National Math Olympiad

Let $p(x)=b_1{x}^3+c_1{x}^2+d_{1}x+e_1$ and $q(x)=b_2{x}^3+c_2{x}^2+d_{2}x+e_2$. Let us write $a=\frac{r}{s}$ where $r$ and $s$ are relatively prime integers and $s>0$. Denote $p(a)=m$ and $q(a)=n$. Then $$b_1\cdot \frac{r^3}{s^3}+c_1\cdot \frac{r^2}{s^2}+d_1\cdot \frac{r}{s}+e_1=m,b_2\cdot \frac{r^3}{s^3}+c_2\cdot \frac{r^2}{s^2}+d_2\cdot \frac{r}{s}+e_2=n.$$ Multiplying both identities by $s^3$ we get $$b_1{r}^3+c_1{r}^{2}s+d_{1}r{s}^2+e_1{s}^3=m{s}^3,b_2{r}^3+c_2{r}^{2}s+d_{2}r{s}^2+e_2{s}^3=n{s}^3.$$ This implies that $s$ divides both $b_1{r}^3$ and $b_2{r}^3$. Since $s$ and $r$ are relatively prime, we can conclude that $s$ divides $b_1$ and $b_2$. Since $b_1$ and $b_2$ are relatively prime, we have $s=1$. So, $a=r$ is an integer.