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29 problems match “fractions”. Clear

Smallest integer from parenthesized division chain ●●●○○ 50 ptsIn the expression $10 : 9 : 8 : 7 : 6 : 5 : 4 : 3 : 2 : 1$ , parentheses are placed so that the value of the expression is an integer. What is the smallest number that could have been obtained?
Least n with many unit-fraction solutions ●●●○○ 50 ptsLet $x$ , $y$ and $n$ be positive integers, and count the solutions $(x, y)$ of $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ with $x \le y$ . For $n=4$ there are exactly three such solutions. Find the least value of $n$ for which the number of solutions exceeds one thousand.
Principal value with real poles ●●●○○ 50 ptsFind the Cauchy principal value of $\int_{-\infty}^{\infty}\frac{x\,dx}{(x-1)^4-1}$ , where the poles at $0$ and $2$ lie on the real axis. Round to two decimal places.
Complex line integral of 1/(1+z^2) ●●●○○ 50 ptsEvaluate $\int_1^{1+i}\frac{1}{1+z^2}\,dz$ along the straight line segment from $z=1$ to $z=1+i$ . Give the real part and the imaginary part of the result separately. Round each to two decimal places.
Elephant carrying bananas to city B ●●●●○ 100 ptsTwo cities $A$ and $B$ are $1000$ km apart. City $A$ has $3000$ bananas and an elephant that can carry at most $1000$ bananas at a time. The elephant eats one banana per km and cannot travel without bananas. Assume the elephant eats continuously, so it can eat fractions of a banana along the way. What is the maximum number of bananas that can be transferred to city $B$ ? Round to two decimal places.
Smallest integer from parenthesised divisions ●●●●○ 100 ptsIn the expression $10:9:8:7:6:5:4:3:2:1$ , where $:$ denotes division, parentheses are placed so that the value of the expression is an integer. What is the smallest value that could have been obtained?
Minimal period of a sum of decimals ●●●●○ 100 ptsPositive rational numbers $a$ and $b$ are written as decimal fractions, each of which has a minimal period of 30 digits. The decimal representation of $a-b$ has a minimal period of length 15. For what smallest natural number $k$ can the decimal representation of $a+kb$ also have a minimal period of length 15?
Least number of factors giving 51/2010 ●●●●○ 100 ptsFind the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots, s_n\}$ consisting of $n$ distinct positive integers such that $\left(1-\frac{1}{s_1}\right)\left(1-\frac{1}{s_2}\right)\cdots\left(1-\frac{1}{s_n}\right)=\frac{51}{2010}.$
Denominators of partial sums ●●●●○ 100 ptsOlya wrote fractions of the form $\frac{1}{n}$ on cards, where $n$ runs over all positive divisors of $6^{100}$ (including $1$ and $6^{100}$ ). She arranged the cards in some order. She then wrote on the board the number on the first card, the sum of the numbers on the first two cards, the sum of the numbers on the first three cards, and so on, up to the sum of the numbers on all the cards. Each sum was written as an irreducible fraction. What is the smallest possible number of different denominators that could appear on the board?