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Nordic Mathematical Contest 2005

problem 1 Find all positive integers equals

such that the product of the digits of

, in decimal notation,

The product of the digits of digits is even; then

is an integer so

, and the product of the is

. Now since is odd, so is odd.

the rightmost digit is even, so

Also the product of the digits of number so because it is odd, that is,

is at most

(trivial to show) so

. On the other hand, it is a non-negative . Now we just have to check . Inspection shows both work.

problem 2 Let

be positive real numbers. Prove that

By Cauchy Schwartz

By Chebyshev and then Nesbitt,

Indeed very easy! I killed this in 10 seconds.

Q.E.D.

Problem3 There are young people sitting around a large circular table. Of these, at most are boys. We say that a girl has a strong position, if, counting from in either direction, the number of girls is always strictly larger than the number of boys ( is herself included in the count). Prove that there is always a girl in a strong position.

It's clear that if the result is true for boys, then it is true for any smaller number

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of boys (just switch some girls into boys till you have girls and boys.

boys, then the result is

true and by switching back the boys into girls it stays true). We'll prove the result for

This can be done by induction. Suppose the result is true for . Now take girls and boys. Take any girl which has a boy for a neighbor (it's clear there is one), and then go round the table till you find another have something like the boys in that interval. Among the . . Now take away girls and , so you'll where is any of boys left there's a girl in

a strong position by hypothesis; it's easy to see she's still strong when we put back

Problem4 The circle is inside the circle , and the circles touch each other at . A line through intersects also at , and also at . The tangent to at intersects at and . The tangents of passing thorugh touch at and . Prove that , , and are concyclic.

Consider the homotety of center to the tangent of and from here at parallel to which tranforms . Hence to then and and we are done. will transform

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