A certain mathematical competition in the UK

Hardy's opposition was to the Mathematical Tripos (the Cambridge undergraduate mathematics degree), as it was prior to its reform in 1909, which Hardy did much to bring about.

The text of "A Mathematician's Apology", with Snow's preface, is here; the relevant paragraphs are

Almost since the time of Newton, and all through the nineteenth century, Cambridge had been dominated by the examination for the old Mathematical Tripos. [...] It had only one disadvantage, as Hardy pointed out with his polemic clarity, as soon as he had become an eminent mathematician and was engaged, together with his tough ally Littlewood, in getting the system abolished: it had effectively ruined serious mathematics in England for a hundred years.

As for why Hardy was so much against it, Snow's preface gives some of the reasons. It was a system which heavily emphasised fluency in intricate calculations rather than conceptual understanding. It was also a very rigid system which was slow to incorporate new developments in the subject (particularly those originating outside Britain). Moreover, the way the examination questions were set prioritised separating the top handful of candidates, rather than testing whether candidates near the bottom end had grasped the essentials (according to the statistics quoted on the Wikipedia page, in the 1854 examination the cut-off mark for a first-class degree was about 10% of the total marks available, and the cut-off for a third-class was about 2% -- seriously!).

Cambridge Mathematical Tripos examination was “a high speed marathon whose like has never been seen before or since” (http://apollonius.math.nthu.edu.tw/d1/disk5/js/history/cambridge.pdf --- "Old Cambridge Days" by Leonard Roth. Another great sources of interesting information about Tripos are the following books: "Mr Hopkins’ men: Cambridge reform and British mathematics in the 19th century" by A.D.D. Craik and "Masters of theory: Cambridge and the rise of mathematical physics" by A. Warwick. Examples below are from these books).

Tripos procedure was quite cruel, some kind of Darwinian natural selection at work, as the following examples illustrate.

C.T. Simpson, the Second Wrangler in 1842, had worked twenty hours a day for a whole week before the examination and during the Tripos he almost broke down from overexertion.

James Wilson, the Senior Wrangler in 1859, experienced a severe mental and physical breakdown immediately after the examination. It took three months him to recover from the illness and after the recovery he found that he had forgotten utterly all the mathematics that he had learned at Cambridge apart from elementary algebra and Euclid.

Nevertheless, according to Leonard Roth, Hardy's position about complete abolishment of Tripos examination was too extremist.

Various generations of Cambridge men have each shaped the examination according to their light, but the work is never complete and probably never will be. Other intending reformers of the Tripos are even now waiting in the wings; indeed some among them would reform it altogether. Such a notion, startling as it may appear, is by no means novel; it was held more than forty years ago, by Hardy himself, who had backed the 1909 reform as only a first stage of the program. Hardy firmly believed that the Tripos was an unmitigated evil, for which one must blame the inferior performance of British pure mathematician vis-a-vis their European colleagues. So, away with the examination.

Now the weakness of this argument resides in its lack of supporting evidence. It would be very difficult to unearth any specific cases of careers which have undoubtedly been ruined or even seriously damaged by the Cambridge mathematical system: on the other hand, the supporters of the status quo can for their part point to a long line of distinguished mathematical physicist, some of whom we have already mentioned, who achieved success either because or in spite of it: looking at their record one could scarcely suppose that they would have done more or better work had they been spared the ordeal of the examination; and, in the past, what an ordeal it was!

Abolitionists are such charming people; their motives are so patently pure, and only rarely do they foresee the full consequence of their projected panaceas. All his lifelong Hardy moved in the highest academic circles and tutored the most talented of young men. Had he troubled to consult any lecturer from a provincial university, or even (it may be) a don from a Cambridge college less exalted than Trinity, he could easily have learned a simple but significant truth: if students know beforehand that a particular subject is not to be examined upon, they will, almost to a man - or a woman - altogether decline to study it. Even Forsyth could have told Hardy that much: for he had been a professor at London University.