geometry problems from Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , with aops links in the names
collected inside aops: here
it didn't take place in 2020
2001 - 06, 2010 - 21
2001-06 without grades
Let the circle $\omega$ pass through the ends of the angle bisector $AL$ of the triangle $ABC$ and touch its side $BC$. Prove that the circle $\omega$ is tangent at point A to the circumcircle of triangle $ABC$.
Given a right parallelepiped $ABCDA_1B_1C_1D_1$. Let the points $E$ and $F$ be orthogonal projections of the point $D$ on the lines $AC$ and $A_1C$, respectively, and the points $P$ and $Q$ be orthogonal projections of the point $C_1$ on the lines $B_1D_1$ and $BD_1$, respectively. Prove that the planes $DEF$ and $C_1PQ$ are perpendicular if and only if $A_1B=BC$.
A circle is circumscribed around triangle $ABC$. Let the point $M$ be the midpoint of the side $AC$, $BD$ be the angle bisector of the triangle $ABC$, and point $D$ lie between points $A$ and $M$. From the midpoint $P$ of the arc $AC$ containing point $B$, a perpendicular $PN$ is drawn on the side $BC$. Prove that the segment $ND$ divides the triangle $ABC$ into two equal figures.
On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $K$ are selected, respectively, such that $CM = DK$. Let $P$ be the intersection point of the segments $MD$ and $BK$. Prove that $MK \perp AP$.
A circle is circumscribed around the acute-angled triangle $ABC$. Tangents to this circle, drawn at points $A$ and $C$, intersect the tangent to the circle constructed at point $B$, at points $M$ and $N$, respectively. Let $BD$ be the altitude of triangle $ABC$. Prove that $\angle MDB = \angle NDB$.
The point $M$ is the midpoint of the hypotenuse $AB$ of the right isosceles triangle $ABC$, and $K$ is the point of the leg $BC$ that $BK=2KC$. The point $N$ is chosen on the leg $AC$ such that for the intersection point $L$ of of the segments $AK$ and $MN$ the ray $LC$ is the bisector of the angle $KLN$. Prove that $KN \parallel AB$.
Let $ABC$ be an equilateral triangle. Find all such triplets of the numbers $x, y,z$ for which
$$|\overrightarrow{AB} + y\overrightarrow{BC} + z\overrightarrow{CA} |=|x\overrightarrow{AB} +\overrightarrow{BC} + z\overrightarrow{CA} | = | x\overrightarrow{AB} + y\overrightarrow{BC} + \overrightarrow{CA} |.$$
The line $m$ intersects the sides $AB$ and $AC$ of the triangle$ ABC$ and ray $BC$ at points $L, M$ and $N$, respectively, and $AL < \frac12 AB$. Let $F, K$ and $T$ be the midpoints of the segments $AM$, $AB$, and $MN$, respectively, and the lines $LF$ and $BC$ are parallel. Prove that the quadrilateral $TCKL$ is a parallelogram.
In the right triangle $ABC$ ($\angle C=90^o$) on the sides $AC$, $BC$ and $AB$, the points $P, Q$ and $R$ different from the vertices are chosen, respectively, so that the equality $\angle CBP + \angle CAQ = \angle RPB + \angle AQR$ takes place. Let the points $S$ and $K$ be the feet of the perpendiculars drawn from the points $Q$ and $P$ respectively on the line $CR$. Prove that $CS = RK$.
Let $O$ be the intersection point of the diagonals of the trapezoid $ABCD$, and $AO^2=AB^2 + OC^2$. On its larger base $AD$, such a point $M$ is chosen that $BM = CM$. Prove that the points $O, C, D$ and $M$ lie on the same circle.
Let the segment $AL$ be the angle bisector of the acute triangle $ABC$, the points $H_B$ and $H_C$ be the orthocenters of the triangles $ABL$ and $ACL$, respectively. Prove that the distance between the points $H_B$ and $H_C$ does not exceed $| AB - AC |$.
On the sides of the triangle $ABC$, equilateral triangles $ABC_1$, $ACB_1$ and $BCA_1$ are constructed externally. Let $AB \cap BA_1 = C_2$, $CB_1 \cap BC_1 = A_2$, $CA_1 \cap AC_1 = B_2$, and the points $A_2$, $B_2$ and $C_2$ are different from the points $D$, $B_1$ and $C_1$, respectively. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ are parallel.
The circle inscribed in the quadrilateral $MKPL$ touches its sides $MK$, $KP$, $PL$ and $LM$ at points $F, R, S$ and $Q$, respectively, and $SF \perp QR$. On the diagonal $KL$ of this quadrilateral, such a point $N$ is chosen that $NF \parallel LM$. Prove that $NF = NS$.
Let the point $M$ be the midpoint of $CD$ of the convex quadrilateral $ABCD$. It is known that the lines $BM$ and $AM$ are perpendicular and $AB = BC + AD$. Prove that the lines $BC$ and $AD$ are parallel.
Let the inscribed circle $\omega$ of the triangle $ABC$ touch its side $AC$ at the point $P$. On the sides $AB$ and $CB$ are chosen different points Q and R from the vertices, respectively, such that the line $QR$ is tangent to the circle $\omega$ at a point $M$, and $BQ + QP = BR + RP$. Prove that the circle inscribed in a triangle $PQR$ is tangent to the circle $\omega$.
In the convex quadrilateral $ABCD$ on the sides $AB$ and $CD$, the points $K$ and $N$ are chosen so that $CK = DK$ and $AN = BN$. Let $BN \cap CK = S$, $AN \cap DK = M$, and $KM = KA$, $NM = ND$. The bisectors of the angles $ABN$ and $DCK$ intersect the segments $KD$ and $NA$ at points $Q$ and $R$, respectively. Prove that the points $S, Q$ and $R$ lie on the same line.
