SAT1000 - P873

Algebra Level pending

If there exists $t \in \mathbb R$ , so that the following system of $n$ equations all holds:

$\begin{cases} \begin{aligned} \lfloor t \rfloor & = 1 \\ \lfloor t^2 \rfloor & = 2 \\ \lfloor t^3 \rfloor & = 3 \\ & \vdots \\ \lfloor t^n \rfloor & = n \end{aligned} \end{cases}$

Then find the maximum of positive integer $n$ .

Have a look at my problem set: SAT 1000 problems

The answer is 4.

1 solution

Alexander McDowell
Jun 19, 2020

Let's look at the bounds for each expression:

$\lfloor t \rfloor = 1$ means that $1 \leq t < 2$ .
$\lfloor t^2 \rfloor = 2$ means that $1.414... \leq t < 1.732...$ .
$\lfloor t^3 \rfloor = 3$ means that $1.4422.. \leq t < 1.5874..$ .
$\lfloor t^4 \rfloor = 4$ means that $1.414... \leq t < 1.49534..$ .
$\lfloor t^5 \rfloor = 5$ means that $1.379... \leq t < 1.430969..$ .

$\lfloor t^4 \rfloor = 4$ is the last condition that can be added for there to exist a solution. Therefore, $\boxed{4}$ is the maximum value of $n$ .

SAT1000 - P873

The answer is 4.

1 solution

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