MathDB

4

Part of 2015 Saint Petersburg Mathematical Olympiad

Problems(3)

Parallelogram geometry

Source: St Petersburg Olympiad 2015, Grade 11, P4

10/17/2017
ABCDABCD is convex quadrilateral. Circumcircle of ABCABC intersect ADAD and DCDC at points PP and QQ. Circumcircle of ADCADC intersect ABAB and BCBC at points SS and RR. Prove that if PQRSPQRS is parallelogram then ABCDABCD is parallelogram
geometrycircumcircleparallelogram
Largest 'Olympic' number not exceeding 2015

Source: St. Petersburg Math Olympiad, 2015, round ii, grade 10, P4

9/30/2016
A positive integer nn is called Olympic, if there exists a quadratic trinomial with integer coeffecients f(x)f(x) satisfying f(f(n))=0f(f(\sqrt{n}))=0. Determine, with proof, the largest Olympic number not exceeding 20152015.
A. Khrabrov
number theoryquadratics
Nice inequality: prove 4x+y+z>=2

Source: St. Petersburg Math Olympiad, 2015, round ii, grade 9, P4

9/30/2016
Positive numbers x,y,zx, y, z satisfy the condition xy+yz+zx+2xyz=1.xy + yz + zx + 2xyz = 1. Prove that 4x+y+z≥2.4x + y + z \ge 2.
A. Khrabrov
inequalitiesinequalities proposedalgebra