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[Solutions] Sharygin Geometry Mathematical Olympiad 2011 (Final Round)

Grade 8

  1. The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles.
  2. Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle. Let this new rectangle be given. Restore the original rectangle using compass and ruler.
  3. The line passing through vertex $A$ of triangle $ABC$ and parallel to $BC$ meets the circumcircle of $ABC$ for the second time at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the perpendiculars from $A_1$, $B_1$, $C_1$ to $BC$, $CA$, $AB$ respectively concur.
  4. Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.
  5. A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB$, $AC$, $BN$, $CM$ is cyclic.
  6. Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.
  7. Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$- and $X$-axis respectively so that angle $PMQ$ is always right. Find the locus of points symmetric to $M$ wrt $PQ$.
  8. Using only the ruler, divide the side of a square table into $n$ equal parts. All lines drawn must lie on the surface of the table.

Grade 9

  1. Altitudes $AA_1$ and $BB_1$ of triangle $ABC$ meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1$, $B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.
  2. In triangle $ABC$, $\widehat B = 2\widehat C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $$\widehat{CAP} = \widehat{PAQ} = \widehat{QAB} = \frac{\widehat{A}}{3}.$$ Prove that $Q$ is the circumcenter of triangle $CPB$.
  3. Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.
  4. Quadrilateral $ABCD$ is inscribed into a circle with center $O$. The bisectors of its angles form a cyclic quadrilateral with circumcenter $I$, and its external bisectors form a cyclic quadrilateral with circumcenter $J$. Prove that $O$ is the midpoint of $IJ$.
  5. It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?
  6. In triangle $ABC$, $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a$, $M_b$ are defined similarly. Prove that points $M_a$, $M_b$, $M_c$ are collinear and lines $AM_a$, $BM_b$, $CM_c$ are parallel.
  7. Circles $\omega$ and $\Omega$ are inscribed into the same angle. Line $\ell$ meets the sides of angles, $\omega$ and $\Omega$ in points $A$ and $F$, $B$ and $C$, $D$ and $E$ respectively (the order of points on the line is $A$, $B$, $C$, $D$, $E$, $F$). It is known that $BC = DE$. Prove that $AB = EF$.
  8. A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?

Grade 10

  1. In triangle $ABC$ the midpoints of sides $AC$, $BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A$, $B$ and the orthocenter of triangle $ABC$.
  2. Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB$, $BC$, $CD$, $DA$ in points $K$, $L$, $M$, $N$ respectively. Points $A′$, $B′$, $C′$, $D′$ are the midpoints of segments $LM$, $MN$, $NK$, $KL$. Prove that the quadrilateral formed by lines $AA′$, $BB′$, $CC′$, $DD′$ is cyclic
  3. Given two tetrahedrons $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$. Consider six pairs of edges $A_iA_j$ and $B_kB_l$, where $(i, j, k, l)$ is a transposition of numbers $(1, 2, 3, 4)$ (for example $A_1A_2$ and $B_3B_4$). It is known that for all but one such pairs the edges are perpendicular. Prove that the edges in the remaining pair also are perpendicular.
  4. Point $D$ lies on the side $AB$ of triangle $ABC$. The circle inscribed in angle $ADC$ touches internally the circumcircle of triangle $ACD$. Another circle inscribed in angle $BDC$ touches internally the circumcircle of triangle $BCD$. These two circles touch segment $CD$ in the same point $X$. Prove that the perpendicular from $X$ to $AB$ passes through the incenter of triangle $ABC$.
  5. The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.
  6. Prove that for any nonisosceles triangle $l_1^2 > \sqrt{3}S > l_2^2$, where $l_1$, $l_2$ are the greatest and the smallest bisectors of the triangle and $S$ is its area.
  7. Point $O$ is the circumcenter of acute-angled triangle $ABC$, points $A_1$, $B_1$, $C_1$ are the bases of its altitudes. Points $A′$, $B′$, $C′$ lying on lines $OA_1$, $OB_1$, $OC_1$ respectively are such that quadrilaterals $AOBC′$, $BOCA′$, $COAB′$ are cyclic. Prove that the circumcircles of triangles $AA1A′$, $BB1B′$, $CC1C′$ have a common point.
  8. Given a sheet of tin $6 \times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2011 (Final Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2011 (Final Round)
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