2013中国女子数学奥林匹克试题及其解答(10)

Now, what values

can

values modulo take modulo ? It is a simple exercise to show all and then

as

can be

can be obtained (just express it

as we can transform the problem to what values modulo

expressed as

works out is if

Therefore we simply require that

of

Note: To

show takes all values modulo modulo , 10

modulo

. and and

modulo , which is known to be all of them). Therefore the only way this problems

and . . For each value of

there are values so the answer should simply

be without relying

on

is not hard, but I'm lazy and felt like reducing it to an already solved

problem.

4.解（yunxiu）

hence

So the answer should be

6.解（crazyfehmy）

If

since

has always a solution

，

so satisfies the problem. . ，

let

then is equivalent

to satisfies

which and since is odd.

be the

and

let in the set for all Now we will show that if elements of the

set then the condition does not satisfy. Let . Consider the

sums

. Since

are also different modulo 's are different

modulo , the numbers 's

. On the other hand, none of 's can be equivalent to

modulo

is a because otherwise we would have two equivalent terms.

Hence

permutation

of and by adding up these equations we

get

which means