Let \(a, b, c, d\) be positive integers such that \(a + b + c + d = 2023\). If \(a: b = 2:5\) and \(c:d=5:2\), then the maximum possible value of \(a + c\) is

The maximum of $a+c$ is $1442$. Put $a=2m,b=5m,c=5n,d=2n$, giving $m+n=289$ and $a+c=1445-3m$. The smallest positive $m$ is $1$, while $1445$ would require $m=0$ and give $a=b=0$.