. In every step we choose two distinct sheets; if the numbers on the two sheets are a and b, then we erase these numbers and write the number a + b on both sheets. Prove that after m2^(m1) steps the sum of the numbers on all sheets >= 4^m 

20150711  20150818 
For which integers k does (x^2  x + k) divide (x^13 + x + 90)? 

20150710  20150711 
For which integers k does (x^2  x + k) divide (x^13 + x +90)? 

20150610  20150710 
A rectangular table has 100 coins with unit radius, placed on it such that none of the coins overlap, and it is impossible to place any more coins on the table without causing an overlap. Using this specific configuration, find a special configuration of 400 coins which covers the table with overlaps. 

20150501  20150610 
http://pastebin.com/kkP2x89d Let a,b be positive integers. Show that if (4ab1) divides ((4a^2)  1)^2 then a=b. 

20150425  20150430 
A number written in base 10 is a string of 3^2013 digit 3s. No other digit appears. Find the highest power of 3 which divides this number. http://pastebin.com/kkP2x89d 

20150422  20150425 
A number written in base 10 is a string of 3^2013 digit 3s. No other digit appears. Find the highest power of 3 which divides this number. 

20150327  20150422 
Let n be a natural number. Prove that [floor(n/1) + floor(n/2) + floor(n/3) + .... + floor(n/n)] + floor(sqrt(n)) is always even 

20150312  20150327 
For each n show that there is a Fibonacci number that ends in at least n zeros. Rest in peace Terry Pratchett, we loved you and your books. 

20150302  20150312 
For each n show that there is a Fibonacci number that ends in at least n zeros. 

20150228  20150301 
In a country there are several cities and several roads. Every road connects to exactly 2 cities. Out of every city there exist at least 3 roads. Prove that there is a cycle, the number of cities in which is not divisible by 3. 

20150221  20150228 
Suppose you are given n blocks, each of which weighs an integral number of pounds, but less than n pounds. Suppose also that the total weight of the n blocks is less than 2n pounds. Prove that the blocks can be divided into two groups, one of which weighs exatly n pounds. 

20150221 (15:02:34  17:02:35) 
Suppose you are given n blocks, eah of which weighs an integral number of pounds, but less than n pounds. Suppose also that the total weight of the n blocks is less than 2n pounds. Prove that the blocks can be divided into two groups, one of which weighs exatly n pounds. 

20150220  20150221 
Suppose you are given n blocks, eah of which weigts an integral number of pounds, but less than n pounds. Suppose also that the total weight of the n bloks is less than 2n pounds. Prove that the blocks can be divided into two groups, one of which weighs exatly n pounds. 

20150213  20150220 
Let n be a fixed positive integer. Find the sum of all positive integers with the following property: In base 2, it has exactly 2n digits consisting of n 1’s and n 0’s. The first digit cannot be 0. 

20150209  20150213 
Suppose that a_0=1 and a_(n+1) = a_n + e^(a_n) for n=0,1,2,... Does a_n  ln(n) have a finite limit as n tends to infinity? 

20150205  20150209 
Each vertex of a finite graph can be colored either black or white. Initially all vertices are black. We are allowed to pick a vertex P and change the color of P and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? 

20150126  20150205 
Let m, n be natural numbers. Show that 4mn − m − n can never be a square. Show that for all n >= 6 there exists natural numbers (a_1,...,a_n) such that 1/a_1^2 + ... + 1/a_n^2 = 1. 

20150103  20150126 
Let m, n be natural numbers. Show that 4mn − m − n can never be a square. 

20141222  20150103 
Given an integer n>=2 prove that the product of all primes lower or equal than n is lower or equal than 4^(n1) 