geometry problems from Kharkiv Masters Tournament in Ukraine, with aops links in the names
collected inside aops
2018-2019
it started in 2018
2018
Cut any triangle into $5$ congruent triangles and a square.
The base $BC$ of an isosceles triangle $ABC$ is larger than its sides. In the triangle, the angle bisector $BE$ is drawn. From the point $C$ is drawn perpendicular on the line $BE$, that crosses the extension of the side $AB$ at the point $P$. The line $PE$ intersects the side $BC$ at the point $S$. Prove that $BS = SP$.
Given a triangle $ABC$, point $M$ is the midpoint of the side $AB$, and point $N$ is the midpoint of the side $AC$. On the side $BC$, such a point $P$ is chosen that $BP: PC = 3: 1$ On the line $PN$, such a point $D$ is marked that $PN = ND$, and on the line $CM$, such a point $E$ is marked that $CM = ME$. Prove that $AE = 4AD$.
Prove that a square can be cut into any number of pentagons larger than one (pentagons can be non-convex).
The bisectors of the angles $B$ and $C$ are drawn in the trapezoid $ABCD$. They intersect the base $AD$ at the points $E$ and $F$, respectively (point $E$ lies between $A$ and $F$, point $F$ lies between $E$ and $D$). The rays $BE$ and $CF$ intersect at the point $G$, and $EF = FG$. Prove that $AD = BC + CD$.
There are $2018$ points on the plane. For each pair of points marked the midpoint of the segment with
the ends at these points. What is the smallest number of different points you could mark?
In the isosceles triangle $ABC$ ($A B=BC$) the angle bisectors $AK$ and $CL$ are drawn. On base $AC$ point $X$ is marked , and on the segment $CX$ point $Y$ is marked such that $AL = AX$ and $CK = CY$. Rays $LX$ and $KY$ intersect at the point $Z$. Find the sum $2\angle AZC + 3\angle ABC$.
Let $AL$ be the angle bisector of triangle $ABC$. The points $Q$ and $N$ are the midpoints of the sides $AB$ and $BC$, respectively. The lines $AL$ and $QN$ intersect at point $F$, points $K$ and $T$ are respectively, the points of tangency of the inscribed and $A$-excircle of this triangle with side $BC$. Find the measure of the angle $KFT$.
In the acute triangle $ABC$ the height $AD$ is drawn. Bisectors of angles $BAD$ and $CAD$ intersect the side $BC$ at points $E$ and $F$, respectively. The circumcircle of triangle $AEF$ intersects sides $AB$ and $AC$ at points $G$ and $H$, respectively. Prove that the lines $EH$, $FG$ and $AD$ intersect at one point,
The extension of the angle bisector $BL$ of the triangle $ABC$ intersects its circumcircle at the point K. The bisector of the exterior angle $B$ intersects the extension of the segment $CA$ at point $A$ at point $N$. Prove that if $BK = BN$, then the segment $LN$ is equal to the diameter of the circumcircle of the triangle.
Given a convex quadrilateral $ABCD$, in which $\angle A = 2 \angle B$. On the side $AB$ there is a point $E$ such that $\angle BCE = \angle DCE = \angle AED$. Prove that $AE + AD = BE$.
An equilateral triangle with side $100$ is divided by parallel sides of straight lines of three directions into many small triangles with side $1$. $77$ vertices of the resulting triangular lattice are painted blue. Prove that there are two blue vertices lying on a line parallel to one of the sides of a large triangle.
There are $2n + 1$ points in a circle: $n$ white, $n$ red and one black. Prove that $2n$ of these points can be connected by $n$ segments so that they do not intersect and none of the segments connects the white and red points.
The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ intersect at the point $E$. Tangent to the circumscribed circle of triangle $ADE$, drawn at point $D$, intersects the line $CB$ at point $F$. Prove that the triangle $CDF$ is isosceles.
Is there a rectangular parallelepiped whose numerical values of volume, total area, and sum of the lengths of all the edges coincide?
In the quadrilateral $ABCD$, $\angle B = \angle C$ and $\angle D = 90^o$ . It is also known that $AB = 2CD$. Prove that the bisector of the angle $ACB$ is perpendicular to $CD$.