Let the $AC$ be chord of the circle $\omega$ , different from the diameter , and let the point $K$ be its midpoint. Chord $BD$ (point $B$ lies on the smaller arc $AC$) intersects the chord $AC$ so that $\angle ABD = \angle KBC$. The ray $DB$ intersects the perpendicular bisector of the chord $AC$ at point $Q$, , which lies outside the circle. Prove that the line $AQ$ is tangent to the circle $\omega$.
Given a triangle $ABC$, in which $AC = BC$. On its sides $AC$ and $BC$, mark points $D$ and $F$, respectively, so that $AD = FD$ and $CD \le CF$. Let the point $M$ be the midpoint of the segment $BF$, and let $E$ be the intersection point of the lines $AB$ and $DM$. Prove that $\angle BEF = \angle DAF$.
The segment $AL$ is the angle bisector of the triangle $ABC$. Tangent to the circumcircle of triangle $ABL$ drawn through point $B$ intersects at point $K$ the tangent to the circumcircle of triangle $ACL$ drawn through point $C$. Prove that the points $A, L$ and $K$ lie on the same line.
Let the point $E$ be the midpoint of the larger base $AD$ of the trapezoid $ABCD$. On the diagonals $AC$ and $BD$, the points $Q$ and $F$ are selected, respectively and it is known that the lines $AF$, $EQ$ and $CD$ intersect at one point. Prove that the lines $AB$, $EF$ and $DQ$ also intersect at one point.
Given a convex quadrilateral $ABCD$, in which $AB = BC =CD$. Let $M$ be the intersection point of the bisector of the external angle at the vertex $B$ with the line $CD$, and let $N$ be the intersection point of the bisector of the external angle at the vertex $C$ with the line $AB$. Prove that $MN \parallel AD$.
The triangle is located inside the square so that the center of the square lies outside this triangle, and there is no vertex of the triangle on the sides of the square. Prove that the length of one of the sides of the triangle is less than the length of the side of the square.
Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. It is known that $BO = OD =BC$ and $\angle BAC = 30^o$. Let the points $M$ and $N$ be the midpoints of the segments $OA$ and $OB$, respectively, and on the side $CD$ a point $P$ be chosen such that $PM = PN$. Prove that $\angle MPN = 60^o$.
A point $N$ is chosen inside the triangle $ABC$ on its median $BM$. Let $K = AN \cap BC$, $L = CN \cap AB$. It is known that a circle can be circumscribed around the quadrilateral $BKNL$. Prove that the line $AC$ is a common tangent to the circles circumscribed around the triangles $ABN$ and $CBN$.
Let the point $O$ be the center of the circumcircle of the acute-angled triangle $ABC$, the segment $BD$ be the altitude of this triangle, $L AO \cap BD$ (the points $L$ and $O$ may coincide), and $K = AO \cap BC $. A line passing through the point $L$ perpendicular to the line $AO$ intersects a circle $\omega$ with center at point $K$ and radius $R=BK$ at points $P$ and $Q$. Prove that the circle $\omega$ is tangent to the sides of the angle $\angle PAQ$.
Inside the convex quadrilateral $ABCD$, a point $F$ is marked such that $\angle FAB = \angle FDC$ and $\angle FBA = \angle FCD$. Let $FM$ and $FN$ be the altitudes of the triangles $FAB$ and $FCD$, respectively. Prove that the line passing through the midpoints of the sides $BC$ and $AD$ bisects the segment $MN$.
On the sides $AB$ and $CD$ of the isosceles trapezoid $ABCD$ are chosen points $N$ and $M$, respectively, different from the vertices, such that $AN = CM$. Let $K= AC \cap NM$, $F = BD \cap NM$. Prove that $NK = FM$.
Prove that a square cannot be cut into several convex hexagons.
Let the point $O$ be the center of the circumcircle of the acute-angled triangle $ABC$, and let the segment $CK$ be the altitude of this triangle. On the sides $AC$ and $BC$, the following points $P$ and $N$ are marked, respectively, that $KP \perp AC$ and $KN \perp BC$. The altitude $PL$ of the triangle $PNC$ intersects the line $OC$ at the point $M$. Prove that the quadrilateral $KNMP$ is a parallelogram.
On the side $CD$ of the right trapezoid $ABCD$ ($\angle A= 90^o$) such a point $M$ is chosen that $\angle AMB <90^o$. The segments $BP$ and $AQ$ are the altitudes of the triangle $BMA$, and $AD = DQ$. Let the point $W$, which lies inside the triangle $PMQ$, be the intersection point of the segments $CP$ and $DQ$. Prove that $W$ is the center of the circumcircle of triangle $PMQ$.
Let $I$ be the center of the inscribed circle of triangle $ABC$. The line $\ell$ passing through point $I $perpendicular to line $AI$ intersects the sides $AB$ and $AC$ at points $F$ and $E$, respectively. Let $M$ be the midpoint of the segment $FI$, $N$ be the midpoint of the segment $EI$, $P$ be the intersection point of the lines $CF$ and $BM$, $Q$ be the intersection of of the lines $BE$ and $CN$. Prove that the points $B, P, Q, C$ lie on the same circle.
Given a triangle $ABC$, in which $AB \ne AC$. On its sides $AB$ and $AC$, are marked points $M$ and $N$, respectively, so that $BM = CN$. The circumcircle of triangle $AMN$ intersects the circumcircle of triangle $ABC$ at point $D$ other than $A$. Prove that $DM = DN$.
On the sides $AB$ and $BC$ of the triangle $ABC$, points $N$ and $M$ are marked, respectively, such that $AN = NM = MC$. Let $Q$ be the intersection point of the segments $AM$ and $CN$. Prove that the center of the inscribed circle of triangle $ABC$ lies on the circumcircle of triangle $ACQ$.
2010-19 grades VII-VIII
In triangle $ABC$, the internal bisectors of angles $A$ and $C$ were drawn. Points $P$ and $Q$ are the feet of the perpendiculars drawn from the vertex $B$ onto these bisectors. Prove that $PQ \parallel AC$.
