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101 | Intermediate | Number Theory | Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55 r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. |
102 | Intermediate | Number Theory | For each positive integer $n$ let $a_n$ be the least positive integer multiple of 23 such that $a_n \equiv 1\left(\bmod 2^n\right)$. Find the number of positive integers $n$ less than or equal to 1000 that satisfy $a_n=a_{n+1}$. |
103 | Intermediate | Number Theory | Let $A$ be an acute angle such that $\tan A=2 \cos A$. Find the number of positive integers $n$ less than or equal to 1000 such that $\sec ^n A+\tan ^n A$ is a positive integer whose units digit is 9 . |
104 | Intermediate | Number Theory | The sum of all positive integers $m$ such that $\frac{13 !}{m}$ is a perfect square can be written as $2^a 3^b 5^c 7^d 11^e 13^f$, where $a, b, c, d, e$, and $f$ are positive integers. Find $a+b+c+d+e+f$. |
105 | Intermediate | Number Theory | For positive integers $a, b$, and $c$ with $a<b<c$, consider collections of postage stamps in denominations $a, b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to 1000 cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c)=97$ for some choice of $a$ and $b$. |
106 | Intermediate | Number Theory | For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2,3,4,5,6,7,8,9$, and 10 . For example, $R(15)=1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n)=R(n+1) ?$ |
107 | Intermediate | Number Theory | For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma\left(a^n\right)-1$ is divisible by 2021 for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$. |
108 | Intermediate | Number Theory | The probability a randomly chosen positive integer $N<1000$ has more digits when written in base 7 than when written in base 8 can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. |
109 | Intermediate | Number Theory | Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5, n$, and $n+1$ cents, 91 cents is the greatest postage that cannot be formed. |
110 | Intermediate | Number Theory | Determine all the sets of six consecutive positive integers such that the product of some two of them, added to the product of some other two of them is equal to the product of the remaining two numbers. |
111 | Olympiad | Number Theory | A deck of $n>1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
For which $n$ does it follow that the numbers on the cards are all equal? |
112 | Olympiad | Number Theory | Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n+17 k$ for some positive integer $k$. |
113 | Olympiad | Number Theory | Two permutations $a_1, a_2, \ldots, a_{2010}$ and $b_1, b_2, \ldots, b_{2010}$ of the numbers $1,2, \ldots, 2010$ are said to intersect if $a_k=b_k$ for some value of $k$ in the range $1 \leq k \leq 2010$. Show that there exist 1006 permutations of the numbers $1,2, \ldots, 2010$ such that any other such permutation is guaranteed to intersect at least one of these 1006 permutations. |
114 | Olympiad | Number Theory | Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0, k_1 \ldots, k_n$, all strictly greater than 1 , such that $k_0 k_1 \cdots k_n-1$ is the product of two consecutive integers. |
115 | Olympiad | Number Theory | $K>L>M>N$ are positive integers such that $K M+L N=(K+L-M+N)(-K+L+M+N)$. Prove that $K L+M N$ is not prime. |
116 | Olympiad | Number Theory | Let $k_1<k_2<k_3<\cdots$; be positive integers, no two consecutive, and let $s_m=k_1+k_2+\cdots+k_m$, for $m=1,2,3, \ldots$ Prove that, for each positive integer $n$, the interval $\left[s_n, s_{n+1}\right)$, contains at least one perfect square. |
117 | Olympiad | Number Theory | Let $a>b>c>d$ be positive integers and suppose that
\begin{align*}
a c+b d=(b+d+a-c)(b+d-a+c) .
\end{align*}
Prove that $a b+c d$ is not prime. |
118 | Olympiad | Number Theory | Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers satisfying $m, n \in\{1,2, \ldots, 1981\}$ and $\left(n^2-m n-m^2\right)^2=1$ |
119 | Olympiad | Number Theory | A permutation of the set of positive integers $[n]=\{1,2, \ldots, n\}$ is a sequence $\left(a_1, a_2, \ldots, a_n\right)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3,5,1,2,4)$ is a permutation of $[5]$. Let $P(n)$ be the number of permutations of $[n]$ for which $k a_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of 2010 . |
120 | Olympiad | Number Theory | Consider an open interval of length $1 / n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p / q$, with $1 \leq q \leq n$, contained in the given interval is at most $(n+1) / 2$. |
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