3. How many distinct triangles satisfy all the following properties:
(i) all three side-lengths are a whole number of centimetres in length;
(ii) at least one side is of length $10 \mathrm{~cm}$;
(iii) at least one side-length is the (arithmetic) mean of the other two side-lengths?
SOLUTION
By (ii), one side has length $10 \mathrm{~cm}$. Let the lengths of the other two sides be $a \mathrm{~cm}$ and $b \mathrm{~cm}$, which, by (i), are positive integers. Without loss of generality, we assume $a \geq b$. Condition (iii) yields three possibilities.
(a) The mean of $a$ and $b$ is 10 , so that $a+b=20$. Now, in order to respect the triangle inequality, the possibilities for $(a, b)$ are $(10,10),(11,9),(12,8),(13,7)$ and $(14,6)$. Note that $(15,5)$ is impossible since it would result in a degenerate $(15,10,5)$ 'triangle'.
(b) The mean of $b$ and 10 is $a$, so $2 a=b+10$. Now the possibilities are $(10,10),(9,8),(8,6)$ and $(7,4)$.
(c) The mean of $a$ and 10 is $b$, so that $2 b=a+10$. Here the possibilities are $(10,10),(12,11)$, $(14,12),(16,13),(18,14),(20,15),(22,16),(24,17),(26,18)$ and $(28,19)$.

Hence, counting the $(10,10,10)$ case only once, there are exactly 17 such triangles.