In triangle $ABC$, the point $H$ is the orthocenter. Let $N$ be a point on the segment $AH$ such that the circumscribed circles of triangles $BNA$ and $CNH$ touch each other. Prove that the circumscribed circles of triangles $CNA$ and $BNH$ also touch each other,
Suppose that in the triangle $ABC$, the bisector of the angle $BAC$ intersects the perpendicular bisector of the side $AC$ at the point $P$ and the perpendicular bisector of $AB$ at the point $Q$. Denote the points of intersection of $BC$ with the circumscribed circles of triangles $AQC$ and $APB$ by $N$ and $M$, respectively. Prove that $CN = BM$.
Points with integer coordinates on the Cartesian plane were painted in two colors. Prove that there exists an infinite one-color subset symmetric with respect to some point of the plane.
Construct a triangle $ABC$ with $\angle C,$ point $C$ and points $K$ and $M$ of intersection of the perpendicular bisectors of the sides $AC$ and $BC$ of the triangle with altitude $CH$.
No research is required.
The circle $\omega'$ lies inside the circle $\omega$ and touches it at the point $N$. Tangent to $\omega'$ at the point $X$ intersects $\omega$ at the points $A$ and $B$. Let $M$ be the midpoint of the arc $AB$ that contains $N$. Prove that the radius of the circumcircle of the triangle $BMX$ is independent of the location of the point $X$.
A circle $\Gamma$ is inscribed in the quadrilateral $ABCD$. The point $E$ is the point of intersection of $\Gamma$ with the diagonal $AC$, the closest to $A$. Tangent to $\Gamma$ at point $F$ intersects the lines $AB$ and $BC$ at points $A_1$ and $C_1$, respectively, and the lines $AD$ and $CD$ at points $A_2$ and $C_2$, respectively. Prove that $A_1C_1 = A_2C_2$.
$3n$ yellow and $n$ blue lines of general position ($n> 2$) are drawn on the plane. Prove that there is a polygon on this plane, all sides of which are yellow.
In triangle $ABC$, point $I$ is the center of the inscribed circle, $I_a$ is the center of the exscribed circle tangent to the side $BC$. Let these inscribed and exscribed circles touch the line $BC$ at the points $P$ and $Q$, respectively, $S$ be the midpoint of the arc $BC$ of the circumcircle $\Omega$ of the triangle $ABC$, which does not contain the point $A$. The circle $\Gamma$ touches the side $BC$ at the point $P$ and the circle $\Omega$ at the point $R$ , and $A$ and $R$ lie on one side of $BC$. The line $RI$ intersects the circle $\Omega$ for the second time at the point $L$. Prove that the lines $LI_a$ and $SQ$ intersect at the circle $\Omega$.
The circle $\omega$ is inscribed in the isosceles trapezoid $ABCD$, Let $M$ be the touchpoint of the circle $\omega$ with the side $CD$. Denote by $K$ and $L$, respectively, the points of intersection of the segments $AM$ and $BM$ with the circle $\omega$. Find the value of the sum $\frac{AM}{AK}+\frac{BM}{BL}$.
There are $4n$ points on the plane, none of which lie on the same line. Consider ${4n \choose 3}$ triangles with vertices at these points. Prove that there exists a point $X$ on the plane belonging to at least $2n^3$ triangles.
On the arc $AB$ of the circumcircle of the triangle $ABC$, the point $M$ is chosen, the projections $X$ and $Y$ of the point $M$ on the lines $AB$ and $BC$ fall on the sides of the triangle, and not on their extension. Let $K$ and $N$ be the midpoints of the segments $XY$ and $AC$, respectively, Find $\angle MKN$.
An acute-angled triangle $ABC$ with orthocenter $H$ is inscribed in a circle $\omega$ with center $O$. The line $d$ passes through $H$ and intersects the smaller arcs $AB$ and $AC$ of the circle $\omega$ at the points $P$ and $Q$, respectively. Let $AA'$ be the diameter of the circle $\omega$ , the lines $A'P$ and $A'Q$ intersect $BC$ at the points $K$ and $L$, respectively. Prove that the points $O, K, L$ and $A'$ lie on the same circle.
A natural number $n$ is given. $n$ straight lines are drawn on the plane, none of which are not parallel and none of the three intersect at one point. All their points of intersection are painted red. Prove that there is a line and such that on both sides of it is not less than $\left[\frac{(n-1) (n-2)}{10}\right]$ red points (red points lying on a straight line and not taken into account). For which $n$ the score cannot be improved?
