Kurchatov 2013-22 (Russia) 54p

 geometry problems from final round of Kurchatov Olympiad  (Russia) with aops links in the names

ОЛИМПИАДА «КУРЧАТОВ»

collected inside aops here

2013 - 2022

2013 Kurchatov Olympiad 7.4

Baron Munchausen has a set of four triangles. He states that the first of the triangles is not equal to the second. And he also says that whichever of the four is lost, from the other three you can make one a larger triangle (no overlaps and holes). Could all the words of the baron be true?

2013 Kurchatov Olympiad  8-9.4

$ABCD$ is trapezoid, $M$ is midpoint of base $AD$ , $E$ is intersection point of the diagonals.

It is known that $AD = 2BC = 6ME$. Prove that $AB^2+ CD^2= BC^2$.

2013 Kurchatov Olympiad 10-11.4

There are $n$ different points on the plane. Prove that they can be denoted by the letters $A_1, A_2, ..., A_n$ so that the vectors $\overrightarrow{A_1A_2}, \overrightarrow{A_2A_3}, ..., \overrightarrow{A_{n-1}A_n}$ are all different.

2014 Kurchatov Olympiad 8.3

A perpendicular bisector was drawn to the leg AB of an isosceles trapezoid $ABCD$. It crossed the segment $BC$ at point $E$. Find the angle $ABC$ if it is known that the lines $AE$ and $CD$ perpendicular.

2014 Kurchatov Olympiad 9.4

Points $M$ and $N$ are marked on the sides of the angle with vertex $Q$, and point $E$ is marked inside the angle so that $QE = MN$, $\angle MQE =  \angle QNM$, $\angle EQN +  \angle QNE =  \angle QMN$. Find $\angle MQN$.

2014  Kurchatov Olympiad 9.5

The coordinate axes and the graph of the function $y=\frac{2}{x}$ are drawn on the plane for $x> 0$. The scale is not specified, but it is known to be the same on both axes. is the same. Unfortunately, a small piece of the graph near the beginning coordinates was accidentally erased (see fig.) Using a compass and a ruler , restore the point with abscissa $1$ on this graph.

2014 Kurchatov Olympiad 10.2

The coordinate axes and the graph function of the funtion $y=\frac{2}{x}$ are drawn on the plane . The scale is not specified, but it is known to be the same on both axes. is the same. Using a compass and a ruler , restore the point at which the abscissa is positive and $2$ less than the ordinate.

2014 Kurchatov Olympiad 10-11.4

Two disjoint circles are inscribed in an angle. They touch one side of the corner at points $K$ and $L$, the other at points $M$ and $N$ (see figure), $C$ is the midpoint of the segment $KL$, $A$ and $B$ are the points of intersection of the segments $CM$ and$CN$ with the circles. Prove that

a) points $A, B, M$ and $N$ lie on the same circle.

b) points $A, B, K$ and $L$ lie on the same circle.

2014  Kurchatov Olympiad 11.2

Divide an isosceles right triangle into two smaller triangles so that some median of one of these triangles is parallel to one of atltides of the second triangle.

2014  Kurchatov Olympiad 11.6

Vasya, an invisible virus, is sitting on a square plate with a side of $1$ cm. He and the doctor Petya take turns. With the next $n$-th move, Petya draws with the vaccine like ink a segment $1$ micron long, and then Vasya must choose a direction and crawl in this straight line a direction of $1 / n$ micron distance (without going beyond the edge of the plate). If Vasya crawls through any of the points with the vaccine or touches it, he will die. Petya can act with any precision. Can Petya within a finite number of moves for sure kill the virus?

2015  Kurchatov Olympiad 7.4

In a right-angled triangle $ABC$, angle bisector $AL$ was drawn and a a point $K$ was marked on the hypotenuse $AB$ such such that $AB = 3BK$. It turned out that the angle $ALK$ is right . Prove that $AL = BL$.

2015 Kurchatov Olympiad 8.3

In the convex quadrilateral $ABCD$, the perpendiculars are drawn to sides $AB, BC$ and $CD$. Inside the quadrilateral, these perpendiculars do not cross . The intersection points of these perpendiculars split the $AD$ side into $4$ equal parts. Prove that $AD\parallel BC$.

2015 Kurchatov Olympiad 9-10.4

On the median $CM$ of triangle $ABC$, a point $D$ is chosen such that $2CD = AB$. Line $BD$ meets side $AC$ at point $E$. Prove that if $DE = CE$, then angle $\angle BMC = 120^o.$

2015 Kurchatov Olympiad 11.4

$ABCD$ is a cyclic quadrilateral, $AB> CD$, $BC> AD$. On sides $AB$ and $BC$ marked points $X$ and $Y$ so that $AX = CD$ and $AD = CY$. $M$ is the midpoint of $XY$. Prove that the angle $AMC$ is right.

