Kharkiv Masters Tournament 2018-19 (Ukraine) 56p

geometry problems from Kharkiv Masters Tournament  in Ukraine, with aops links in the names

collected inside aops 

juniors and seniors

2018-2019

it started in 2018

2018

2018 1st Kharkiv Masters Tournament VII 3.2

Cut any triangle into $5$ congruent triangles and a square.

2018 1st Kharkiv Masters Tournament VII 3.7

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 $BS = SP$.

2018 1st Kharkiv Masters Tournament VII 4.2

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 $AE = 4AD$.

2018 1st Kharkiv Masters Tournament VII 4.8

Prove that a square can be cut into any number of pentagons larger than one (pentagons can be non-convex).

2018 1st Kharkiv Masters Tournament VIII-IX p3

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 $AD = BC + CD$.

2018 1st Kharkiv Masters Tournament VIII-IX p6

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?

2018 1st Kharkiv Masters Tournament VIII-IX 1.5

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 $2\angle AZC + 3\angle ABC$.

2018 1st Kharkiv Masters Tournament VIII-IX 1.8

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 $KFT$.

2018 1st Kharkiv Masters Tournament VIII-IX 2.5

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,

2018 1st Kharkiv Masters Tournament VIII-IX 2.8

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.

2018 1st Kharkiv Masters Tournament VIII-IX 3.2

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 $AE + AD = BE$.

2018 1st Kharkiv Masters Tournament VIII-IX 3.5

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.

2018 1st Kharkiv Masters Tournament VIII-IX 3.7,  X-XI 3.7

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.

2018 1st Kharkiv Masters Tournament VIII-IX 3.8

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.

2018 1st Kharkiv Masters Tournament VIII-IX 4.1

Is there a rectangular parallelepiped whose numerical values of volume, total area, and sum of the lengths of all the edges coincide?

2018 1st Kharkiv Masters Tournament VIII-IX 4.2

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 $CD$.

2018 1st Kharkiv Masters Tournament VIII-IX 4.5

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,

2018 1st Kharkiv Masters Tournament X-XI p3

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 $CN = BM$.

2018 1st Kharkiv Masters Tournament X-XI p6

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.

2018 1st Kharkiv Masters Tournament X-XI 1.7

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.

2018 1st Kharkiv Masters Tournament X-XI 2.5

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 $X$.

2018 1st Kharkiv Masters Tournament X-XI 2.8

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 $A_1C_1 = A_2C_2$.

2018 1st Kharkiv Masters Tournament X-XI 2.10

$3n$ 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.

2018 1st Kharkiv Masters Tournament X-XI 3.2

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 $\Omega$.

2018 1st Kharkiv Masters Tournament X-XI 3.8

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 $\frac{AM}{AK}+\frac{BM}{BL}$.

2018 1st Kharkiv Masters Tournament X-XI 3.9

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.

2018 1st Kharkiv Masters Tournament X-XI 4.2

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 $\angle MKN$.

2018 1st Kharkiv Masters Tournament X-XI 4.6

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.

2018 1st Kharkiv Masters Tournament X-XI 4.8

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

2019 2nd Kharkiv Masters Tournament VIII p3

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 $BQ = 2PH$.

2019 2nd Kharkiv Masters Tournament VIII 1.5

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 $60^o$.

2019 2nd Kharkiv Masters Tournament VIII 1.6

Construct a right triangle given the feet of its angle bisectors.

No research is required.

2019 2nd Kharkiv Masters Tournament VIII 2.7

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.

2019 2nd Kharkiv Masters Tournament VIII 2.8

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 $AL = AB$.

2019 2nd Kharkiv Masters Tournament VIII 3.6

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.

2019 2nd Kharkiv Masters Tournament VIII 3.8

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 $AXB$.

2019 2nd Kharkiv Masters Tournament VIII 4.4

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 $AC = TB$.

2019 2nd Kharkiv Masters Tournament VIII 4.7

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 $DCF$.

2019 2nd Kharkiv Masters Tournament IX p3

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.

2019 2nd Kharkiv Masters Tournament IX p5

$80$ 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.

2019 2nd Kharkiv Masters Tournament IX 1.5

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.

2019 2nd Kharkiv Masters Tournament IX 1.6

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 $RCB$.

2019 2nd Kharkiv Masters Tournament IX 1.9, X-XI 1.8

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.)

2019 2nd Kharkiv Masters Tournament IX 2.8

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 $MXY$

2019 2nd Kharkiv Masters Tournament IX 3.8

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 $M$.

2019 2nd Kharkiv Masters Tournament IX 3.10

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 $$\frac{R^4}{S_1^2}+\frac{R^4}{S_2^2}+\frac{R^4}{S_3^2}\ge 16$$

2019 2nd Kharkiv Masters Tournament IX 4.4

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 $BC$.

2019 2nd Kharkiv Masters Tournament X-XI p4

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 $BC$.

2019 2nd Kharkiv Masters Tournament X-XI 1.5

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 $\ell$.

2019 2nd Kharkiv Masters Tournament X-XI 2.7

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 $TBC$.

2019 2nd Kharkiv Masters Tournament X-XI 2.8

The center of the circumcircle of triangle $ABC$ lies on its inscribed circle. Find the sum of the cosines of the angles of triangle $ABC$.

2019 2nd Kharkiv Masters Tournament X-XI 3.5

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 $AQ = 3QH$.

2019 2nd Kharkiv Masters Tournament X-XI 4.A3

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 $CD$.

2019 2nd Kharkiv Masters Tournament X-XI 4.A6

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 $\omega$.

2019 2nd Kharkiv Masters Tournament X-XI 4.A6.1

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 $IO$.

2019 2nd Kharkiv Masters Tournament X-XI 4.B10

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?

source: https://mathedu.kharkiv.ua/examples/seflkm/