2019
In the square $ABCD$ on the side $CD$, the point $P$ is chosen. The point $Q$ on the side $AD$ is such that $BQ$ is the bisector of the angle $PBA$. From the point $P$, the perpendicular $PH$ on the line $BQ$ is drawn. Prove that $BQ = 2PH$.
The median $BM$ is drawn in the triangle $ABC$. On the side $AB$, the point $K$ is marked so that $\angle BMK = 90^o$. It turned out that $BK = BC$. Find the angle $ABM$ if the angle $CBM$ is $60^o$.
Construct a right triangle given the feet of its angle bisectors.
No research is required.
In triangle $ABC$, the point $H$ is the orthocenter. The point $D$ is chosen and the segment $AC$, and the point $E$ on the line $BC$ such that $BC \perp DE$. Prove that $EH \perp BD$ if and only if the line $BD$ intersects the segment $AE$ in its midpoint.
The points $K$ and $L$ are marked on the sides $AB$ and $BC$ of the triangle $ABC$, respectively. The segments $AL$ and $CK$ intersect at the point $P$. It turned out that $KC = BC$ and $PC = PB = BL$. Prove that $AL = AB$.
A circle is circumscribed around an isosceles triangle $ABC$ with base $AC$. On an arc $AB$ that does not contain a point $C$, an arbitrary point $D$ is marked. Let the point $E$ be symmetric to the point $C$ with respect to the line $BD$. Prove that the points $A, D$ and $E$ lie on the same line.
On the hypotenuse $AB$ of a right triangle $ABC$, points M and N are marked such that $CB = NB$ and $AC = MA$. The point $X$ is chosen inside the triangle $ABC$ so that the triangle $MNX$ is an isosceles rectangle with a right angle $X$. Find the measure of the angle $AXB$.
The point $M$ is the midpoint of the hypotenuse $AB$ of the right triangle $ABC$. The perpendicular bisector of $AB$ intersects the side $BC$ at the point $K$. The line perpendicular to $CM$ and passing through the point $K$ intersects the ray $CA$ at the point $P$ (point $A$ lies between $C$ and $P$). The lines $CM$ and $BP$ intersect at the point $T$. Prove that $AC = TB$.
On the base of $BC$ of an isosceles triangle ABC with an angle at the vertex of $100^o$ the point $D$ is marked so that $AC = DC$ and on the side $AB$ the point $F$ is marked so that $DF\parallel AC$. Find the measure of the angle $DCF$.
The angle bisector $AD$ is drawn in the acute-angled triangle $ABC$. A line perpendicular to $AD$ passing through point $B$ intersects the circumcircle of the triangle $ABD$ at the point $E$ for the second time. Prove that the points $A, E$ and the center of the circumcircle of the triangle $ABC$ lie on the same line.
$80$ different points are marked on the line: $60$ yellow and $20$ blue. It turned out that there are
at least two yellow dots on any segment with blue ends. Prove that at least $\frac23$ of all segments
with yellow ends have at least two blue dots.
On the sides $AB$ and $BC$ of the parallelogram $ABCD$, the points $E$ and $F$ are marked so that $AC \parallel EF$. The right triangles $BEX$ and $BFY$ are constructed on the outside. Prove that the triangles $ADX$ and $CDY$ are congruent.
In the triangle $ABC$ with $AB = AC$, $M$ is the midpoint of $BC$. Circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. The line $MP$ intersects the line $AB$ at the point $Q$, and the point $R$ on $AP$ is such that $QR \parallel BP$. Prove that $CP$ is the bisector of the angle $RCB$.
There are $2019$ points on the plane, none of which lie on the same line. What is the minimum number of lines that must be drawn to ensure that all points are separated from each other? (Two points are separated if there is at least one line with respect to which they lie on opposite sides.)
Two intersecting circles $\omega_1$ and $\omega_2$ are given. The lines $AB$ and $CD$ are common tangents to these circles (points $A$ and $C$ lie on $\omega_1$, and points $B$ and $D$ lie on $\omega_2$). We know $M$ is the the midpoint of the segment $AB$. Tangents drawn from point $M$ to $\omega_1$ and $\omega_2$, other than $AB$, intersect $CD$ at points $X$ and $Y$. Prove that $IC = ID$ if I is the center of triangle $MXY$
The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.
