Contestar vs. Responder

Contestar vs. Responder
 

It's my impression that these two words are very similar in meaning. My students (mostly Central American and Mexican) tell me that I should use "contestar" more often when we're talking about answering a question that they have about the mathematics.

Anyway, I just did an exercise in my workbook that required me to translate a sentence using the command form for ustedes. The setting is a boss giving a training session to new hotel employees, telling them how to treat guests.

English sentence: Answer their questions.
My translation: Contesten sus preguntas.
The book's answer: Respondan a sus preguntas.

So my questions are:
1) Is my use of contestar here okay, too?
2) Does contestar require "a"? Should I have written "Contesten a sus preguntas"?

Thank you!!

Yes. At least in Chile.

At least for me responder = contestar
I think around here we usually don't use "a". In the example you gave I would say it without "a", but both ways are ok.

"2. Cuando significa ‘dar respuesta a la pregunta, llamada o comunicación de alguien’, puede construirse con complemento directo o con un complemento precedido de la preposición a: «Jamás contestó las preguntas que hacían sobre su vida» (Azuela Casa [Méx. 1983]); «El procesado contestó a sus preguntas» (País [Esp.] 3.6.97); «La gerencia del Wyndham no contestó las llamadas de este diario» (NDía [P. Rico] 23.10.97); «Nadie contestaba a las llamadas del timbre» (Mundo [Esp.] 15.1.96)."

Contestar takes a direct object. It doesn't need to be followed by the preposition, but it is in certain circumstances, like when a question you asked has gone unanswered and you command "contesten a mi pregunta".
Responder doesn't take a direct object.

Thanks, all - that's very helpful!