Slični zadaci

IMO Shortlist 2001 problem G5

Let $ABC$ be an acute triangle. Let $DAC,EAB$ , and $FBC$ be isosceles triangles exterior to $ABC$ , with $DA=DC, EA=EB$ , and $FB=FC$ , such that

$\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB.$

Let $D'$ be the intersection of lines $DB$ and $EF$ , let $E'$ be the intersection of $EC$ and $DF$ , and let $F'$ be the intersection of $FA$ and $DE$ . Find, with proof, the value of the sum

$\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.$

Slični zadaci

Lista Tekst Dva stupca

Zadaci

IMO Shortlist 2001 problem G3

Let $ABC$ be a triangle with centroid $G$ . Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$ .

IMO Shortlist 2001 problem G4

Let $M$ be a point in the interior of triangle $ABC$ . Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$ . Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define

$p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.$

Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

IMO Shortlist 2001 problem G6

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$ , $BP$ , $CP$ meet the sides $BC$ , $CA$ , $AB$ (or extensions thereof) in $D$ , $E$ , $F$ , respectively. Suppose further that the areas of triangles $PBD$ , $PCE$ , $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

IMO Shortlist 2001 problem G7

Let $O$ be an interior point of acute triangle $ABC$ . Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$ . Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$ , and $CA_1B_1$ .

IMO Shortlist 2007 problem G6

Determine the smallest positive real number $k$ with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1$ , $B_1$ , $C_1$ , and $D_1$ lie on sides $AB$ , $BC$ , $CD$ , and $DA$ , respectively. Consider the areas of triangles $AA_1D_1$ , $BB_1A_1$ , $CC_1B_1$ and $DD_1C_1$ ; let $S$ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$ . Then we always have $kS_1\ge S$ .

Author: unknown author, USA

IMO Shortlist 2009 problem G6

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$ ) intersect at point $P$ . Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$ , respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$ , respectively. Prove that the perpendicular from $E_1$ on $CD$ , the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.

Proposed by Ukraine