The radius of the circumcircle of triangle $ABC$ is equal to $R$, the point $I$ is the incenter of the triangle. Denote by $S_1$, $S_2$ and $S_3$ the areas of the triangles $ABI$, $BCI$ and $CAI$, respectively. Prove that$$\frac{R^4}{S_1^2}+\frac{R^4}{S_2^2}+\frac{R^4}{S_3^2}\ge 16$$
Find the radius of the circumcircle of triangle $ABC$, in which the altitude and angle bisector drawn from the vertex $A$ are equal to $h$ and $\ell$, respectively, if it is known that the distance between the feet of thid altitude and that angle bisector is equal to the distance between the foot of this angle bisector and the midpoint of side $BC$.
Triangles $ABC$ and $XYZ$ have a common inscribed circle, lines $BC$ and $YZ$ coincide, and line $AX$ is parallel to $BC$. Prove that the common chord of the circles circumscribed around the triangles $ABC$ and $XYZ$ contains the point of contact of the common inscribed circle with the line $BC$.
The two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. The line $\ell$ passes through point $B$ and intersects the circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Let $H_c$ and $H_d$ be the orthocenters of triangles $ABC$ and $ABD$, respectively, and let $A_c$ and $A_d$ be points diametrically opposed to $A$ in circles $\omega_1$ and $\omega_1$, respectively. The lines $H_cH_d $,$A_cA_d$ intersect at the point $E$, and the lines $H_cA_d$ and $H_dA_c$ intersect at the point $F$. Prove that the midpoint of the segment $EF$ lies on the line $\ell$.
In the triangle $ABC$ the inequality $AB> A$C holds. The foor of the altitude from the vertex $A$ to the side $BC$ is the point D. The point of intersection of the bisector of the angle $B$ of the triangle $ABC$ with the line $AD$ is the point $K$. The foot of the perpendicular from the point $B$ to $CK$ is the point $M$. The lines $BM$ and $AK$ intersect at the point $N$. A line passing through the point $N$ parallel to $DM$, intersects $AC$ at the point $T$. Prove that $BM$ is the bisector of the angle $TBC$.
The center of the circumcircle of triangle $ABC$ lies on its inscribed circle. Find the sum of the cosines of the angles of triangle $ABC$.
In an acute-angled triangle ABC, the point $O$ is the center of the circumcircle, $H$ is the orthocenter, and $M$ is the midpoint of the side $BC$. Segments $BE$ and $CF$ are the altitudes of the triangle. The point $P$ lies on the line $EF$ such that $PH \perp HO$. The point $Q$ lies on the segment $AH$, and $PQ \perp HM$. Prove that $AQ = 3QH$.
The two circles $\gamma_1$ and $\gamma_2$ intersect at points $A$ and $B$. The points $P, Q$ are chosen on the circles $\gamma_1$ and $\gamma_2$, respectively, so that $AP = AQ$. The segment $PQ$ intersects the circles $\gamma_1$ and $\gamma_2$ at the points $M$ and $N$, respectively. Point $C$ is the midpoint of the arc $BP$ of $\gamma_1$, which does not contain point $A$, and point $D$ is the midpoint of the arc $BQ$ of circle $\gamma_2$, which does not contain point $A$. The lines $CM$ and $DN$ intersect at point $E$. Prove that the line $AE$ is perpendicular to the line $CD$.
A circle $\omega$ with center $I$ is inscribed in an isosceles acute-angled triangle $ABC$. Denote by $B_1$ and $C_1$ the projections of points $B$ and $C$ on the line $AE$, respectively. Points $X$ and $Y$ are chosen and the segments $BC$ so that $\angle B_1XC_1 = \angle B_1YC_1 = 90^o$. Prove that the circle circumscribed around the triangle $AXY$ touches $\omega$.
In a non-equilateral triangle $ABC$, point $I$ is the center of the inscribed circle, and point $O$ is the center of the circumscribed circle. The line $s$ passes through $I$ and is perpendicular to the line $IO$. The line $I$ is symmetric to the line $BC$ with respect to $s$ and intersects the segments $AB$ and $AC$ at the points $K$ and $L$, respectively (both points $K$ and $L$ are different from $A$). Prove that the center of the circumcircle of triangle $AKL$ lies on the line $IO$.
The bases of the trapezoid are $a$ and $b$. its sides are perpendicular. What is the largest value of the area of a triangle formed by the intersection of the midline of the trapezoid and its diagonals?
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