Found problems: 283
(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
\[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]
If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$.Let $f(n)$ be the largest positive integer $ k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that :
[b](a)[/b] $f(n)<3\sqrt{n}$ for all positive integers $n$ .
[b](b)[/b] If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$.
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
Let $n$ be a positive integer such that $n \geq 2$. Let $x_1, x_2,..., x_n$ be $n$ distinct positive integers and $S_i$ sum of all numbers between them except $x_i$ for $i=1,2,...,n$. Let $f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}.$
Determine maximal value of $f(x_1,x_2,...,x_n)$, while $(x_1,x_2,...,x_n)$ is an element of set which consists from all $n$-tuples of distinct positive integers.
Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S, $x \neq y$ } $\ge$ 2 + max S.
Prove that $\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2} $.
Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime.
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 5$
For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$
Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.
For positive integers $a$, $b$, and $c$, define \[ f(a,b,c)=\frac{abc}{\text{gcd}(a,b,c)\cdot\text{lcm}(a,b,c)}. \] We say that a positive integer $n$ is $f@$ if there exist pairwise distinct positive integers $x,y,z\leq60$ that satisfy $f(x,y,z)=n$. How many $f@$ integers are there?
[i]Proposed by Michael Ren[/i]
Let $x, y$, and $z$ be real numbers satisfying $x + y + z = xyz.$ Prove that
\[x(1 - y^2)(1 - z^2) + y(1 -z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz.\]
Hey,
This problem is from the VTRMC 2006.
3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6).
As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation.
Consider the following,
01 = 1 mod 10
01 = 1 mod 10
02 = 2 mod 10
03 = 3 mod 10
05 = 5 mod 10
08 = 6 mod 10
13 = 3 mod 10
21 = 1 mod 10
34 = 4 mod 10
55 = 5 mod 10
89 = 9 mod 10
Now, consider that between the first appearance and second apperance of $ 5 mod 10$, there is a difference of five terms. Following from this we see that the third appearance of $ 5 mod 10$ occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships.
$ F(55) \equal{} F(05) \plus{} 5({2}^{2})$
This is pretty much as far as we got, any ideas?
Given quadrilateral $ ABCD$. The circumcircle of $ ABC$ is tangent to side $ CD$, and the circumcircle of $ ACD$ is tangent to side $ AB$. Prove that the length of diagonal $ AC$ is less than the distance between the midpoints of $ AB$ and $ CD$.
How many numbers are there that appear both in the arithmetic sequence $10,
16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$
Suppose that $p$ is a prime number.
Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have
\[ n|a^{\frac{\varphi(n)}{p}}-1 \]
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?
[b]b.)[/b] For which $n>2$ is there exactly one set having this property?
When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is
$\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$
Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
$\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$
$\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$
$\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$
$\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$
$\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$
Determine all possible values of $m+n$, where $m$ and $n$ are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\]
Let $x$ and $y$ be positive integers such that $2(x+y)=gcd(x,y)+lcm(x,y)$. Find $\frac{lcm(x,y)}{gcd(x,y)}$.
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 $a_1, \ldots , a_k.$
For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.
Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that
a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$,
b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$.
Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a [i]winning number[/i] if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.
[b]Note:[/b] The number $\operatorname{gcd}(a, b)$ denotes the [i]greatest common divisor[/i] of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the [i]least common multiple[/i] of $a$ and $b$.