Bonjour,
1 ) On a 0 < x < 10 soit x ∈ ]0 ; 10[
2 ) AT = 2x × x / 2 = x²
AC = (10-x)²
3 ) AC ≥ AT ⇔ (10 - x)² ≥ x²
⇔ (10 - x)² - x² ≥ 0
⇔ (10 - x - x) (10 - x + x) ≥ 0
⇔ 10 - 2x ≥ 0
⇔ x ≤ 5
4 ) TO + OI + IT ≥ NO + OU + US + SN
⇔ x + 2 √((2x)² + (x/2)²) ≥ 4 (10 - x)
⇔ x + 2 √(4x² + x²/4) ≥ 4 (10 - x)
⇔ x + √(16x² + x²) ≥ 4 (10 - x)
⇔ x + x √17 + 4x ≥ 40
⇔ x (5 + √17) ≥ 40
⇔ x (5 + √17) (5 - √17) ≥ 40 (5 - √17)
⇔ x (25 - 17) ≥ 40 (5 - √17)
⇔ x ≥ 5 (5 - √17)
⇔ x ≥ 25 - 5√17
5) (1) et (2) ⇔ 25 - 5√17 ≤ x ≤ 5
