2021 JMPSC Accuracy Problems/Problem 9: Difference between revisions
Created page with "==Problem== If <math>x_1,x_2,\ldots,x_{10}</math> is a strictly increasing sequence of positive integers that satisfies <cmath>\frac{1}{2}<\frac{2}{x_1}<\frac{3}{x_2}< \cdots..." |
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==Solution== | ==Solution== | ||
Say we take <math>x_1,x_1,x_3,...,x_{10}</math> as <math>4,5,6,...,13</math> as an example. The first few terms of the inequality would then be: | |||
<cmath>\frac{1}{2}<\frac{2}{4}<\frac{3}{5}<\frac{4}{6}</cmath> | |||
But <math>\frac{3}{5}<\frac{4}{6}</math>, reaching a contradiction. | |||
A contradiction will also be reached at some point when <math>x_1\geq 4</math> or when <math>x_1\leq 2</math>, so that must mean <math>x_1=3</math>. | |||
<math>\implies 3+4+5+...+12=\frac{10\cdot 15}{2}=\boxed{75}</math> | |||
<math>\linebreak</math> | |||
~Apple321 | |||
Revision as of 10:12, 11 July 2021
Problem
If 
is a strictly increasing sequence of positive integers that satisfies ![\[\frac{1}{2}<\frac{2}{x_1}<\frac{3}{x_2}< \cdots < \frac{11}{x_{10}},\]](http://latex.artofproblemsolving.com/a/d/1/ad108b31c7b45c0ccab98e7ff977b0ef95f8c89f.png)
find 
.
Solution
Say we take 
as 
as an example. The first few terms of the inequality would then be:
![\[\frac{1}{2}<\frac{2}{4}<\frac{3}{5}<\frac{4}{6}\]](http://latex.artofproblemsolving.com/3/8/8/38840b3754b4082d8437320d392037b6a34dae42.png)
But 
, reaching a contradiction.
A contradiction will also be reached at some point when 
or when 
, so that must mean 
.
~Apple321