In triangle $ABC$, the bisector of angle $A$ is drawn and and has the second intersection point $W$ with the circumcircle . $O$ is the center of the circumcircle. It is known that $OW = CW = AC$. Find all the angles of triangle $ABC$.
$A$ and $B$ are arbitrary points on the sides of a right angle with apex $O$. Point $C$ lies inside the corner $AOB$. Prove that the perimeter of triangle $ABC$ is greater than $2OC$.
$AL$ is the bisector in triangle $ABC$. Circles $\omega_1$ and $\omega_2$ are circumscribed around triangles $ABL$ and $ACL$, respectively. The external bisector of angle $B$ intersects $\omega_1$ at point $K$. The external bisector of angle $C$ intersects $\omega_2$ at point $N$. Find the angle between lines $AL$ and $KN$.
Reconstruct triangle $ABC$ given vertex $C$, the point $N$ of intersection of the straight line containing the altitude ${{h} _ {a}} $ with the circle circumscribed about $\Delta ABC$, as well as the straight line $OH$ passing through the intersection point $H$ of altitudes of $\Delta ABC$ and the center $O$ of the circumscribed circle around it.
(V. Satko)
In triangle $ABC$, the angle $A$ is $60 {} ^ \circ$. $BL$ and $CN$ angle are bisectors in this triangle. The perpendicular bisector of $LN$ meets $BC$ at point $T$. Prove that triangle $LNT$ is equilateral.
(A.Karlyuchenko)
Side $BC$ of equilateral triangle $ABC$ was divided by points $K$ and $H$ into three equal parts. Point $M_1$ on the $AC$ side is such that $AM_1: M_1C = 1: 2$. Find the sum of the angles $AKM_1$ and $AHM_1$.
In a five-pointed star, the bisectors of its four angles pass through one point. Prove that the fifth angle bisector also passes through the same point.
(Plotnikov M.)
Points $B$ and $C$ are given on a semicircle with diameter $AD$. Point $A$ was connected to the midpoint $P$ of segment $BC$, and then we doubled the constructed segment beyond point $P$ to obtain point $Q$. Prove that $C$ is the orthocenter of triangle $BQD$.
(Revako O.)
In the triangle $ABC$, an arbitrary point $X$ is chosen. Prove that the intersection point of the bisector of $\angle A $, with the bisector of $\angle ACX$, as well as the intersection point of the bisectors of $\angle BXC$ with $\angle ABX$, and the intersection point of $CX$ with side $AB$ are collinear.
In the square $ABCD$ on the sides $BC$ and $CD$ select the points $K$ and $N$ respectively so that $\angle KAN = 45 {} ^ \circ$. The intersection points of the diagonal of the square $BD$ and the segments $AK$ and $AN$ are $P$ and $T$, respectively. Prove that the area of $KANC$ is equal to twice the sum of the areas of the triangles $ABP$ and $ATD$.
(M. N. Rozhkova)
The point $X$ slides along the diameter of the semicircle. From this point the rays $XY$ and $XZ$ are drawn at an angle $\alpha$ to the diameter of the semicircle. Prove that all circles circumscribed about triangles $XYZ$ have a common point.
(A. A. Shamovich)
Given an isosceles triangle $ABC$ in which $\angle A=120^o$. Points $K$ and $N$ divides the base $BC$ into three equal parts: $BK=KN=NC$. Prove that the triangle $AKN$ is equilateral.
(M. Plotnikov - E. Diomidov)
A square $ADKB$ is built on the side $AB$ of an equilateral triangle $ABC$ to the outside. Let $X$ be the intersection point of $CD$ and $AK$. Prove that $CX=XK$.
(V. Podkhalyuzin)
Given a rectangular plaque $ABCD$. The points $E$ and $F$ are taken on the sides $BC$ and $AD$ respectively. Using only a ruler without divisions, determine which of the segments $AE$ , or $CF$ is larger ? (it is not allowed to make notches on the ruler)
(M. N. Rozhkova).
Given an acute-angled triangle $ABC$ with orthocenter $H$. Point $L$ is the foot of the bisector of angle $A$. Point $N$ is the middle of the arc $BHC$ of the circumcircle $\omega$ of triangle $BHC$ . $Q$ is the intersection point of lines $NL$ and $AH$. Prove that point $Q$ lies on circle $\omega$.
(A.V. Karlyuchenko)
On the sides $AC$ and $AB$ of triangle $ABC$, points $K$ and $N$ are selected, respectively, such that $CK = KN = NB$. Prove that $\angle L_2BK = \angle NCL_3$, where $L_2$ and $L_3$ are the feet of the bisectors of angles $B$ and $C$ respectively.
(A. Grishchenko)
Two circles $\omega_1$ and $\omega_2$ are given, tangent externally. They have a common tangent $AB$ (point $A \in \omega_1$, point $B \in \omega_2$). Circle with center $B$ and radius $BA$ intersects $\omega_2$ at points $D$ and $E$. Prove that the line $DE$ touches $\omega_1$.
(V. Podkhalyuzin)
On sides $AB$ and $BC$ of square $ABCD$, construct the equilateral triangles $AQB$ and $BTC$ on the outside. The entire drawing except for the points $Q$ and $T$ was erased. Restore the first initial square.
(A. Medvedev)
For a triangle $ABC$, it is known that $\angle ABC = 2\angle ACB$. Prove that $BI = AC - AB$, where $I$ is the incenter of triangle $ABC$.
(M. Plotnikov)
In the tangential pentagon $ABCDE$, the sides $BC, CD$ and $DE$ are equal. Let $K$ be the touch point the inscribed circle of the given pentagon with side $CD$, and $O$ is the center of this circle. Prove that points $A, O, K$ are collinear.
(M. Plotnikov)
In triangle $ABC$, the bisector of angle $A$ is drawn up to the intersection with the side $BC$ at the point $L$. Points $E$ and $F$ on sides $AC$ and $AB$ are taken respectively so that $EF \parallel BC$ and $CE + FB =BC$. A circle is drawn through the points $E, L, F$, which intersects $AL$ at the point $Q$. Prove that the point $Q$ is the incenter of the triangle $AEF$.
