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[Solutions] Iranian Geometry Olympiad 2017

Elementary

  1. Each side of square $ABCD$ with side length of $4$ is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Just write the numbers without proof.
  2. Find the angles of triangle $ABC$.
  3. In the regular pentagon $ABCDE$, the perpendicular at $C$ to $CD$ meets $AB$ at $F$. Prove that $AE+AF=BE$.
  4. $P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle clockwise if the increasing order of them is in clockwise order. Can the number of clockwise triangles be exactly $2017$?
  5. In the isosceles triangle $ABC$ ($AB=AC$), let $l$ be a line parallel to $BC$ through $A$. Let $D$ be an arbitrary point on $l$. Let $E$, $F$ be the feet of perpendiculars through $A$ to $BD$, $CD$ respectively. Suppose that $P$, $Q$ are the images of $E$, $F$ on $l$. Prove that $AP+AQ\le AB$.

Intermediate

  1. Let $ABC$ be an acute-angled triangle with $A=60^{\circ}$. Let $E$, $F$ be the feet of altitudes through $B$, $C$ respectively. Prove that $$CE-BF=\frac{3}{2}(AC-AB).$$
  2. Two circles $\omega_1,\omega_2$ intersect at $A,B$. An arbitrary line through $B$ meets $\omega_1$, $\omega_2$ at $C$, $D$ respectively. The points $E,F$ are chosen on $\omega_1$, $\omega_2$ respectively so that $CE=CB$, $BD=DF$. Suppose that $BF$ meets $\omega_1$ at $P$, and $BE$ meets $\omega_2$ at $Q$. Prove that $A$, $P$, $Q$ are collinear.
  3. On the plane, $n$ points are given ($n>2$). No three of them are collinear. Through each two of them the line is drawn, and among the other given points, the one nearest to this line is marked (in each case this point occurred to be unique). What is the maximal possible number of marked points for each given $n$?
  4. In the isosceles triangle $ABC$ ($AB=AC$), let $l$ be a line parallel to $BC$ through $A$. Let $D$ be an arbitrary point on $l$. Let $E$, $F$ be the feet of perpendiculars through $A$ to $BD$, $CD$ respectively. Suppose that $P$, $Q$ are the images of $E$, $F$ on $l$. Prove that $AP+AQ\le AB$.
  5. Let $X$, $Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ($X$ is between $B$, $Y$). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$. Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$, and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$. Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$.

Advanced Level

  1. In triangle $ABC$, the incircle, with center $I$, touches the sides $BC$ at point $D$. Line $DI$ meets $AC$ at $X$. The tangent line from $X$ to the incircle (different from $AC$) intersects $AB$ at $Y$. If $YI$ and $BC$ intersect at point $Z$, prove that $AB=BZ$.
  2. We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one.
  3. Let $O$ be the circumcenter of triangle $ABC$. Line $CO$ intersects the altitude from $A$ at point $K$. Let $P,M$ be the midpoints of $AK$, $AC$ respectively. If $PO$ intersects $BC$ at $Y$, and the circumcircle of triangle $BCM$ meets $AB$ at $X$, prove that $BXOY$ is cyclic.
  4. Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1$, $\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$.
  5. Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B'$, $C'$, $D'$ are defined similarly. Prove that $A'$, $B'$, $C'$, $D'$ are coplanar and the plane $A'B'C'D'$ touches $S$.

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MOlympiad: [Solutions] Iranian Geometry Olympiad 2017
[Solutions] Iranian Geometry Olympiad 2017
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