Found problems: 71
A student divides an integer $m$ by a positive integer $n$, where $n \le 100$, and claims that $\frac{m}{n}=0.167a_1a_2...$ . Show the student must be wrong.
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
By placing parentheses in the expression $1:2:3$ we can get two different number values: $(1 : 2) : 3 = \frac16$ and $1 : (2 : 3) = \frac32$. Now brackets are placed in the expression $1:2:3:4:5:6:7:8$. Multiple bracket pairs are allowed, whether or not in nest form.
(a) What is the largest numerical value we can get, and what is the smallest?
(b) How many different number values can be obtained?
What is the value of $\dfrac{11!-10!}{9!}$?
$\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.
Determine all sequences of equal ratios of the form
\[
\frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}
\]
which simultaneously satisfy the following conditions:
$\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$.
$\bullet$ The common value of the ratios is a natural number.
Rational number $\frac{m}{n}$ admits representation
$$\frac{m}{n} = 1+ \frac12+\frac13 + ...+ \frac{1}{p-1}$$
where p $(p > 2)$ is a prime number. Show that the number $m$ is divisible by $p$.
Describe all ordered sets of four real numbers $(a, b, c, d)$ for which the values $a + b, b + c, c + d, d + a$ are all non-zero and \[\frac{a+2b+3c}{c+d}=\frac{b+2c+3d}{d+a}=\frac{c+2d+3a}{a+b}=\frac{d+2a+3b}{b+c}.\]
Let $n \in N$. Find the maximum number of irreducible fractions a/b (i.e., $gcd(a, b) = 1$) which lie in the interval $(0,1/n)$.
Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction.
Prove that $h$ is remarkable if and only if it is prime.
(Recall that an common fraction has an integer numerator and a natural denominator.)
What is the value of
$$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
$\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$
Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$.
A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\mu$. How does the minimum $\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$?
$\textbf{(A)}\mu \propto s^{0} \qquad
\textbf{(B)}\mu \propto s^{-\frac{1}{2}}\qquad
\textbf{(C)}\mu \propto s^{-1}\qquad
\textbf{(D)}\mu \propto s^{-2}\qquad
\textbf{(E)}\text{none of these}$
Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that:
$$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
Let $a, b, c, d$ be reals such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7$ and $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12$.
Find the value of $w =\frac{a}{b}+\frac{c}{d}$
.
Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that
$$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$
Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.
For each integer $n$, determine the smallest real number $M_n$ such that
\[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\]
for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
$\textbf{(A)}\hspace{.05in} \dfrac1{24}\qquad \textbf{(B)}\hspace{.05in}\dfrac1{12} \qquad \textbf{(C)}\hspace{.05in}\dfrac18 \qquad \textbf{(D)}\hspace{.05in}\dfrac16 \qquad \textbf{(E)}\hspace{.05in}\dfrac14 $
Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions:
$$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992}$$