PALO ALTO, California — Cementing its standing as the most selective institution of higher education in the country, Stanford University announced this week that it had once again received a record-setting number of applications and that its acceptance rate — which had dropped to a previously uncharted low of 5 percent last year — plummeted all the way to its inevitable conclusion of 0 percent.

With no one admitted to the class of 2020, Stanford is assured that no other school can match its desirability in the near future.

“We had exceptional applicants, yes, but not a single student we couldn’t live without,” said a Stanford administrator who requested anonymity. “In the stack of applications that I reviewed, I didn’t see any gold medalists from the last Olympics — Summer or Winter Games — and while there was a 17-year-old who’d performed surgery, it wasn’t open-heart or a transplant or anything like that. She’ll thrive at Yale.”

News of Stanford’s unprecedented selectiveness sent shock waves through the Ivy League, along with Amherst, Northwestern and at least a dozen other elite schools where, as a consequence, there could be substantial turnover among underperforming deans of admission.