2015 Kurchatov Olympiad 11.6

To prepare mashed potatoes, Kolya, the chef, needs to get the specified amount of peeled potatoes as soon as possible. Not caring about saving peelings, he cuts cubes from spherical potatoes, with each stroke clearing one edge of the knife. Can he complete the task faster when the same frequency of knife strokes if you cut out any other polyhedra? (formally: is it true that of all polyhedra cut from of the given sphere, the largest ratio of volume to the number of faces is for the inscribed cube?)

2016 Kurchatov Olympiad 7.2

If the width of the rectangle is increased by $3$ cm, and the length is reduced by $3$ cm, it's area will not change. And how will the area change if, instead, the original reduce the width of the rectangle by $4$ cm, and increase the length by $4$ cm?

2016 Kurchatov Olympiad 7.4

The segments $KL$ and $MN$ intersect at point $T$. It is known that the triangle $KNT$ is equilateral and $KL = MT$. Prove that triangle $LMN$ is isosceles.

2016 Kurchatov Olympiad 8.1

If the width of the rectangle is increased by $30\%$, and the length is decreased by $20\%$, it's perimeter will not change. And the perimeter will decrease or increase, and by what percentage, if instead of that the original rectangle has its width reduced by $20\%$, and the length increase by $30\%$?

2016 Kurchatov Olympiad 8.2

The diagonals of parallelogram $ABCD$ meet at point $O$. In triangles $OAB, OBC, OCD$ medians $OM, OM', OM''$ and angle bisectors $OL, OL', OL''$ are drawn respectively. Prove that angles $MM'M''$ and $LL'L''$ are equal.

2016 Kurchatov Olympiad 9.3

You are given an isosceles triangle $ABC$. On the lateral side $AB$, a point $M$ is marked such that $CM = AC$. Then, on the lateral side $BC$, a point $N$ was marked such that $BN = MN$, and the angle bisector $NH$ was drawn at triangle $CNM$. Prove that $H$ lies on the median $BK$ of the triangle $ABC$.

2016 Kurchatov Olympiad 10.3

Given a triangle $ABC$. From point $P$ inside it, the perpendiculars $PA', PB', PC'$ are drawn on the sides $BC,CA,AB$ respectively. Then from point $P$ perpendiculars $PA'', PB''$ are drawn on the sides $B'C'$ and $C'A'$ respectively. Prove that $PA\cdot PA'\cdot PA''= PB\cdot PB'\cdot PB''$.

2016 Kurchatov Olympiad 11.1

Two right-angled triangles have the same area and perimeter. Is it obligatory are these triangles congruent?

2016 Kurchatov Olympiad 11.5

Two opposite vertices $M$ and $M '$ were chosen from the cube and flat sections $ABC$ and $A'B'C'$ cut off two triangular pyramids $MABC$ and $M'A'B'C '$. It turned out to be an octahedron (see fig.) Three distances turned out to be are pairwise different: between straight lines $AB$ and $A'B '$, between straight lines $BC$ and $B'C'$ and between straight lines $AC$ and $A'C'$. Prove that lines $AA ', BB'$ and $CC '$ have a common point.

2017 Kurchatov Olympiad 7.5

All angles of a convex hexagon $ABCDEF$ are equal. Prove that $AB - DE = EF - BC = CD - FA$.

2017 Kurchatov Olympiad 8.2

The diagonals of the trapezoid $ABCD$ ($AD \parallel BC$) meet at point $S$. It is known that that $AD \perp  AC$ and $BS = 2CD$. Prove that $\angle CDB = 2\angle ADB$.

2017 Kurchatov Olympiad 11.3

Points $X$ and $Y$ are chosen on sides $AB, AD$ of square $ABCD$ so that $AX =DY$. Lines $BC$ and $DX$ meet at point $P$, lines $CD$ and $BY$ at point $Q$. Prove that points $P, Q, A$ are collinear.

2017 Kurchatov Olympiad 11.6

Let $A$ and $B$ be different points belonging to the line of intersection of the perpendicular planes $\pi_1$ and $\pi_2$. Point $C$ belongs to the plane $\pi_2$, but does not belong to $\pi_1$. Let $P$ denote the intersection point of the bisector of the angle $ACB$ with line $AB$ and through $\omega$ a circle with diameter $AB$ in the plane $\pi_1$. The plane $\pi_3$ containing $CP$ meets the circle $\omega$ at points $D$ and $E$. Prove that $CP$ is the bisector of the angle $DCE$.

2018 Kurchatov Olympiad 7.3

Points $X$ and $Y$ are marked on the sides $AB, BC$ of triangle $ABC$, respectively, so that $AY = AB$ and $CX = CB$. The straight line passing through the vertex $A$ parallel to the side $BC$ intersects the straight line, passing through $C$ parallel to side $AB$, at point $D$. Prove that $DX = DY$.

