Found problems: 39
In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?
The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*}
Calculate $f(14,52)$.
Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.
If the Highest Common Divisor of $ 6432$ and $ 132$ is diminished by $ 8$, it will equal:
$ \textbf{(A)}\ \minus{}6 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ \minus{}2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
What is the units digit of $ 13^{2003}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.
Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$.
For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows.
(i) $a_1=<a>$
(ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$,
if $a_n=0$, then $a_{n+1}=0$.
(1) For $a=\sqrt{2}$, find $a_n$.
(2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$.
(3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$.
[i]2011 Tokyo University entrance exam/Science, Problem 2[/i]
Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.
By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder.
As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$.
Find $m$ and $n$.
Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties:
[list]
[*] $f(1,1)=0$.
[*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1;
[*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$.
[/list]
Prove that
$$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible?
$ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$
Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.