Consider four natural numbers $a , b , c$ and $d$ satisfying $1 < a < b < c < d$. Suppose that two sets using these numbers, $A = \{ a , b , c , d \}$ and $B = \left\{ a ^ { 2 } , b ^ { 2 } , c ^ { 2 } , d ^ { 2 } \right\}$, satisfy the following two conditions: (i) Just two elements belong to the intersection $A \cap B$, and the sum of these two elements is greater than or equal to 15, and less than or equal to 25. (ii) The sum of all the elements belonging to the union $A \cup B$, is less than or equal to 300. We are to find the values of $a , b , c$ and $d$. First, set $A \cap B = \{ x , y \}$, where $x < y$. Since $x \in B$ and $y \in B$, it follows from (i) that $y = \mathbf { A B }$ and that $x$ is either $\mathbf{C}$ or $\mathbf { D }$. (Write the answers in the order $\mathbf { C } < \mathbf { D }$.) Here, when we consider (ii), we see that $x = \mathbf { E }$. Hence $A$ includes the elements $\mathbf { F }$, $\mathbf{F}$ and $\mathbf{F}$. Furthermore, when we denote the remaining element of $A$ by $z$, from (ii) we see that $z$ satisfies $$z ^ { 2 } + z \leqq \mathbf { G H } .$$ Hence we have $z = \mathbf { I }$. From the above we obtain $$a = \mathbf { J } , \quad b = \mathbf { K } , \quad c = \mathbf { L } \text { and } d = \mathbf { M N } .$$
Consider four natural numbers $a , b , c$ and $d$ satisfying $1 < a < b < c < d$. Suppose that two sets using these numbers, $A = \{ a , b , c , d \}$ and $B = \left\{ a ^ { 2 } , b ^ { 2 } , c ^ { 2 } , d ^ { 2 } \right\}$, satisfy the following two conditions:\\
(i) Just two elements belong to the intersection $A \cap B$, and the sum of these two elements is greater than or equal to 15, and less than or equal to 25.\\
(ii) The sum of all the elements belonging to the union $A \cup B$, is less than or equal to 300.
We are to find the values of $a , b , c$ and $d$.
First, set $A \cap B = \{ x , y \}$, where $x < y$. Since $x \in B$ and $y \in B$, it follows from (i) that $y = \mathbf { A B }$ and that $x$ is either $\mathbf{C}$ or $\mathbf { D }$. (Write the answers in the order $\mathbf { C } < \mathbf { D }$.) Here, when we consider (ii), we see that $x = \mathbf { E }$. Hence $A$ includes the elements $\mathbf { F }$, $\mathbf{F}$ and $\mathbf{F}$.
Furthermore, when we denote the remaining element of $A$ by $z$, from (ii) we see that $z$ satisfies
$$z ^ { 2 } + z \leqq \mathbf { G H } .$$
Hence we have $z = \mathbf { I }$.\\
From the above we obtain
$$a = \mathbf { J } , \quad b = \mathbf { K } , \quad c = \mathbf { L } \text { and } d = \mathbf { M N } .$$