Found problems: 30
1975 Chisinau City MO, 87
Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.
2015 JBMO Shortlist, NT1
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
2001 Estonia National Olympiad, 5
A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column.
(a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table.
(b) Show that there is such a table not all of whose entries are $0$.
1986 Tournament Of Towns, (127) 2
Does there exist a number $N$ so that there are $N - 1$ infinite arithmetic progressions with differences $2 , 3 , 4 ,..., N$ , and every natural number belongs to at least one of these progressions?
2001 Singapore MO Open, 3
Suppose that there are $2001$ golf balls which are numbered from $1$ to $2001$ respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither $5$ nor $8$. How many such golf balls the box can contain at most? Justify your answer.
2005 Singapore Senior Math Olympiad, 1
The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.
1983 Tournament Of Towns, (039) O1
Numbers from $1$ to $1000$ are arranged around a circle. Prove that it is possible to form $500$ non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than $749$.
(AA Razborov, Moscow)
2013 NZMOC Camp Selection Problems, 2
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).
2004 Estonia National Olympiad, 2
The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.
2003 Junior Balkan Team Selection Tests - Moldova, 5
Prove that each positive integer is equal to a difference of two positive integers with the same number of the prime divisors.
1985 Tournament Of Towns, (091) T2
From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .
2019 Peru EGMO TST, 3
For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$, not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$, not necessarily different.
For example, if $A = \{3,1,-1\}$
$\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$.
$\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$.
Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$.
2019 Dutch Mathematical Olympiad, 5
Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera.
At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins.
(a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty.
(b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?
2011 Tournament of Towns, 1
The numbers from $1$ to $2010$ inclusive are placed along a circle so that if we move along the circle in clockwise order, they increase and decrease alternately. Prove that the difference between some two adjacent integers is even.
1986 Tournament Of Towns, (119) 1
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.
1949-56 Chisinau City MO, 44
Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.
1987 Bundeswettbewerb Mathematik, 4
Place the integers $1,2 , \ldots, n^{3}$ in the cells of a $n\times n \times n$ cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?
1991 All Soviet Union Mathematical Olympiad, 545
The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$
1975 Chisinau City MO, 101
Prove that among any $k + 1$ natural numbers there are two numbers whose difference is divisible by $k$.
2017 Junior Balkan Team Selection Tests - Romania, 2
Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most
$n$.
2025 Philippine MO, P1
The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?
2013 Balkan MO Shortlist, N7
Two distinct positive integers are called [i]close [/i] if their greatest common divisor equals their difference. Show that for any $n$, there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.
2015 JBMO Shortlist, NT3
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee
2006 Tournament of Towns, 3
(a) Prove that from $2007$ given positive integers, one of them can be chosen so the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$. (2)
(b) One of $2007$ given positive integers is $2006$. Prove that if there is a unique number among them such that the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$, then this unique number is $2006$. (2)
1997 Swedish Mathematical Competition, 6
Assume that a set $M$ of real numbers is the union of finitely many disjoint intervals with the total length greater than $1$. Prove that $M$ contains a pair of distinct numbers whose difference is an integer.