## $hide=mobile ### Grade 8 1. In trapezoid$ABCD$angles$A$and$B$are right,$AB = AD$,$CD = BC + AD$,$BC < AD$. Prove that$\widehat{ADC} = 2\widehat{ABE}$, where$E$is the midpoint of segment$D$. 2. A circle passing through$A$,$B$and the orthocenter of triangle$ABC$meets sides$AC$,$BC$at their inner points. Prove that$60^0 < \widehat{C} < 90^0$. 3. In triangle$ABC$we have$AB = BC$,$\widehat{B} = 20^0$. Point$M$on$AC$is such that$AM : MC = 1 : 2$, point$H$is the projection of$C$to$BM$. Find angle$\widehat{AHB}$. 4. Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes. 5. Two equal hard triangles are given. One of their angles is equal to$\alpha$(these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to$\alpha/2$. (No instruments are allowed, even a pencil.) 6. Lines$b$and$c$passing through vertices$B$and$C$of triangle$ABC$are perpendicular to sideline$BC$. The perpendicular bisectors to$AC$and$AB$meet$b$and$c$at points$P$and$Q$respectively. Prove that line$PQ$is perpendicular to median$AM$of triangle$ABC$. 7. Point$M$on side$AB$of quadrilateral$ABCD$is such that quadrilaterals$AMCD$and$BMDC$are circumscribed around circles centered at$O_1$and$O_2$respectively. Line$O_1O_2$cuts an isosceles triangle with vertex$M$from angle$CMD$. Prove that$ABCD$is a cyclic quadrilateral. 8. Points$C_1$,$B_1$on sides$AB$,$AC$respectively of triangle$ABC$are such that$BB_1 \perp CC_1$. Point$X$lying inside the triangle is such that$\widehat{XBC} = \widehat{B_1BA}$,$\widehat{XCB} = \widehat{C_1CA}$. Prove that$\widehat{B_1XC_1} = 90^0 − \widehat{A}$. ### Grade 9 1. Circles$\alpha$and$\beta$pass through point$C$. The tangent to$\alpha$at this point meets$\beta$at point$B$, and the tangent to$\beta$at$C$meets$\alpha$at point$A$so that$A$and$B$are distinct from$C$and angle$ACB$is obtuse. Line$AB$meets$\alpha$and$\beta$for the second time at points$N$and$M$respectively. Prove that$2MN < AB$. 2. A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram. 3. Let$100$discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least$15$of these discs. 4. A fixed triangle$ABC$is given. Point$P$moves on its circumcircle so that segments$BC$and$AP$intersect. Line$AP$divides triangle$BPC$into two triangles with incenters$I_1$and$I_2$. Line$I_1I_2$meets$BC$at point$Z$. Prove that all lines$ZP$pass through a fixed point. 5. Let$BM$be a median of nonisosceles right-angled triangle$ABC$($\widehat B = 90^0$), and$H_a$,$H_c$be the orthocenters of triangles$ABM$,$CBM$respectively. Prove that lines$AH_c$and$CH_a$meet on the medial line of triangle$ABC$. 6. The diagonals of convex quadrilateral$ABCD$are perpendicular. Points$A_0$,$B_0$,$C_0$,$D_0$are the circumcenters of triangles$ABD$,$BCA$,$CDB$,$DAC$respectively. Prove that lines$AA_0$,$BB_0$,$CC_0$,$DD_0$concur. 7. Let$ABC$be an acute-angled, nonisosceles triangle. Altitudes$AA_0$and$BB_0$meet at point$H$, and the medians of triangle$AHB$meet at point$M$. Line$CM$bisects segment$A_0B_0$. Find angle$C$. 8. A perpendicular bisector to side$BC$of triangle$ABC$meets lines$AB$and$AC$at points$A_B$and$A_C$respectively. Let$O_a$be the circumcenter of triangle$AA_BA_C$. Points$O_b$and$O_c$are defined similarly. Prove that the circumcircle of triangle$O_aO_bO_c$touches the circumcircle of the original triangle. ### Grade 10 1. Let$K$be an arbitrary point on side$BC$of triangle$ABC$, and$KN$be a bisector of triangle$AKC$. Lines$BN$and$AK$meet at point$F$, and lines$CF$and$AB$meet at point$D$. Prove that$KD$is a bisector of triangle$AKB$. 2. Prove that an arbitrary triangle with area$1$can be covered by an isosceles triangle with area less than \sqrt{2}. 3. Let$A_1$,$B_1$and$C_1$be the midpoints of sides$BC$,$CA$and$AB$of triangle$ABC$. Points$B_2$and$C_2$are the midpoints of segments$BA_1$and$CA_1$respectively. Point$B_3$is symmetric to$C_1$wrt$B$, and$C_3$is symmetric to$B_1$wrt$C$. Prove that one of common points of circles$BB_2B_3$and$CC_2C_3$lies on the circumcircle of triangle$ABC$. 4. Let$AA_1$,$BB_1$,$CC_1$be the altitudes of an acute-angled, nonisosceles triangle$ABC$, and$A_2$,$B_2$,$C_2$be the touching points of sides$BC$,$CA$,$AB$with the correspondent excircles. It is known that line$B_1C_1$touches the incircle of$ABC$. Prove that$A_1$lies on the circumcircle of$A_2B_2C_2$. 5. Let$BM$be a median of right-angled nonisosceles triangle$ABC$($\widehat B = 90^0$), and$H_a$,$H_c$be the orthocenters of triangles$ABM$,$CBM$respectively. Lines$AH_c$and$CH_a$meet at point$K$. Prove that$\widehat{MBK}$= 90^0$.
6. Let $H$ and $O$ be the orthocenter and the circumcenter of triangle $ABC$. The circumcircle of triangle $AOH$ meets the perpendicular bisector to $BC$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ concur.
7. Let $SABCD$ be an inscribed pyramid, and $AA_1$, $BB_1$, $CC_1$, $DD_1$ be the perpendiculars from $A$, $B$, $C$, $D$ to lines $SC$, $SD$, $SA$, $SB$ respectively. Points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are distinct and lie on a sphere. Prove that points $A_1$, $B_1$, $C_1$ and $D_1$ are complanar.
8. Does there exist a rectangle which can be divided into a regular hexagon with sidelength $1$ and several equal right-angled triangles with legs $1$ and $\sqrt 3$?

ltr
item