# Concours General de Mathematiques "Minko Balkanski" - May 6th 2006

All the answers must be given in English or in French. The clarity and precision will be taken into account for the final grade.
Time limit: 4 hours

Exercise 1.
We consider a circle C with center O and line L that does not meet C. Let P be the orthogonal projection of O on L. Let Q be a point on L distinct from P and let QA and QB be the two tangent lines to C passing through Q (the points A and B belonging to C). We denote by K the intersection point of AB and OP. Finally, let M and N be the orthogonal projections of P on QA and QB. Show that the line MN passes through the middle of the line segment KP.

Exercise 2.
Show that for all real positive numbers x, y and z the following inequality holds:

( 2x + y + z )2 + ( 2z + x + y )2 + ( 2y + z + x)2 <= 8
2x2+(y+z)22y2+(z+x)22z2+(x+y)2

Exercise 3.
For a natural n, we denote by f(n) the number of integers between 1 and n that are coprime to n (Euler’s function). We recall that if n = p1a1 … pkak is the prime factorization of n, then f(n) = (p1a1 – p1a1-1) … (pkak – pkak-1).

a) Show that f(n) is never an odd integer > 1.
b) Show that f(n) < > 14.
c) Show that there exist infinitly many even natural numbers not equal to f(n) for any n.
d) Find all the natural numbers n such that f(n) = 215.

Exercise 4.
Let AI and AM be the angle bisector and median from A in triangle ABC. The line orthogonal to AI passing through I intersect the lines AB, AM and AC at the points D, P et E, respectively. Let Q be the unique point of the line AI such that DQ is orthogonal to AD.

a) Show that EQ is orthogonal to AE.
b) Show that PQ is orthogonal to BC. Изтегли файла (.doc)

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Дани Кондова ( 2847) на 09 Юли 2006 ... МОГА!!! А всъщност е много лесно Записки.инфо ( 2777) на 09 Юли 2006
Много си добра! Как успя да напишеш тея дроби?? Вземи драсни един кратък самоучител за писане на математични формули като статична подстраница. 