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[Solutions] Sharygin Geometry Mathematical Olympiad 2013 (Final Round)

Grade 8

  1. Let $ABCDE$ be a pentagon with right angles at vertices $B$ and $E$ and such that $AB = AE$ and $BC = CD = DE$. The diagonals $BD$ and $CE$ meet at point $F$. Prove that $FA = AB$.
  2. Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C$ and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.
  3. Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?
  4. The diagonals of a convex quadrilateral $ABCD$ meet at point $L$. The orthocenter $H$ of the triangle $LAB$ and the circumcenters $O_1$, $O_2$, and $O_3$ of the triangles $LBC$, $LCD$, and $LDA$ were marked. Then the whole configuration except for points $H$, $O_1$, $O_2$, and $O_3$ was erased. Restore it using a compass and a ruler.
  5. The altitude $AA_0$, the median $BB_0$, and the angle bisector $CC_0$ of a triangle $ABC$ are concurrent at point $K$. Given that $A_0K = B_0K$, prove that $C_0K = A_0K$.
  6. Let $\alpha$ be an arc with endpoints $A$ and $B$. A circle $\omega$ is tangent to segment $AB$ at point $T$ and meets $\alpha$ at points $C$ and $D$. The rays $AC$ and $TD$ meet at point $E$, while the rays $BD$ and $TC$ meet at point $F$. Prove that $EF$ and $AB$ are parallel.
  7. In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.
  8. Let $P$ be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$; the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear.

Grade 9

  1. All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$; the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets $\omega$ for the second time at point $D_1$. Prove that $B_1D_1 || AE$.
  2. Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.
  3. Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $$S_{AOB} + S_{COD} \leq 2(S_{AOD} + S_{BOC}).$$
  4. A point $F$ inside a triangle $ABC$ is chosen so that $\widehat{AFB} = \widehat{BFC} = \widehat{CFA}$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1$, $B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.
  5. Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $BK : CK = FS : ES$.
  6. A line $\ell$ passes through the vertex $B$ of a regular triangle $ABC$. A circle $\omega_a$ centered at $I_a$ is tangent to $BC$ at point $A_1$, and is also tangent to the lines $\ell$ and $AC$. A circle $\omega_c$ centered at $I_c$ is tangent to $BA$ at point $C_1$, and is also tangent to the lines $\ell$ and $AC$. Prove that the orthocenter of triangle $A_1BC_1$ lies on the line $I_aI_c$.
  7. Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points $X$ are collinear as $R$ changes.
  8. Three cyclists ride along a circular road with radius $1 km$ counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1 km$?

Grade 10

  1. A circle $k$ passes through the vertices $B$, $C$ of a scalene triangle $ABC$. $k$ meets the extensions of $AB$, $AC$ beyond $B$, $C$ at $P$, $Q$ respectively. Let $A_1$ is the foot the altitude drop from $A$ to $BC$. Suppose $A_1P=A_1Q$. Prove that $\widehat{PA_1Q}=2\widehat{BAC}$.
  2. Let $ABCD$ is a tangential quadrilateral such that $AB=CD>BC$. $AC$ meets $BD$ at $L$. Prove that $\widehat{ALB}$ is acute.
  3. Let $X$ be a point inside triangle $ABC$ such that $$XA.BC=XB.AC=XC.AC.$$ Let $I_1$, $I_2$, $I_3$ be the incenters of $XBC$, $XCA$, $XAB$. Prove that $AI_1$, $BI_2$, $CI_3$ are concurrent.
  4. Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper.
  5. Let ABCD is a cyclic quadrilateral inscribed in $(O)$. $E$, $F$ are the midpoints of arcs $AB$ and $CD$ not containing the other vertices of the quadrilateral. The line passing through $E$, $F$ and parallel to the diagonals of $ABCD$ meet at $E$, $F$, $K$, $L$. Prove that $KL$ passes through $O$.
  6. The altitudes $AA_1$, $BB_1$, $CC_1$ of an acute triangle $ABC$ concur at $H$. The perpendicular lines from $H$ to $B_1C_1$, $A_1C_1$ meet rays $CA$, $CB$ at $P$, $Q$ respectively. Prove that the line from $C$ perpendicular to $A_1B_1$ passes through the midpoint of $PQ$.
  7. In the space, five points are marked. It is known that these points are the centers of five spheres, four of which are pairwise externally tangent, and all these $15$ four are internally tangent to the fifth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fifth (the largest) sphere. Find the ratio of the greatest and the smallest radii of the spheres.
  8. In the plane, two fixed circles are given, one of them lies inside the other one. For an arbitrary point $C$ of the external circle, let $CA$ and $CB$ be two chords of this circle which are tangent to the internal one. Find the locus of the incenters of triangles $ABC$.

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2013 (Final Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2013 (Final Round)
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https://www.molympiad.net/2017/06/sharygin-geometry-olympiad-2013-solutions.html
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