Found problems: 15460
2016 AMC 12/AHSME, 24
There are exactly $77,000$ ordered quadruples $(a,b,c,d)$ such that $\gcd(a,b,c,d)=77$ and $\operatorname{lcm}(a,b,c,d)=n$. What is the smallest possible value of $n$?
$\textbf{(A)}\ 13,860 \qquad
\textbf{(B)}\ 20,790 \qquad
\textbf{(C)}\ 21,560 \qquad
\textbf{(D)}\ 27,720 \qquad
\textbf{(E)}\ 41,580$
2014 Nordic, 3
Find all nonnegative integers $a, b, c$ such that
$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$
1997 German National Olympiad, 2
For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$.
Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.
2012 India Regional Mathematical Olympiad, 2
Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.
2023 LMT Fall, 16
Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?
[i]Proposed byMuztaba Syed[/i]
2019 CMIMC, 9
Let $a_0=29$, $b_0=1$ and $$a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1}$$ for $n\geq 1$. Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$.
2008 Regional Olympiad of Mexico Center Zone, 1
Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.
2017 Latvia Baltic Way TST, 16
Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.
2014 Purple Comet Problems, 27
Five men and fi ve women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1995 AMC 12/AHSME, 27
Consider the triangular array of numbers with $0,1,2,3,...$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows $1$ through $6$ are shown.
\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular}
Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by $100$?
$\textbf{(A)}\ 12\qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 50 \qquad
\textbf{(D)}\ 62 \qquad
\textbf{(E)}\ 74$
2022 LMT Fall, 4
Find the least positive integer ending in $7$ with exactly $12$ positive divisors.
2002 Hungary-Israel Binational, 1
Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.
1998 IMO Shortlist, 5
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
1961 All Russian Mathematical Olympiad, 009
Given $a, b, p$ arbitrary integers. Prove that there always exist relatively prime (i.e. that have no common divisor) $k$ and $l$, that $(ak + bl)$ is divisible by $p$.
2023 Kurschak Competition, 1
Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.
2011 Northern Summer Camp Of Mathematics, 2
Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]
2020 German National Olympiad, 4
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.
2018 IMO Shortlist, N3
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
1987 Tournament Of Towns, (159) 3
Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .
1990 Swedish Mathematical Competition, 1
Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.
2024 Thailand Mathematical Olympiad, 9
Prove that for all positive integers $n$, there exists a sequence of positive integers $a_1,a_2,\dots,a_n$ and $d_1,d_2,\dots,d_n$ satisfying all of the following three conditions.
[list]
[*] $\binom{2a_i}{a_i}$ is divisible by $d_i$ for all $i=1,2,\dots,n$
[*] $d_{i+1}=d_i+1$ for all $i=1,2,\dots, n-1$
[*] $d_i\neq m^k$ for all $i=1,2,\dots, n$ and positive integers $m$ and $k$ such that $k\geq 2$
[/list]
1990 Irish Math Olympiad, 2
A sequence of primes $a_n$ is defined as follows: $a_1 = 2$, and, for all $n \geq 2$,$
a_n$ is the largest prime divisor of $a_1a_2...a_{n-1} + 1$. Prove that $a_n \neq 5$
for all n.
I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks
2022 Baltic Way, 16
Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition
$$ f(a) + f(b) \mid (a + b)^2$$
for all $a,b \in \mathbb{Z^+}$
1993 China Team Selection Test, 1
Find all integer solutions to $2 x^4 + 1 = y^2.$
TNO 2008 Senior, 8
Two mathematicians discuss two positive integers. One of them states that the square of the ratio between their product and their sum is exactly one more than this ratio. What is the smaller of these two numbers?