For all $x \in\left(0, \frac{\pi}{2}\right)$, the largest positive integer $n$ that satisfies the inequality $\sin ^{n} x+\cos ^{n} x>\frac{1}{2}$ is
10. 3.

For $x \in\left(0, \frac{\pi}{2}\right)$, we have $\sin x+\cos x>\sin ^{2} x+\cos ^{2} x>\sin ^{3} x+\cos ^{4} x>\sin ^{4} x+\cos ^{4} x=1-$ $\frac{1}{2} \sin ^{2} 2 x \geqslant \frac{1}{2}$, hence $n \geqslant 3$. Also, for $n \geqslant 1$, we have $\sin ^{n} \frac{\pi}{4}+\cos ^{n} \frac{\pi}{4}=\left(\frac{1}{\sqrt{2}}\right)^{n}+\left(\frac{1}{\sqrt{2}}\right)^{n} \leqslant 2\left(\frac{1}{\sqrt{2}}\right)^{4}=\frac{1}{2}$, thus $n<4$. Therefore, the largest integer $n=3$.