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