MathDB

3

Part of 2010 Tuymaada Olympiad

Problems(4)

Tuymaada 2010, Junior League, Problem 3

Source:

7/18/2010
Let f(x)=ax2+bx+cf(x) = ax^2+bx+c be a quadratic trinomial with aa,bb,cc reals such that any quadratic trinomial obtained by a permutation of ff's coefficients has an integer root (including ff itself). Show that f(1)=0f(1)=0.
quadraticsalgebrapolynomialVietaabsolute valuealgebra unsolved
Tuymaada 2010, Junior League, Problem 7

Source:

7/18/2010
Let ABCABC be a triangle, II its incenter, ω\omega its incircle, PP a point such that PI⊥BCPI\perp BC and PA∥BCPA\parallel BC, Q∈(AB),R∈(AC)Q\in (AB), R\in (AC) such that QR∥BCQR\parallel BC and QRQR tangent to ω\omega. Show that ∠QPB=∠CPR\angle QPB = \angle CPR.
geometryincentergeometric transformationreflectionsymmetrygeometry unsolved
Equal number of 1s, 2s, & 3s in a circular sequence

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

7/31/2011
Arranged in a circle are 20102010 digits, each of them equal to 11, 22, or 33. For each positive integer kk, it's known that in any block of 3k3k consecutive digits, each of the digits appears at most k+10k+10 times. Prove that there is a block of several consecutive digits with the same number of 11s, 22s, and 33s.
combinatorics unsolvedcombinatorics
Equal distances between pairs of orthocenters in cyclic quad

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

7/31/2011
In a cyclic quadrilateral ABCDABCD, the extensions of sides ABAB and CDCD meet at point PP, and the extensions of sides ADAD and BCBC meet at point QQ. Prove that the distance between the orthocenters of triangles APDAPD and AQBAQB is equal to the distance between the orthocenters of triangles CQDCQD and BPCBPC.
geometrycircumcirclegeometric transformationreflectioncyclic quadrilateralgeometry unsolved