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Part of 2012 France Team Selection Test

Problems(2)

AP, BC and OH are concurrent iff AH=HN

Source: French TST 2012

8/2/2012
Let ABCABC be an acute-angled triangle with AB≠ACAB\not= AC. Let Γ\Gamma be the circumcircle, HH the orthocentre and OO the centre of Γ\Gamma. MM is the midpoint of BCBC. The line AMAM meets Γ\Gamma again at NN and the circle with diameter AMAM crosses Γ\Gamma again at PP. Prove that the lines AP,BC,OHAP,BC,OH are concurrent if and only if AH=HNAH=HN.
geometrycircumcircletrigonometryparallelogramgeometric transformationreflectionratio
The roots are the coefficients

Source: French TST 2012

8/2/2012
Determine all non-constant polynomials Xn+an−1Xn−1+⋯+a1X+a0X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0 with integer coefficients for which the roots are exactly the numbers a0,a1,…,an−1a_0,a_1,\ldots ,a_{n-1} (with multiplicity).
algebrapolynomialalgebra unsolved