Found problems: 116
1983 IMO Longlists, 51
Decide whether there exists a set $M$ of positive integers satisfying the following conditions:
(i) For any natural number $m>1$ there exist $a, b \in M$ such that $a+b = m.$
(ii) If $a, b, c, d \in M$, $a, b, c, d > 10$ and $a + b = c + d$, then $a = c$ or $a = d.$
PEN P Problems, 40
Show that [list=a][*] infinitely many perfect squares are a sum of a perfect square and a prime number, [*] infinitely many perfect squares are not a sum of a perfect square and a prime number. [/list]
PEN P Problems, 26
Let $a, b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x, y, z \in \mathbb{N}_{0}$
1983 IMO Shortlist, 15
Decide whether there exists a set $M$ of positive integers satisfying the following conditions:
(i) For any natural number $m>1$ there exist $a, b \in M$ such that $a+b = m.$
(ii) If $a, b, c, d \in M$, $a, b, c, d > 10$ and $a + b = c + d$, then $a = c$ or $a = d.$
2001 Romania Team Selection Test, 4
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
PEN P Problems, 22
Show that an integer can be expressed as the difference of two squares if and only if it is not of the form $4k+2 \; (k \in \mathbb{Z})$.
PEN P Problems, 16
Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.
1969 IMO Shortlist, 7
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
2015 India IMO Training Camp, 1
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
PEN P Problems, 10
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [list=a] [*] Prove that $S(n)\leq n^{2}-14$ for each $n\geq 4$. [*] Find an integer $n$ such that $S(n)=n^{2}-14$. [*] Prove that there are infinitely many integers $n$ such that $S(n)=n^{2}-14$. [/list]
1975 IMO, 2
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
PEN P Problems, 19
Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.
PEN P Problems, 42
Prove that for each positive integer $K$ there exist infinitely many even positive integers which can be written in more than $K$ ways as the sum of two odd primes.
2001 IMO Shortlist, 6
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?
2002 IMO Shortlist, 6
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that
- for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and
- the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$
Prove that \[{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.\]
PEN P Problems, 37
Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$, where $n$ is an integer greater than $1$. Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.)
1992 IMO, 3
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
PEN P Problems, 20
If an integer $n$ is such that $7n$ is the form $a^2 +3b^2$, prove that $n$ is also of that form.
PEN P Problems, 3
Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.
1969 IMO Shortlist, 25
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$
PEN P Problems, 31
A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. [list=a] [*] Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$. [*] Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=1996$. [/list]
PEN P Problems, 23
Show that there are infinitely many positive integers which cannot be expressed as the sum of squares.
1977 IMO, 2
Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$
2016 Bosnia and Herzegovina Team Selection Test, 4
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.
1992 IMO Longlists, 77
Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$, then there exist two among the chosen integers such that one of them is a factor of the other.