2018 Kurchatov Olympiad 8.1

An overlapping square and circle are drawn on the plane. Together they cover an area of $2018$ cm$^2$. The intersection area is $137$ cm$^2$ .The area of the circle is $1371$ cm$^2$ . What is the perimeter of the square?

2018 Kurchatov Olympiad 8.4

In triangle $ABC$, point $D$ is chosen on the side $AC$, and point $E$ on the side $BC$ so that relations $CD = AB$, $BE = BD$, $AB \cdot AC = BC^2$. Find $\angle DEA$ if it is known that $\angle DBC = 40^o$.

2018 Kurchatov Olympiad 9.4

In an acute-angled triangle $ABC$, a straight line $\ell$ is drawn through the vertex $A$, perpendicular to the median, starting from the vertex $A$. The extensions of the altitudes $BD$ and $CE$ of the triangle intersect the straight line $\ell$ at the points $M$ and $N$. Prove that $AM = AN$.

2018 Kurchatov Olympiad 10.5

The inscribed circle of triangle $ABC$ touches sides $AB$ and $AC$ at points $D$ and $E$, respectively. Point $I_A$ is the center of the $A$-excircle of triangle $ABC$, and points $K$ and $L$ are the midpoints of the segments $DI_A$ and $EI_A$ , respectively. Lines $BK$ and $CL$ meet at point $F$ lying inside the angle $BAC$. Find $\angle BFC$ if $\angle BAC = 50^o$.

2018 Kurchatov Olympiad 11.6

A tetrahedron $ABCD$ with acute-angled edges is inscribed in a sphere centered at $O$. The straight line passing through point $O$ perpendicular to the plane $ABC$ intersects the sphere at point $E$ such that $D$ and $E$ lie on opposite sides relative to plane $ABC$. Line $DE$ intersects plane $ABC$ at point $F$, lying inside the triangle $ABC$. It turned out that $\angle ADE =  \angle BDE$, $AF \ne BF$ and $\angle AFB = 80^o$. Find the value for $\angle ACB$.

2019 Kurchatov Olympiad 6.2

A square with a side of $1$ m is cut into three rectangles with equal perimeters. What can these perimeters be equal to? Indicate all possible options and explain why there are no others.

2019 Kurchatov Olympiad 7.4

Points $X$ and $Y$ are chosen on the sides $AB$ and $AC$ of triangle $ABC$, respectively, so that $\angle AYB = \angle AXC = 134^o$ . On ray $YB$, point $M$ was marked beyond point $B$, and on ray $XC$ beyond point $C$ point $N$. It turned out that $MB = AC$, $AB = CN$. Find $\angle MAN$.

2019 Kurchatov Olympiad 8.4

Distances from a point $P$, which lies inside an equilateral triangle, to its vertices are $3, 4$ and $5$. Find the area of the triangle.

2019 Kurchatov Olympiad 9-10.3

The circle $\omega$ centered at point $I$ is inscribed in a convex quadrilateral $ABCD$ and touches side $AB$ at point $M$, and side $CD$ at point $N$, while $\angle BAD + \angle ADC <180^o$. On the line $MN$, a point $K \ne M$ is chosen such that $AK = AM$. In what ratio does the straight line $DI$ divide the segment $KN$? Give all the possible answers and prove that there are no others.

2019 Kurchatov Olympiad 11.6

In tetrahedron $ABCD$, the following equalities hold:

$\angle BAC + \angle BDC = \angle ABD + \angle ACD$, $\angle BAD + \angle BCD = \angle ABC + \angle ADC$. Prove that the center of the circumscribed sphere of the tetrahedron lies on the line connecting the midpoints of the edges $AB$ and $CD$.

2020 Kurchatov Olympiad 7.4

The angle bisector $BL$ is drawn in right-angled triangle $ABC$ with right angle $A$. Point $E$ is selected on segment $BC$, and that segment $CL$ is point $D$ so that $\angle LDE = 90^o$, $AL = DE$. Prove that $AB = LD + BE$.

2020 Kurchatov Olympiad 8.4

In a right-angled triangle $ABC$ with a right angle $A$, the altitude $AH$ is drawn. On the extension of the hypotenuse $BC$ beyond the point $C$, there is a point $X$ such that $HX = \frac{BH + CX}{3}$. Prove that $2\angle ABC = \angle AXC$.

2020 Kurchatov Olympiad 9.3

In triangle $ABC$ with angles $\angle A = 35^o,  \angle B = 20^o$ and $\angle C = 125^o$, point $O$ is the center of the circumscribed circle . Prove that points $O, A, B, C$ are the vertices of a trapezoid.