(A. Nikolaev,O. Revako)
In triangle $ABC$, point $N$ is the midpoint of the arc $BAC$ of the circumscribed circle of triangle $ABV$ , $I$ is the center of the inscribed circle of triangle $ABC$, $M$ is the midpoint of the side $BC$. Prove that the angle $BIM$ is equal to the angle formed by straight lines $NI$ and $CI$.
(M.Plotnikov)
In triangle $ABC$, we doubled the median $CM_3$ and obtained point $T$. Points $H_1, H_2$ are the feet of the altitudes drawn from the vertices $A, B$, respectively, $H$ is the orthocenter of triangle $ABC$. Prove that lines $TH$ and $H_1H_2$ are perpendicular.
(A.Karlyuchenko)
Let $K$ be the point of tangency of the circle inscribed in the triangle $ABC$ to the side $AC$. A perpendicular is drawn from the point $K$ ον the side $BC$, which intersects the bisector of the angle $DIA$ at the point $E$. Prove that $KI = KE$, where $I$ is the center of the triangle $ABC$.
(O. Karlyuchenko)
Let $H$ be the foot of the altitude drawn from the vertex $A$ on the side $BC$ of the triangle$ ABC$. The point $L$ is the foot of the bisector of $\angle AHC$ in the corresponding triangle, and $M$ is the midpoint of the segment $AB$. Reconstruct the triangle $ABC$ given the points $H, L$ and $M$.
(O. Grishchenko)
Reconstruct the triangle ABC given the line $a$, which contains the side $BC$ of the triangle, and the points $W$ and $D$. The point $W$ is the intersection point of the line containing the bisector of $\angle BAC$, with the circumscribed circle around triangle$ ABC$ , and $D$ is diametrically opposite the vertex $A$, point of the circumscribed circle.
(S. Yakovlev)
In the isosceles triangle $ABC$ ($AB = AC$), the points $M$ and $N$ are the feet of the medians and altitudes drawn from the vertices $C$ and $B$, respectively, and $O$ is the center of the circle circumscribed around the triangle $AMN$. $H$ is the midpoint of BC. It turned out that $OH$ is the bisector of the angle $BON$. Find the angles of triangle $ABC$.
(M. Plotnikov)
For point $B$ on the segment $AC$ there are points $D$ and $E$ such that $AD = BD = BC$ and $AB = BE = EC$. The bisector of the angle $DBE$ intersects $DE$ at the point $F$. Prove that $AF = FC$.
(M. Plotnikov)
The points $A$ and $B$ are given on the plane. A segment $AB$ and any arc with ends in $A$ and $B$, less than $180^o$, are drawn. Inside the obtained segment, an arbitrary point $K$ was chosen. Construct a line through the point $K$, which intersects the arc at the point $X$, and the segment at the point $Y$ so that $XY = YB$.
(E. Diomidov)
$AL$ and $BF$ are angle bisectors in a right triangle $ABC$ ($AB$ is hypotenuse). $LK$ and $FN$ are perpendiculars from points $L$ and $F$ on $AB$. Find the length of the segment $KN$ if it is known that the radius of the inscribed circle of the triangle $ABC$ is equal to $r$.
(A. Kornienko)
The two circles have no common points and are outside each other. $VW$ and $TU$ are their external tangents, and $SP$ is internal tangent (points $S$ and $P$ lie on the segments $VW$ and $TU$ respectively). Prove that $VW = TU = SP$.
(Sangaku, Japanese temple geometry)
The cheerful hedgehog has $8$ angles (not necessarily equal): $1,2,3,4, 5, \alpha, \beta, \phi$ (see figure). Prove that$$\angle 1 + \angle 2 + \angle 3 +\angle 4 + \angle 5 = \alpha+ \beta+ \phi.$$
In the triangle $ABC$, the inscribed circle touches the sides $AB, BC$ and $AC$ at points $M, N$ and $K$ respectively. The points $H$ and $F$ are selected on the side $BC$ in such a way that the $AN$ is parallel to $MN$ and $AF$ is parallel to $KN$. $NK$ intersects $AN$ at point $E$, $AF$ intersects $MN$ at point $D$. Prove that the segment $ED$ is parallel to $BC$.
In the triangle $ABC$, $H$ is the intersection point of altitudes , and $W$ is the intersection point of the extension of the bisector of angle $A$ with the circumscribed circle. It turned out that $AH = HW$. Find the angle $BAC$.
(T. Batsenko)
In the triangle $ABC$, a line $AS$ is drawn, symmetric to the median with respect to the bisector of the angle $BAC$ (point $S$ lies on the side $BC$). $\omega$, $\omega_1$ and $\omega_2$ are the circles circumcribed around the triangles $ABC$, $BAS$ and $CAS$ respectively. Tangent to $\omega$ at point $A$ intersects $\omega_1$ and $\omega_2$ at points $P$ and $Q$ respectively. Prove that $PA = QA$.
(M. Plotnikov)
Reconstruct the acute triangle $ABC$ at angle $A$ and segments $H_2H_3$ and $BH_3$, where the points $H_2$ and $H_3$ are the feet of the altitudes of the triangle, which are drawn from the vertices $B$ and $C$, respectively.
(V. Pavlyuk)
Equilateral triangles $LAB$ and $LCD$ are constructed on the sides of the angle $KLM$ in such a way that the rays $LB$ and $LD$ lie inside the angle $ALC$. Prove that $AD = BC$.
(M. Plotnikov).
In an equilateral triangle $ABC$, the point $M$ is the midpoint of the side $AB$. On the ray $CM$ , select point $D$ is such that $CD > 2CM$. On the extension of the altitude drawn on the side $AC$ of the triangle $ABC$, the point $E$ is taken so that $\angle ADC =\angle ADE$. Find the value of the angle $DAE$.
(D. Ershov)
Let's call a "mermaid" a device that builds the geometric location of points equidistant from the point and this line. Construct the bisector of the given angle $ABC$ with an inaccessible vertex using a "mermaid" and a ruler.
