# Difference between revisions of "1978 AHSME Problems/Problem 18"

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==Solution== | ==Solution== | ||

− | + | Adding <math>\sqrt{n - 1}</math> to both sides, we get | |

+ | <cmath>\sqrt{n} < \sqrt{n - 1} + 0.01.</cmath> | ||

+ | Squaring both sides, we get | ||

+ | <cmath>n < n - 1 + 0.02 \sqrt{n - 1} + 0.0001,</cmath> | ||

+ | which simplifies to | ||

+ | <cmath>0.9999 < 0.02 \sqrt{n - 1},</cmath> | ||

+ | or | ||

+ | <cmath>\sqrt{n - 1} > 49.995.</cmath> | ||

+ | Squaring both sides again, we get | ||

+ | <cmath>n - 1 > 2499.500025,</cmath> | ||

+ | so <math>n > 2500.500025</math>. The smallest positive integer <math>n</math> that satisfies this inequality is <math>\boxed{2501}</math>. |

## Latest revision as of 21:17, 30 September 2021

## Problem

What is the smallest positive integer such that ?

## Solution

Adding to both sides, we get Squaring both sides, we get which simplifies to or Squaring both sides again, we get so . The smallest positive integer that satisfies this inequality is .