Found problems: 700
Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.
Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$.
Determine the smallest value of $T = x^3 + y^3$.
Find all positive integers $a,b,c$ and prime $p$ satisfying that
\[ 2^a p^b=(p+2)^c+1.\]
Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.
Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.
Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers?
$
\textbf{(A)}\ 59170
\qquad\textbf{(B)}\ 59149
\qquad\textbf{(C)}\ 59130
\qquad\textbf{(D)}\ 59121
\qquad\textbf{(E)}\ 59012
$
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)f(a+b)-ab$ is a perfect square for all $a, b \in \mathbb{N}$.
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$, and $b < p$, such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$.
[i]Noam Elkies[/i]
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$
[list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good.
[*][b](b)[/b] Show that all 2010-good pairs are very good.[/list]
[i]Proposed by Okan Tekman, Turkey[/i]
Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.
Find an integer $n$, where $100 \leq n \leq 1997$, such that
\[ \frac{2^n+2}{n} \]
is also an integer.
How many interger tuples $(x,y,z)$ are there satisfying $0\leq x,y,z < 2011$, $xy+yz+zx \equiv 0 \pmod{2011}$, and $x+y+z \equiv 0 \pmod{2011}$ ?
$\textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 4021 \qquad\textbf{(E)}\ 4023$
Suppose that for a prime number $p$ and integers $a,b,c$ the following holds:
\[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\]
Prove that $p\mid a,b,c$.
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]
Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$).
Prove that there are infi nite number of [i]Primes of the Year[/i].
Let $x, y$ be integers and $p$ be a prime for which
\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
For how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13?$
Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$
[i]Vlad Matei[/i]
Let \( p \neq 3 \) be a prime number. Prove that there exist natural numbers \( a \), \( b \), \( c \), \( d \), none of which are divisible by \( p \), such that \( a^2 + 3b^5 + 5c^6 + 7d^7 \) is divisible by \( p^{1000} \).
Let $ m,\ n \geq 3$ be positive odd integers. Prove that $ 2^{m}\minus{}1$ doesn't divide $ 3^{n}\minus{}1$.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
Find all positive integers $n$, such that there exists a positive integer $m$ and primes $1<p<q$ such that $q-p \mid m$ and $p, q \mid n^m+1$.
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.