MathDB

4

Part of 2008 Iran MO (3rd Round)

Problems(5)

x+y+z=3

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Let x,y,z\in\mathbb R^{\plus{}} and x\plus{}y\plus{}z\equal{}3. Prove that: \frac{x^3}{y^3\plus{}8}\plus{}\frac{y^3}{z^3\plus{}8}\plus{}\frac{z^3}{x^3\plus{}8}\geq\frac19\plus{}\frac2{27}(xy\plus{}xz\plus{}yz)
inequalitiesinequalities proposedCauchy Inequality
Writing as sum of two squares

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Let u u be an odd number. Prove that \frac{3^{3u}\minus{}1}{3^u\minus{}1} can be written as sum of two squares.
number theory proposednumber theory
In terms of Fibonacci sequence

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Let S S be a sequence that: \left\{ \begin{array}{cc} S_0\equal{}0\hfill\\ S_1\equal{}1\hfill\\ S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1) \end{array} \right. such that Fn F_n is Fibonacci sequence such that F_1\equal{}F_2\equal{}1. Find Sn S_n in terms of Fibonacci numbers.
functioncombinatorics proposedcombinatorics
Find BAC

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Let ABC ABC be an isosceles triangle with AB\equal{}AC, and D D be midpoint of BC BC, and E E be foot of altitude from C C. Let H H be orthocenter of ABC ABC and N N be midpoint of CE CE. AN AN intersects with circumcircle of triangle ABC ABC at K K. The tangent from C C to circumcircle of ABC ABC intersects with AD AD at F F. Suppose that radical axis of circumcircles of CHA CHA and CKF CKF is BC BC. Find ∠BAC \angle BAC.
geometrycircumcirclepower of a pointradical axisgeometry proposed
Algebraic Sets

Source: Iranian National Olympiad (3rd Round) 2008

9/20/2008
=A subset S S of R2 \mathbb R^2 is called an algebraic set if and only if there is a polynomial p(x,y)∈R[x,y] p(x,y)\in\mathbb R[x,y] such that S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\} Are the following subsets of plane an algebraic sets? 1. A square http://i36.tinypic.com/28uiaep.png 2. A closed half-circle http://i37.tinypic.com/155m155.png
algebrapolynomialgeometryperimeteranalytic geometryfunctionparameterization