Found problems: 283
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$.
Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .
How many ordered pairs of positive integers $(m, n)$ satisfy the system
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2,
\\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
Prove that for any integer $n \ge 3$, the least common multiple of the numbers $1,2, ... ,n$ is greater than $2^{n-1}$.
[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.
[b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.
Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.
Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.)
Boris Frenkin
$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?
Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?
How many integers between $1000$ and $2000$ have all three of the numbers $15$, $20$, and $25$ as factors?
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that
$$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$
depending only on $r$ and $s.$
Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$.
Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .
If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.
Which of the following is the largest?
$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{3}{8} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{7}{24}$
A number which when divided by $ 10$ leaves a remainder of $ 9$, when divided by $ 9$ leaves a remainder of $ 8$, by $ 8$ leaves a remainder of $ 7$, etc., down to where, when divided by $ 2$, it leaves a remainder of $ 1$, is:
$ \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 419 \qquad\textbf{(C)}\ 1259 \qquad\textbf{(D)}\ 2519 \qquad\textbf{(E)}\ \text{none of these answers}$
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.
Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.
The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list:
\[\begin{tabular}[t]{lccc}
& & AJHSME & 1989 \\
& & & \\
1. & & JHSMEA & 9891 \\
2. & & HSMEAJ & 8919 \\
3. & & SMEAJH & 9198 \\
& & ........ &
\end{tabular}\]
What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time?
$\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
Given a positive integer $ n>3$, prove that the least common multiple of the products $ x_1x_2\cdots x_k$ ($ k\geq 1$) whose factors $ x_i$ are positive integers with $ x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n$, is less than $ n!$.
(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also
\[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\]
(b) Show that there are no two positive integers $a$ and $b$ such that
\[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]
[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.
[b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
Find all integer positive numbers $n$ such that:
$n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.
Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points)
Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.