(T. Timoshkevich)
Inside a circle, a point $K$ is selected that does not belong to it's chord $AB$. Construct a point $X$ on the given circle such that $\angle KXB =\angle XBA$.
(E. Diomidov.)
The two hypotenuses $DB$ and $CA$ of right triangles $DAB$ and $CBA$ intersect at point $X$. From point $X$ we draw the perpendicular on $AB$ and obtain point $H$. Prove that $CH$ is the bisector of the angle $DIA$ if $AD = AC$.
(O. Karlyuchenko)
An arbitrary point $X$ lies inside the triangle $ABC$. The bisectors of the angles $BAC$ and $ACX$ intersect at the point $T$, and the lines that contain the bisectors of the angles $BXC$ and $XBQ$ intersect at the point $N$. Let $Q$ be the intersection point of the ray $CX$ with the side $AB$. Prove that the points $T, N, Q$ belong to one line.
(Alexey Karlyuchenko)
Point $W$ is the point of intersection of the bisector of bisector of the angle $A$ of the triangle $ABC$ and the circle circumscribed around it. Let $M_2, M_3$ be midpoints of sides $AC, AB$ respectively. On the segment $AW$ are marked points $D$ and $E$ such that $M_2E$ and $M_3D$ are perpendiculars bisectors of the sides $AC$ and $AB$ respectively. Prove that $AD = EW$.
(Oleg Cherkasky)
In a right triangle $ABC$, $\angle A = 60^o$. Point $N$ is the midpoint of the hypotenuse $AB$. The radius of the circle circumscribed around the triangle $ANC$ is $R$. Find the length of the ecathetus $BC$.
(Alexey Pakhomov)
A circle is circumscribed around triangle $ABC$. Inside the triangle , a point $X$ is taken, such that the lines $AX$ and $CX$ intersect the opposite sides of the triangle $ABC$ and the circumscribed circle at points $A_1$ and $A_2$, $C_1$ and $C_2$ respectively and $XA_1 = A_1A_2$, $XC_1 = C_1C_2$. Prove that the point $X$ coincides with the orthocenter of triangle $ABC$.
(O. Cherkasky)
In the triangle $ABC$ we draw a line $AT$ such that $\angle CAT = \angle ABT$. $AT$ intersects the circumscribed circle at point $D$. On the segment $AD$ we took the point$ K$ such that $\angle BAT = \angle ACK$. Prove that $AT = KD$.
The triangle $\vartriangle ABC$ is given on the plane. The circle $k$, with the cent at the point $K$, passes through the points $B$ and $C$ and intersects the sides $AB$ and $AC$ at the points $B`$ and $C`$, respectively. Let $M$ and $N$ be diametrically opposite points to point $A$ in the circles circumscribed around the triangles $\vartriangle ABC$ and $\vartriangle AB`C`$, respectively. Prove that $K$ is the midpoint of the segment $MN$.
Seventh-grader Petryk played in the sand on the beach. He first drew an equilateral triangle $ABC$. Subsequently he marked points $D$ and $E$ on the sides $AB$ and $BC$, respectively, using small shells so that they divide the sides to which they belong, in a ratio of $2: 1$ and $1: 2$, counting from the vertex $B$. An unexpected wave washed the triangle out of the sand, but the shells remained in place. Help Petrik to reconstruct the triangle $ABC$ by performing geometric constructions with the help of a compass and a ruler.
(G. Filippovsky)
In an equilateral triangle $ABC$ through the point $A_1$ on the side $BC$ are drawn lines, parallel to the other two sides of the triangle, which intersect $AC$ and $AB$ at points $B_1$ and $C_1$. Through point $B_2$ on the side $AC$ are drawn lines parallel to the other two sides, which intersects $BC$ and $AB$ at points $A_2$ and $C_2$. Prove that the centers of the circumscribed circles triangles $A_1B_1C_1$ and $A_2B_2C_2$ coincide.
(Jury of the Rusanov Olympiad)
In triangle $ABC$, point $I$ is the incenter . It turned out that the radii of the circumscribed circles $ABC$ and $BIC$ are equal. Find the angle $BAC$.
(O. Shamovich)
$AD$ is the part of the diameter of the circumscribed circle $ABC$. The points $P$ and $Q$ are located on the sides $AC$ and $AB$, respectively, so that $PD = CD$, $QD = BD$. Prove that $PQ\parallel BC$ .
(O. Cherkassky)
Let lines $\ell_1,\ell_2,\ell_3$ be parallel. The straight line $\ell$ intersects $\ell_1,\ell_2,\ell_3$ at points $A,B,C$ respectively. The points $E$ and $D$ lies on the lines $\ell_1,\ell_3$ respectively such that $BE = BC$, $AB = DB$. Prove that the midpoint of the angle bisector $BL$ of the $BDE$ triangle lies at the same distance from the straight lines $\ell_1$ and $\ell_3$.
(M. Plotnikov)
Captain Flint hid the tresures on the vertices $B,C,D$ on the convex $ ABCD$. On the treasure map are marked the point A and the three point of intersection of the angles bisectors of the angles $B, C, D$. Will Jim Gokins, who was able to use the map, be able to find these treasures?
(W. Bryman)
Opposite sides of a convex hexagon are parallel in pairs. Four of them are equal to $10$ cm, the fifth is equal to $11$ cm. Find the length of the sixth side.
(M. Hasin)
Construct a triangle $ABC$ with vertices $C$ and $B$ and an arbitrary point $K$ on side $AC$, if it is known that the medians $BF$ and $CN$ are perpendicular.
(G. Filippovsky)
The sides of one isosceles triangle are equal to $a, a, b$, and another's $b, b, a$, where $a > b$. The angle at the apex of the first triangle is equal to the external angle at the apex of the second. Find this angle.
