Found problems: 698
For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!\plus{}1$ and $ (n\plus{}1)!$. Find, without proof, a formula for $ f(n)$.
Let $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$. There is a unique value of $ y$ such that $ 0 < y < x$ and $ x^x \equal{} y^y$. What is the value of $ y$? Express your answer in the form $ \sqrt[c]{\frac{a}{b}}\,$, where $ a$ and $ b$ are relatively prime positive integers and $ c$ is a prime number.
You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$.
[i]Proposed by Aaron Lin[/i]
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]
If $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} = \frac{m}{n}$ for relatively prime integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a\plus{}9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\ge\frac12$ can be written as $ \frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.
Is there an infinite sequence of positive integers where no two terms are relatively prime, no term divides any other term, and there is no integer larger than $1$ that divides every term of the sequence?
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy
\[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]
For how many ordered pairs of positive integers $ (m,n)$, $ m \cdot n$ divides $ 2008 \cdot 2009 \cdot 2010$ ?
$\textbf{(A)}\ 2\cdot3^7\cdot 5 \qquad\textbf{(B)}\ 2^5\cdot3\cdot 5 \qquad\textbf{(C)}\ 2^5\cdot3^7\cdot 5 \qquad\textbf{(D)}\ 2^3\cdot3^5\cdot 5^2 \qquad\textbf{(E)}\ \text{None}$
Let $p$ a prime number and $d$ a divisor of $p-1$. Find the product of elements in $\mathbb Z_p$ with order $d$. ($\mod p$).
(10 points)
Already the complete problem:
Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers:
$\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$
One idea?
:arrow: [b][url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad[/url][/b]
Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.
One hundred natural numbers whose greatest common divisor is $1$ are arranged around a circle. An allowed operation is to add to a number the greatest common divisor of its two neighhbors. Prove that we can make all the numbers pairwise copirme in a finite number of moves.
Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that
\[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]
Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" .
(1) Is $(s,t)$ a good pair ?
(2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".
Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.
When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\textstyle\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$.
[i]Proposed by Eugene Chen[/i]
A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x \plus{} 1$ and $ \frac {x}{x \plus{} 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning.
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
When $12{}^1{}^8$ is divided by $18{}^1{}^2$, the result is $(\tfrac{m}{n})^3$, where $m$ and $n$ are relatively prime integers. Find $m-n$.
Show that there are infinitely many noncongruent triangles which satisfy the following conditions:
i) the side lengths are relatively prime integers;
ii)the area is an integer number;
iii)the altitudes' lengths are not integer numbers.
Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.)
[i]David Yang.[/i]
For $\pi\leq\theta<2\pi$, let
\[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots
\] and
\[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta
+\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$.
Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.