2. For a real number $a$, let $[a]$ denote the greatest integer not greater than $a$. Find all integers $y$ for which there exists a real number $x$ such that $\left[\frac{x+23}{8}\right]=[\sqrt{x}]=y$.
Let $y$ be such a number. Then $\sqrt{x} \geq[\sqrt{x}]=y$. Since $\sqrt{x} \geq 0$, it follows that $y=[\sqrt{x}] \geq 0$, so we can square the inequality to get $x \geq y^{2}$. Additionally, $\frac{x+23}{8}<$ $\left[\frac{x+23}{8}\right]+1=y+1$ or $x<8 y-15$. From this, it follows that $y^{2}<8 y-15$ or $(y-3)(y-5)<0$, which means $3<y<5$. Since $y$ must be an integer, $y=4$. In this case, there indeed exists an $x$ that satisfies the condition, for example, $x=16$.

## SCORING CRITERIA

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-18.jpg?height=65&width=1633&top_left_y=1892&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-18.jpg?height=57&width=1631&top_left_y=1945&top_left_x=218)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-18.jpg?height=57&width=1631&top_left_y=1996&top_left_x=218)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-18.jpg?height=54&width=1631&top_left_y=2046&top_left_x=218)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-18.jpg?height=55&width=1631&top_left_y=2097&top_left_x=218)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-18.jpg?height=57&width=1633&top_left_y=2147&top_left_x=220)

If the contestant generally observes that $[a] \leq a<[a]+1$, they receive 2 points, 1 point for each inequality.

If the contestant correctly evaluates the expressions $\left[\frac{x+23}{8}\right]$ and $[\sqrt{x}]$ for different values of $y$ and thus intuitively justifies that the only solution is $y=4$, they receive 5 points. For a reasoned conclusion (e.g., using a graph or proving a different rate of growth of the functions inside [ ]), that $y=4$ is the only solution, they receive an additional 2 points, for a total of 7 points. In the case of a partial solution along this path or errors in the process, the relevant clauses at the beginning of the criteria are appropriately considered.

If the contestant simply guesses the solution $y=4$, they receive 1 point.