(S. Yakovlev)
In the triangle $ABC$ denote the midpoint of the segment between the vertex $A$ and the orthocenter of the triangle $ABC$ and the midpoint of the segment between the feet of the altitudes drawn from the vertices $B$ and $C$. Prove that these two points and the foot of the median drawn from the vertex $A$ lie on the same line.
In an acute-angled triangle $ABC$, the radius of the circumcircle is $R$. The median $AM_1$ is doubled (beyond the point $M_1$) to obtain the point $F$. Find the length of the segment $FH$, where $H$ is the orthocenter of the triangle $ABC$.
(G. Filippovsky)
In the triangle $ABC$, from the incenter $I$ draw a perpendicular on the side $AB$, which intersects it at the point $K$. From the foot of the bisector of the angle $A$ draw another perpendicular to the same side $AB$, which intersects it at the point $T$. Prove that the segment $IT$ is divisible by line $KW$ in half, where $W$ is the point of intersection of the extension of the bisector of the angle $A$ with the circumcircle of the triangle $ABC$.
(D. Basov)
An isosceles right triangle $ABC$ ($\angle C=90^o$) is given. Inside the triangle on the perpendicular bisector of $AC$ is taken a point K such that the angle $ABK$ is equal to $15^o$. Find the measure of angle B$KC$.
(O. Shamovich)
Given an angle equal to $138^o$ and a template of angle $7^o$. Using this template and drawing no more than $5$ lines, construct a bisector of angle $138^o$.
(Yu. Rabinovych)
It is known that in the quadrilateral $ABCD$ (AB>AD), $AB \parallel CD$, $BC\parallel AD$. Point $E$ lies on the side$ AB$ such that $AE=AD$ . In the extension of DA beyond point A, we take a point F such that $BF \perp DE$. Prove that the angles $AEF$ and $CAD$ are equal.
(M. Plotnikov)
Let $BH_2$ and $CH_3$ be the altitudes of the acute triangle $ABC$, in which $AB <AC$. The line $H_2H_3$ intersects the extension of the side $BC$ at the point $D$. The segment $AD$ intersects the circle circumscribed around the triangle $AH_2H_3$ at the point $F$. Prove that a circle can be circumscribed around the quadrilateral $FH_3BD$.
(G. Filippovsky)
In the triangle $ABC$, the point $W$ is the midpoint of the arc $BC$ of the circumscribed circle, $WF$ and $WE$ are the angle bisectors of the triangles $ACW$ and $ABW$, respectively. The line $FE$ intersects the rays $WC$ and $WB$ at points $P$ and $Q$, respectively. Prove that $WA = WP = WQ$.
(M. Kursky)
Given an acute triangle $ABC$ with an angle $\angle A = 60^o$. Altitudes drawn from vertices $B$ and $C$, intersect the bisector of the angle $A$ at points $T$ and $Q$, respectively. The bisector of angle $A$ intersects the circumcribed around the triangle ABC circle at point $D$. Prove that $AT = QD$.
(A. Brovchenko)
2010-19 team grades IX-X (not these grades in 2021)
You are given a ball, a sheet of paper, a ruler and a compass, with which you can draw circles both on the sheet and on the ball. Plot the radius of the ball.
The chords $AB$ and $AC$ are drawn in the circle $\omega$. The circle $\omega_1$ touches the circle $\omega$, as well as the chords $AB$ and $AC$ at points $D$ and $E$, respectively. The circle $\omega_2$ touches the circle $\omega_1$ internally, and the circle $\omega_1$ touches the circle externally at point $D$. The circle $\omega_3$ touches the circle $\omega_1$ internally, and the circle $\omega_1$ touches the externally at point $E$. Prove that the circles $\omega_1$, $\omega_2$, $\omega_3$ have a common external tangent line.
(O. Karlyuchenko)
Arbitrary points $D$ and $E$ are taken on the sides $AB$ and $AC$ of triangle $ABC$, respectively. The line passing through $D$ and parallel to $AC$, and the line passing through $E$ and parallel to $AB$, intersect at point $G$. Find all the points $G$ obtained in this way, for which the equality also holds:$$\sqrt {{{S} _ {BDG }}} + \sqrt {{{S} _ {CEG}}} = \sqrt {{{S} _ {ABC}}} $$
(S. Yakovlev)
Line $\ell$ is the perpendicular bisector of the bisector $AL_1$ of triangle $ABC$. The external and internal bisectors of angle $B$ intersect $\ell$ at points $B_1$ and $B_2$, and the external and internal bisectors of angle $C$ intersect at points $C_1$ and $C_2$. Prove that $\angle {{B} _ {1}} A {{B} _ {2}} = \angle {{C} _ {1}} A {{C} _ {2}} $
(A. Karlyuchenko)
Reconstruct the triangle $ABC$ given the angle $A$ (given by the rays $AB$ and $AC$) and the straight line passing through the midpoint of the side $BC$ and the incenter of the triangle $ABC$.
(Kushnir I.A.)
Let $h_1, h_2, h_3$ be the distances from point $X$ to the sides of triangle $ABC$. Find the locus of points $X$ inside the triangle $ABC$, for which a triangle can be formed from the segments $h_1, h_2, h_3$.
(Karlyuchenko A.V.)
The cyclic $n$-gon is divided by disjoint diagonals into ($n - 2$) triangles. Let ${{r} _ {1}}$, ${{r} _ {2}}$ , $...$, ${{r} _ {n-2}} $ be the radii inscribed in these triangles are circles. Prove that the sum $\left ({{r} _ {1}} + {{r} _ {2}} + \ldots + {{r} _ {n-2}} \right)$ does not depend on the way of dividing the $n$-gon into triangles.
(Sangaku)
Through the midpoint of the chord $AB$ of the circle $\omega$, draw the chord $CD$. The tangents to the circle, drawn at points $A$ and $B$, meet at point $E$. Prove that the angles $AEC$ and $BED$ are equal.
(Karlyuchenko OA)
At the height $AH_1$ of triangle $ABC$, there was such a point $K$ that the sum of the distances from it to sides $AB$ and $AC$ turned out to be equal to $AK$. Prove that in triangle $ABC$ the sum of the altitudes drawn on sides $AB$ and $AC$ is equal to the sum of the diameters of the inscribed and circumscribed circles of the triangle ABC.
