MathDB

1

Part of 2015 Iran MO (3rd round)

Problems(3)

Inequality Iran 2015

Source: Iranian third round - Algebra exam 2015 (problem1)

9/8/2015
x,y,zx,y,z are three real numbers inequal to zero satisfying x+y+z=xyzx+y+z=xyz. Prove that ∑(x2−1x)2≥4 \sum (\frac{x^2-1}{x})^2 \geq 4
Proposed by Amin Fathpour
inequalities
primes less than 1394

Source: Iranian third round number theory P1

9/6/2015
Prove that there are infinitely natural numbers nn such that nn can't be written as a sum of two positive integers with prime factors less than 13941394.
number theoryalgebra
intersection of two circles

Source: Iranian third round 2015 geometry problem 1

9/10/2015
Let ABCDABCD be the trapezoid such that AB∥CDAB\parallel CD. Let EE be an arbitrary point on ACAC. point FF lies on BDBD such that BE∥CFBE\parallel CF. Prove that circumcircles of △ABF,△BED\triangle ABF,\triangle BED and the line ACAC are concurrent.
geometrytrapezoidcircumcircle