Subject 3. Determine an arithmetic progression of six terms $t_{1}, t_{2}, \ldots, t_{6}$ such that the sum of the squares of the terms $t_{1}, t_{2}, t_{3}, t_{4}$ is equal to both the product $t_{1} t_{2} t_{3}$ and the product $t_{5} t_{6}$.

Mathematical Gazette 1964
Subject 3. $\left.a^{2}+(a+r)^{2}+(a+2 r)^{2}+(a+3 r)^{2}=(a+4 r)(a+5 r) \mathbf{( 1 p}\right)$ $\Rightarrow(a-r)(a+2 r)=0(\mathbf{2 p})$. Case $a=r$, the numbers are $r, 2 r, 3 r, 4 r, 5 r, 6 r$ (1p); $r^{2}+(2 r)^{2}+(3 r)^{2}+(4 r)^{2}=r \cdot 2 r \cdot 3 r \Rightarrow r=0$ or $r=5$ (1p). Case $a=-2 r$, the numbers are $-2 r,-r, 0, r, 2 r, 3 r(\mathbf{1} \mathbf{p}) \cdot(-2 r)^{2}+(-r)^{2}+0^{2}+r^{2}=$ $(-2 r)(-r) 0 \Rightarrow r=0(\mathbf{1} \mathbf{p})$