MathDB

4

Part of 1995 IMO Shortlist

Problems(4)

4xyz = a^2x + b^2y + c^2z + abc

Source: IMO Shortlist 1995, A4, , Titu Andreescu

3/26/2005
Find all of the positive real numbers like x,y,z, x,y,z, such that : 1.) x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c 2.) 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.
functioninequalitiesIMO Shortlistoptimizationsystem of equations133109
All 6 vertices of the two triangles lie on single circle

Source: IMO Shortlist 1995, G4, Iran PPCE 1997, P2

8/10/2008
An acute triangle ABC ABC is given. Points A1 A_1 and A2 A_2 are taken on the side BC BC (with A2 A_2 between A1 A_1 and C C), B1 B_1 and B2 B_2 on the side AC AC (with B2 B_2 between B1 B_1 and A A), and C1 C_1 and C2 C_2 on the side AB AB (with C2 C_2 between C1 C_1 and B B) so that
\angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.
The lines AA1,BB1, AA_1,BB_1, and CC1 CC_1 bound a triangle, and the lines AA2,BB2, AA_2,BB_2, and CC2 CC_2 bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
trigonometrygeometrycircumcirclehexagonIMO Shortlist
z + y² + x³ = xyz , x = gcd(y,z)

Source: IMO Shortlist 1995, N4

1/6/2005
Find all x,y x,y and z z in positive integer: z \plus{} y^{2} \plus{} x^{3} \equal{} xyz and x \equal{} \gcd(y,z).
algebranumber theoryequationgreatest common divisorIMO Shortlist
Power of two sequence inequality

Source: IMO Shortlist 1995, S4

8/18/2004
Suppose that x1,x2,x3,… x_1, x_2, x_3, \ldots are positive real numbers for which x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n for n \equal{} 1, 2, 3, \ldots Prove that ∀n, \forall n, 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.
inequalitiesfunctionalgebrapolynomialIMO Shortlist