Arun scored 25, 13, 12 and 22 points in the $6^{\text{th}}$, $7^{\text{th}}$, $8^{\text{th}}$ and $9^{\text{th}}$ basketball games, respectively. His points-per game average was higher after nine games than it was after the first five games. If his average after ten games was greater than 19, the least number of points that he could have scored in the $10^{\text{th}}$ game, was

Let $S_5$ be the sum of scores in the first 5 games.

Let $x$ be the score in the $10^{\text{th}}$ game.

The sum of scores for games 6, 7, 8, and 9 is:

$25 + 13 + 12 + 22 = 72$

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The average after 9 games is higher than the average after 5 games.

Average after 5 games: $\frac{S_5}{5}$

Average after 9 games: $\frac{S_5 + 72}{9}$

Setting up the inequality:

$\frac{S_5 + 72}{9} > \frac{S_5}{5}$

$5(S_5 + 72) > 9(S_5)$

$5S_5 + 360 > 9S_5$

$360 > 4S_5$

$90 > S_5$

The sum of the scores of the first 5 games must be less than 90.

Since basketball scores are integers, the maximum possible integer value for $S_5$ is 89.

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The average after 10 games is greater than 19.

The sum of all 10 games is $S_5 + 72 + x$.

$\frac{S_5 + 72 + x}{10} > 19$

$S_5 + 72 + x > 190$

$S_5 + x > 118$

$x > 118 - S_5$

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To find the least number of points $x$, the largest possible value of $S_5$ needs to be subtracted from 118.

The maximum integer value for $S_5$ is 89.

$x > 118 - 89$

$x > 29$

Since $x$ must be an integer strictly greater than 29, the least possible integer value is 30.

Therefore, the least number of points that could have been scored in the $10^{\text{th}}$ game is 30.