[ Quadratic inequalities and systems of inequalities ] [ Examples and counterexamples. Constructions ]

Find all prime numbers $p$, for each of which there exists a natural number $m$ such that

$\sqrt{m}+\sqrt{m+p}$ is also a natural number.
Let $p$ be odd, that is, $p=2 n+1$, where $n$ is a natural number. Then the condition of the problem is satisfied for $m=n^{2}$. It remains to consider the case $p=2$. Let $\sqrt{m}+\sqrt{m+2}$ be a natural number. Then the number $(\sqrt{m}+\sqrt{m+2})^{2}=2 m+2+2 \sqrt{m^{2}+2 m}$, and therefore, $2 \sqrt{m^{2}+2 m}$ is an integer. But $(2 m+1)^{2}<4\left(m^{2}+2 m\right)$ $<(2 m+2)^{2}$, that is, $2 m+1<2 \sqrt{m^{2}+2 m}<2 m+2$. Contradiction.

## Answer

$p-$ any odd prime number.

![](https://cdn.mathpix.com/cropped/2024_05_06_6fdbe810ffc70cc86b90g-23.jpg?height=166&width=2041&top_left_y=217&top_left_x=0)

Author: $\underline{\text { Frankin B.R. }}$

In a chess tournament, each participant played two games with each of the others: one with white pieces, the other with black. At the end of the tournament, it turned out that all participants scored the same number of points (1 point for a win, $1 / 2$ point for a draw, 0 points for a loss). Prove that there will be two participants who won the same number of games with white pieces.

## Solution

In total, $n(n-1)$ points were played in the tournament. Therefore, each participant scored $n-1$ points. Each chess player played $n-1$ games with white pieces, and the number of games won by them with white pieces is one of $n$ numbers: $0, \ldots, n-1$. Suppose all won a different number of games with white pieces. Then all possible variants from 0 to $n-1$ are realized.

Consider two participants in the tournament: $A$, who won $n-1$ games with white pieces, and $B$, who did not win any such games. What could be the result of the game that $A$ played against $B$ with black pieces? On the one hand, $A$ scored $n-1$ points playing with white pieces, so he lost all his games with black pieces, including this one. But $B$ did not win any games with white pieces, so he could not have won this game either. Contradiction.

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