(A.V. Karlyuchenko)
From vertex $A$ of triangle $ABC$, straight lines $\ell$ and $k$ are drawn, symmetric wrt bisectors of angle $A$. Let $K, L, M, N$ be the feet of the perpendiculars drawn on $\ell$ and $k$ from vertices $B$ and $C$. Prove that points $K, L, M, N$ lie on the same circle with center on $BC$.
(A.V. Karlyuchenko)
In a non-isosceles triangle, three different points were found such that the sums the distances from them to the sides of the triangle are respectively equal. Prove that these three points are collinear.
(A. Potapenko)
An acute-angled triangle $ABC$ and its Euler circle are given. Two different circles are drawn through vertices $A$ and $B$, tangent to Euler's circle at points $C_1$ and $C_2$. Points $B_1, B_2, A_1, A_2$ are obtained by similar constructions. Prove that three straight lines $C_1C_2$, $B_1B_2$, $A_1A_2$ intersect at one point belonging to the Euler line.
(D. Soskin)
A square with side $1$ is cut into one hundred rectangles of the same perimeter $p$. What is the largest possible value of $p$?
Let $O$ be the center of the circumscribed circle of triangle $ABC$. Line $AO$ and $BO$ intersect sides $BC$ and $AC$ at points $A_1$ and $B_1$, and intersect the circumcircle at points $A_2$ and $B_2$ respectively. Lines $A_1B_2$ and $A_2B_1$ meet at point $H$. Prove that $CH \perp AB$.
(B.Podkhalyuzin)
In triangle $ABC$ , altitudes $AH_1$, $BH_2$,$CH_3$ are drawn. On ray $H_3H_1$ behind point $H_1$, some point $D$ is selected. The circumscribed circle of the triangle $DH_1C$ intersects the segment $AC$ at point $E$. Prove that the center of the circumcircle of triangle $DEH_2$ lies on altitude $AH_1$.
(D. Khilko)
Given a non-isosceles triangle $ABC$. Perpendicular bisector on the side $BC$ intersects lines $AC$ and $AB$ at points $B_A$ and $C_A$ respectively; similarly defined are points $A_B$, $C_B$ and $B_C$, $A_C$. Prove that the circumcircles of triangles $AA_B A_C$, $BB_A B_C$ and $CC_AC_B$ intersect at one point that lies on the circumcircle of triangle $ABC$.
(M. Plotnikov)
Restore the triangle $ABC$ , given the vertex A, the point $M_1$, the midpoint of the segment $BC$, and the point $K_1$ the touchpoint of the inscribed circle with the side $BC$.
(O. Karlyuchenko)
On the line passing through the center of the circle $O$, fixed points $P$ and $Q$ are fixed. For an arbitrary diameter of the circle $AB$, the point $X$ is the intersection of the lines $AP$ and $BQ$. Find the locus of the points $X$.
(E. Diomidov, V. Kalashnikov)
In the tetrahedron $ABCD$ the plane angles $ADB$, $BDC$ and $CDA$ are equal to each other. The points $A_1$, $B_1$ and $C_1$, are selected on sides $BC, AC$ and $AB$ respectively, that triangles $DAA_1$, $DBB_1$ and $DCC_1$ have the smallest possible perimeters. Prove that lines $AA_1$, $BB_1$ and $CC_1$ intersect at one point.
(M. Plotnikov)
In the triangle $ABC$ denote $I$ the center of the inscribed circle, $M_1$ the midpoint of the side $BC$, $T_1$ is the point of contact of the $A$-exscribed circle with the side $BC$, $N$ is the midpoint of the arc $BAC$ of the circumscribed circle triangle. Line $IM_1$ intersects for the second time the circle circumscribed around the triangle $BIC$, at the point $K$. Prove that line $ KT_1$ divides the segment $NM_1$ in half.
(D. Hilko)
Inside the circle is a circle of twice smaller radius. All possible chords AB of the larger circle touching the smaller one are drawn. For each of them the midpoint C of the smaller arc AB is taken. Let X be a point symmetric to C wrt the line AB. Find the locus of points X.
(E. Diomidov).
In an arbitrary triangle $ABC$, point $M_1$ is the midpoint of the side $BC$, point $L_1$ is the foot of the angle bisector from vertex $A$, point $W$ is the intersection point of the bisector $AL_1$ with the circumscribed circle, points $J_b$ and $J_c$ are the centers of exscribed circles tangent to sides $b$ and $c$, respectively. Prove that
a) points $J_b, J_c, L_1, M_1$ lie on one circle;
b) $L_1$ is the orthocenter of the triangle $J_cWJ_b$.
(D. Basov, Y. Biletsky)
The altitudes $BB_1$ and $CC_1$ are drawn in the acute-angled triangle $ABC$. Point $P$ moves on the circumscribed circle around the triangle $AB_1C_1$ . Lines $PB_1$ and $PC_1$ intersect $BC$ at points $X$ and $Y$, respectively. Prove that all triangles $AXY$ have a fixed orthocenter.
(D. Hilko)
Prove that $EF, BC$ and $KL$ intersect at one point if and only if $BAE$ and $CAF$ are isogonals of the angle $BAC$.
(M. Plotnikov)
Let $ABC$ be an acute-angled triangle, $AH$ and $BF$ be its altitudes, and let $S$ be the circumscribed circle of quadrilateral $ABHF$, and $S_1$ be a circle that touches the altitudes of $AH$ and $BF$, as well as the arc $AB$ of the circle $S$, which does not contain points $H$ and $F$. Let $M$ and $N$ be the points of tangency of the circle $S_1$ with the altitudes $AH$ and $BF$, respectively. Using a compass and a ruler, construct the triangle $ABC$, given the vertices $A$ and $B$ and the line $MN$.
