Found problems: 116
PEN P Problems, 18
Let $p$ be a prime with $p \equiv 1 \pmod{4}$. Let $a$ be the unique integer such that \[p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}\] Prove that \[\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),\] where $\left(\frac{k}{p}\right)$ denotes the Legendre Symbol.
1975 IMO Shortlist, 11
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, 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]
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.
1966 IMO Shortlist, 11
Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?
1977 Germany Team Selection Test, 3
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 $p \neq q.$
1983 IMO Longlists, 27
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 $x,y,z$ are non-negative integers.
PEN P Problems, 9
The integer $9$ can be written as a sum of two consecutive integers: 9=4+5. Moreover it can be written as a sum of (more than one) consecutive positive integers in exactly two ways, namely 9=4+5= 2+3+4. Is there an integer which can be written as a sum of $1990$ consecutive integers and which can be written as a sum of (more than one) consecutive positive integers in exactly $1990$ ways?
1992 IMO Shortlist, 21
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 $S(n)=n^{2}-14.$
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 $a = d.$
1989 IMO Shortlist, 15
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
1969 IMO Longlists, 18
$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$
PEN P Problems, 33
Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.
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, 14
Let $n$ be a non-negative integer. Find all non-negative integers $a$, $b$, $c$, $d$ such that \[a^{2}+b^{2}+c^{2}+d^{2}= 7 \cdot 4^{n}.\]
1992 IMO Longlists, 45
Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.
PEN P Problems, 35
Prove that every positive integer which is not a member of the infinite set below is equal to the sum of two or more distinct numbers of the set \[\{ 3,-2, 2^{2}3,-2^{3}, \cdots, 2^{2k}3,-2^{2k+1}, \cdots \}=\{3,-2, 12,-8, 48,-32, 192, \cdots \}.\]
1992 IMO Longlists, 34
Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that
\[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]
1996 IMO Shortlist, 3
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
PEN P Problems, 34
If $n$ is a positive integer which can be expressed in the form $n=a^{2}+b^{2}+c^{2}$, where $a, b, c$ are positive integers, prove that for each positive integer $k$, $n^{2k}$ can be expressed in the form $A^2 +B^2 +C^2$, where $A, B, C$ are positive integers.
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 $q^2 + r = 1977.$
1989 IMO Longlists, 50
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
PEN P Problems, 5
Show that any positive rational number can be represented as the sum of three positive rational cubes.
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.)
PEN P Problems, 29
Show that the set of positive integers which cannot be represented as a sum of distinct perfect squares is finite.