[Solutions] Sharygin Geometry Mathematical Olympiad 2014 (Final Round)

Grade 8

1. The incircle of a right-angled triangle $ABC$ touches its catheti $AC$ and $BC$ at points $B_1$ and $A_1$, the hypotenuse touches the incircle at point $C_1$. Lines $C_1A_1$ and $C_1B_1$ meet $CA$ and $CB$ respectively at points $B_0$ and $A_0$. Prove that $AB_0 = BA_0$.
2. Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b || L_aL_b$. Is it necessarily true that $AC = BC$?
3. Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\widehat{MAN} = 15^0$ and $\widehat{BAN} = 45^0$ . Find the value of angle $\widehat{ABM}$.
4. Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)?
5. A triangle with angles of $30$, $70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.)
6. Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$.
7. Two points on a circle are joined by a broken line shorter than the diameter of the circle. Prove that there exists a diameter which does not intersect this broken line.
8. Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.

Grade 9

1.  Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $$(AD − BC)(AB − CD) > 0.$$
2. In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$. Prove that line $XY$ passes through the midpoint of $AC$.
3. An acute angle $A$ and a point $E$ inside it are given. Construct points $B$, $C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$.
4. Let $H$ be the orthocenter of a triangle $ABC$. Given that $H$ lies on the incircle of $ABC$, prove that three circles with centers $A$, $B$, $C$ and radii $AH$, $BH$, $CH$ have a common tangent.
5. In triangle $ABC$ ($\widehat B = 60^0$), $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.
6. Let $I$ be the incenter of triangle $ABC$, and $M$, $N$ be the midpoints of arcs $ABC$ and $BAC$ of its circumcircle. Prove that points $M$, $I$, $N$ are collinear if and only if $AC + BC = 3AB$.
7. Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent.
8. A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$?.

Grade 10

1. The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle.
2. A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and $WY$ meet $AB$ at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.
3. Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)?
4. Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a$, $B_c$, $C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b$, $A_c$, $B_a$, $B_c$, $C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be?
5. The altitude from one vertex of a triangle, the bisector from the another one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.)
6. The incircle of a non-isosceles triangle $ABC$ touches $AB$ at point $C_0$. The circle with diameter $BC_0$ meets the incircle and the bisector of angle $B$ again at points $A_1$ and $A_2$ respectively. The circle with diameter $AC_0$ meets the incircle and the bisector of angle $A$ again at points $B_1$ and $B_2$ respectively. Prove that lines $AB$, $A_1B_1$, $A_2B_2$ concur.
7. Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron.
8. Given is a cyclic quadrilateral $ABCD$. The point $L_a$ lies in the interior of $\Delta BCD$ and is such that its distances to the sides of this triangle are proportional to the lengths of corresponding sides. The points $L_b$, $L_c$, and $L_d$ are defined analogously. Given that the quadrilateral $L_aL_bL_cL_d$ is cyclic, prove that the quadrilateral $ABCD$ has two parallel sides.

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