(D. Soskin)
Let $\omega$ be the circumcircle of the acute-angled triangle $ABC$. The bisector of angle $A$ intersects for the second time $\omega$ at the point $W, AD$ is the diameter $\omega$. Let $H'$ be a point symmetric to the orthocenter $H$ of the triangle $ABC$ wrt the line $AW$. Prove that the center of the circumcircle of the triangle $WDH'$ belongs to the line $BC$.
(D. Hilko)
Let $M_1$ and $M_3$ be the midpoints of the sides $BC$ and $AB$ of the equilateral triangle $ABC$, respectively. An arbitrary point $X$ is chosen on the line $AM_1$, and a point $Y$ is chosen so that $XB$ is a bisector of angle $YXA$. Let the lines $XY$ and $CM_3$ intersect at the point $Z$. Find the angle $ZBX$.
(D. Ershov)
Given a parallelogram $ABCD$ with center at the point $O$. The circle $ABO$ intersects the sides $BC$ and $DA$ at points $P$ and $Q$, respectively. The circles $PCO$ and $QDO$ intersect for the second time at the point $K$. Prove that the orthocenter of the triangle $KPQ$ lies on the perpendicular drawn from the point $O$ on the line $BC$.
(M. Plotnikov)
The country of Flatland has the shape of a square with a side of $100$ km, in which it is located seven big cities. The Government of Flatland has allocated a budget sufficient to build $400$ km railways. Will this be enough to unite all the cities of the country into a single railway network? Justify the answer.
In the the triangle $ABC$ with $AC + AB = 2BC$, $M_1$ is midpoint of $BC$, $I$ is incenter, $M$ is the centroid. The lines $MI$ and $M_1I$ intersect the altitude $AH_1$ at points $K$ and $T$, respectively. Find the ratio $AT: TK: KH_1$.
(O. Karlyuchenko)
The trapezoid $ABCD$ ($BC \perp AD$) is inscribed in a circle. On the diagonal $BD$ is randomly selected point $E$. The line $CE$ intersects the side $AB$ at point $K$. Denote by $\omega$ the circumscribed circle around the triangle $CED$. The perpendicular drawn from point $C$ on $AD$ intersects $\omega$ for second time at point $N$, and the circumcircle of triangle $BKE$ intersects $\omega$ for second time at point $M$. Prove that the line $TM$ passes through the center of the circumscribed circle of the trapezoid $ABCD$, if the point $M$ lies inside the triangle $ABD$.
(D. Hilko)
Given a triangle $ABC$. Circles $\omega_1$ and $\omega_2$ pass through vertex $A$ and touch the side $BC$ at vertices $B$ and $C$, respectively. On the arc B$C$ of the circumcircle of the triangle $ABC$ take the point $F$. $FB$ intersects again $\omega_1$ at the point $K$, and $FC$ intersects again $\omega_2$ at the point $T$. Prove that the orthocenters of the triangles $ABC$, $AKF$ and $AFT$ lie on the same line.
(M. Plotnikov)
Find the largest number $q$ for which there is a point $X$ in the plane of an equilateral triangle $ABC$ such that $AX: BX: CX = 1: q: q^2$.
(W. Bryman)
Captain Flint hid the treasures at the vertices $B, C$, and $D$ of the convex quadrilateral $ABCD$, and on the map he marked point $A$ and three points of intersection of the bisectors of the angles $B, C$, and $D$ of the quadrilateral. Will Jim Hawkins, who got the map, be able to find these treasures?
(W. Bryman)
On the sides $AB, BC$ and $CA$ of the triangle $ABC$ mark the points $D, E$ and $F$, respectively, so that $AD: DB = BE: EC = CF: FA = 1: 2$. The sides of the triangle $T_1$ lie on the perpendicular bisectors of$AD$, $BE$ and $CF$ , and the sides of triangle $T_2$ are the perpendicular bisectors to $DB$ ,$EC$ and $AF$ . Find the ratio of the areas $S (T_1): S (T_2)$.
(W. Bryman)
Three secants $A-B-C$, $A-N-F$, $A-D-E$ are drawn passing through external point $A$ to the circle, so that points $B, C, D, E, N, F$ lie on the circle and the angles $CAF$ and $FAE$ are equal to $60^o$. Prove the equality $AB + AC + AD + AE = AN + AF$.
(O. Cherkassky)
Given a triangle $ABC$, a circle circumcribed around it and the center of the inscribed circle. Without using a compass, use a ruler (without divisions) to construct the centroid of the triangle $ABC$.
(G. Filippovsky)
The two circles intersect at points $A$ and $B$. A third circle with center at point$ O$ touches the first circle at point $F$ externally and the second at point $G$ internally. Prove that if $A,F,G$ are collinear, then the angle $ABO$ is right.
(Yu. Biletsky)
The opposite sides of the quadrilateral $ABCD$ intersect at points $E$ and $F$, and a circle $\omega$ is inscribed in $ABCD$. Let $P$ and $Q$ be the points of tangency to the circle of two circles $\omega_1$ and $\omega_2$, both of which pass through the points $E$ and $F$. Prove that the line $PQ$ passes through the intersection point of the diagonals of the quadrilateral $ABCD$.
(M. Plotnikov)
In the isosceles triangle $ABC$ on the sides $AB$ and $AC$, the points $T$ and $P$ are taken, respectively, and $TB = PA$. Circles with centers $T$ and $P$ and radii $TB$ and $PC$ are drawn, respectively, as well as a circle through points $A, T$ and $P$. Prove that these three circles have a common point.
(Yu. Biletsky)
Let $M_1$ and $M_2$ be the midpoints of the two sides of the triangle $ABC$, $L_1$ and $L_2$ be the feet of the corresponding angle bisectors, and let $W_1$ and $W_2$ be the points of intersection of the extensions of these angle bisectors with the circumscribed circle. It turned out that the lines $W_1M_2$, $W_2M_1$ and $L_1L_2$ intersect at one point. Determine the type of triangle $ABC$.
(O. Karlyuchenko, A. Bashkirevich)
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