899/IMO. problem / Math - Solved Math Problems

MR-899. problem

Solve the following equation:

$t! = t^{3} - t$

Source: IMO: International Mathematical Olympiad - 0.0.)

Složeni zadaci

$t! = t^{3} - t$

$t \cdot (t - 1)! = t^{3} - t$

$t \cdot (t - 1)! = t \cdot (t^{2} - 1)$

$t \cdot (t - 1)! = t \cdot (t^{2} - 1)$

$(t - 1) \cdot (t - 2)! = (t - 1) \cdot (t + 1)$

$(t - 1) \cdot (t - 2)! = (t - 1) \cdot (t + 1)$

$(t - 2)! = t + 1$

$k = t - 2$

$t = k + 2$

$k! = k + 2 + 1$

$k! = k + 3$

$k \cdot (k - 1)! = k + 3$

$k \cdot (k - 1)! - k = 3$

$k [(k - 1)! - 1] = 3$

$(I) \to 1 \cdot 3 = 3; (I I) \to 3 \cdot 1 = 3$

First case is impossible.

Second case:

$3 \cdot 1 = 3$

$k = 3$

$3 = 3 \cdot [(3 - 1)! - 1]$

$3 = 3 \cdot (2! - 1)$

$3 = 3 \cdot 1$

$t = k + 2$

$t = 3 + 2$

$t = 5$