Q:

SF novel in which the presidency of the world depends on your IQ score

A friend is trying to identify a science fiction novel (not a short story) which he read no later than 1980 — more likely in the '60s or '70s. The thing he remembers is that the presidency of the world is dependant on one's IQ score.
He is very clear that it was a novel, so it is not a misremembering of (say) Harrison Bergeron.

A:

This could be The World of Ā (later published as The World of Null-A, spelling out the pronunciation of the "Ā" glyph), by A. E. van Vogt.  It was serialized in 1945 and first published as a novel in 1948.
The plot summary, per Wikipedia:

Gilbert Gosseyn (pronounced go sane), a man living in an apparent utopia where those with superior understanding and mental control rule the rest of humanity, wants to be tested by the giant Machine that determines such superiority. However, he finds that his memories are false. In his search for his real identity, he discovers that he has extra bodies that are activated when he dies (so that, in a sense, he cannot be killed), that a galactic society of humans exists outside the Solar system, a large interstellar empire wishes to conquer both the Earth and Venus (inhabited by masters of non-Aristotelian logic), and he has extra brain matter that, when properly trained, can allow him to move matter with his mind.

The Games Machine mentioned in the summary is responsible for determining which people are intellectually advanced enough to receive important positions (including the presidency of Earth) or to be allowed to settle on Venus.

