▪ Romania National Μath Olympiad 2012

IX

1. The altitude dropped onto the hypotenuse of a triangle intersects the bisectors and at and respectively. Prove that the line passing through the midpoints of the segments and is parallel to the line . 

2. Find all functions with the following property: for any open bounded interval , the set is an open interval having the same length with . 

3 Prove that if is a natural number and are positive real numbers, then:

4. On a table there are piles having pencils respectively. A move consists in choosing two piles having and pencils respectively, and transferring pencils from the first pile to the second one. Find the necessary and sufficient condition for , such that there exists a succession of moves through which all pencils are transferred to the same pile.

X

1. Let Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set . 

2. Let , and be three complex numbers such that and . Prove that:

for any ,

3. Let with . Prove that:

a) , for

b)

4. Let and be two natural numbers, . Find the number of injective functions

such that there exists a unique number for which

XI

1. Let be two functions such that is monotonic, surjective and , for any .

a) Prove that is continuous and that there exists some with .

b) Prove that the set is a closed interval.

2. Let and be two natural numbers such that and . Prove that if the matrix has exactly minors of order equal to , then . 

3. Let such that and . Prove that:

4. Find all differentiable functions for which and for any . 

XII

1. Let be a continuous function such that , for any natural number . Prove that is a periodic function. 

2. Let be a ring and let be a surjective endomorphism of such that for any , where , . Prove that:

a) and , for any

b) If is a division ring and is different from the identity function, then is commutative. 

3. Let be the set of integrable functions such that for any . Define the function by

Determine the following two sets:

a) , where , if and , if

b)

4. Let and be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial , when covers the set of - degree polynomials with complex coefficients.

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