2020 Kurchatov Olympiad 10.5

Point $H$ is the orthocenter of an acute-angled triangle $ABC$. Point $G$ is such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be a point on the line $GH$ such that $AC$ divides $HI$ in half. Line $AC$ meets the circumcircle of triangle $GCI$ at points $C$ and $J$. Prove that $IJ = AH$.

2020 Kurchatov Olympiad 11.4

Let $SABCD$ be a regular rectangular pyramid with base $ABCD$. On the segment $AC$, a point $M$ was found such that $SM = MB$ and the planes $SBM$ and $SAB$ are perpendicular. Find the ratio $AM: AC$.

2021 Kurchatov Olympiad 7.3

The altitude $AD$ is drawn in an acute-angled triangle $ABC$. It turned out that $AB + BD = DC$. Prove that $\angle B = 2\angle C$.

2021 Kurchatov Olympiad 8.3

On the side $BC$ of an acute-angled triangle $ABC$, mark the point $D$ such that $AB + BD = DC$. Prove that $\angle ADC = 90^o$ if it is known that $\angle B = 2\angle C$.

2021 Kurchatov Olympiad 9.4

A point $M$ is marked on the side $AC$ of triangle $ABC$ such that $AM = AB + MC$. Prove that the perpendicular to $AC$ passing through $M$ is divides the arc $BC$ of the circumcircle $ABC$ in half.

2021 Kurchatov Olympiad 10.3

Given a rectangular trapezoid $ABCD$ with a right angle $A$ ($BC \parallel AD$). It is known that $BC = 1$, $AD = 4$. Point X is marked on the side $AB$, and point $Y$ on side $CD$such that $XY = 2$, $XY \perp  CD$. Prove that the circumscribed circle of the triangle $XCD$ touches $AB$.

2021 Kurchatov Olympiad 11.4

The diagonals of the trapezoid $ABCD$ ($AD \parallel BC$) meet at point $O$. On $AB$ mark a point $E$ such that line $EO$ is parallel to the base of the trapezoid. It turned out that $EO$ is the bisector of the angle $CED$ . Prove that the trapezoid is right.

2022 Kurchatov Olympiad 7.1

The figure shows three squares. Find the marked angles, if the other two angles in the figure are known.

2022 Kurchatov Olympiad 7.5

In triangle $ABC$, the angle at vertex $B$ is $120^o$, point $M$ is the midpoint of side $AC$. On sides $AB$ and $BC$, points $E$ and $F$ are selected, respectively, such that $AE = EF = FC$. Find $\angle EMF$.

2022 Kurchatov Olympiad 8.4

Given a triangle $ABC$ such that $\angle BAC = 2\angle BCA$. Point $L$ lies on side $BC$ is that $\angle BAL = \angle CAL$. Point $M$ is the midpoint of side $AC$. Point $H$ lies on segment $AL$ is such that $MH \perp AL$. There is a point $K$ on the side of $BC$ such that triangle $KMH$ is equilateral. Prove that points $B, H$ and $M$ lie on the same line.

2022 Kurchatov Olympiad 9.5

Diagonals of convex quadrilateral $ABCD$ intersect at point $O$. Points $P$ and $Q$ are the midpoints of segments $AC$ and $BD$, respectively. On the segments $OA$, $OB$, $OC$, $OD$ points $A_1$, $B_1$, $C_1$, $D_1$ respectively such that $AA_1=CC_1$, $BB_1=DD_1$.

$\bullet$ The circumscribed circles of triangles $AOB$ and $COD$ intersect at points $K$ and $O$.

$\bullet$ The circumscribed circles of triangles $A_1OB_1$ and $C_1OD_1$ intersect at points $M$ and $O$.

Prove that the points $K, M, P, Q$ lie on the same circle.

2022 Kurchatov Olympiad 10.2

Given an acute-angled non-isosceles triangle $ABC$, point $O$ is the center of its circumscribed circle. The extension of altitude $BH$ of triangle $ABC$ intersects it's circumscribed circle at point $N$. On the sides $AB$ and $BC$ the points $X$ and $Y$ are marked, respectively, such that $OX \parallel AN$ and $OY\parallel CN$ . The circumscribed circle of the triangle $XBY$ intersects the segment $BH$ at the point $Z$. Prove that $XY \parallel OZ$.

2022 Kurchatov Olympiad 11.5

A point $P$ inside an acute triangle $ABC$ is such that $\angle BAP =\angle CAP$. Point $M$ is the midpoint of side $BC$. The line $MP$ intersects the circumscribed circles of triangles $ABP$ and $ACP$ at points $D$ and $E$, respectively (point $P$ lies between the points $M$ and $E$, the point $E$ lies between the points $P$ and $D$). It turned out that $DE = MP$. Prove that $BC = 2BP$.

sources:

http://olimpiadakurchatov.ru/

https://olimpiada.ru/activity/389

https://postypashki.ru/