"intractable challenge" is correct and usable in written English.
It refers to a problem or issue that is difficult or impossible to solve or overcome. For example, "The refugee crisis is an intractable challenge that the world is still struggling to address."
An even subtler and more intractable challenge is identifying bees once they've been collected.
Yet the engineering of functionally appropriate human-device interfaces represents an intractable challenge in the field.
Despite the mounting pressure on Isis in recent weeks, its grip on the two cities remains unrelenting, posing an intractable challenge to forces seeking to liberate them.
Modern biology suggests that combinations of drugs are more effective than a single antibiotic, however, screening all combinations is an intractable challenge.
"Yet obtaining a little extra money to bring in a lot of extra money remains an intractable challenge for the I.R.S., and that is unfortunate," the report said.
Meniscal injury is one of the most common knee joint injuries, which remains an intractable challenge in clinical practice to date.
In academia, the focus is on problems that can be solved, proved and published — not messy, intractable challenges.
Nuclear energy is a risky and high cost source of electricity that produces intractable challenges of militarisation, waste and proliferation.
Among the many seemingly intractable challenges China faces, the economy, environment and corruption are at the top of the list.
Both have confronted the same issues (Islam, secularism, sexuality) and both have stumbled into public controversies when their soft-spoken styles collided with intractable challenges.
Six months into his governorship, Mr. Christie in his speech on Wednesday took on several of the state's longest-running and most intractable challenges.
