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High Standards 2009-22 VII+ (Высшая проба) (Russia) 51p

geometry problems from the 2nd / final Round Russian High Standards Olympiad - Высшая проба
with aops links in the names

collected inside aops here

2009 - 2022


The height $AA_1$ is drawn in an acute-angled triangle $ABC$. $H$ is the intersection point of the altitudes of the triangle $ABC$. It is known that $AH = 3, A_1H = 2$, and the radius of the circle circumscribed around the triangle $ABC$ is $4$. Find the distance from the center of this circle to $H$.

Six straight lines are drawn through the center of a sphere of radius $\sqrt2$, parallel to the edges of some regular tetrahedron. The intersection points of these lines with the sphere are the vertices of the convex polyhedron. Calculate the volume and surface area of this polyhedron.

2010 High Standards Olympiad 9.3 10.3
Circles $S_1, S_2 , S_3$ with centers $O_1,O_2 ,O_3$ respectively pass through point $F$. It is known that the second intersection point of circles $S_2$ and $S_3$ lies on the straight line $FO_1$, and the second intersection point of the circles $S_1$ and $S_3$ lies on the straight line $FO_2$. Prove that the second intersection point of circles $S_1$ and $S_2$ lies on line $FO_3$.

A double-sided ruler allows you to draw a straight line through two given points, as well as a straight line parallel to this one, at a distance of $3$ cm. Divide the segment into three equal parts using a double-sided ruler, without using a compass.

A Cartesian coordinate system is given on the plane. Does there exist a circle on which exactly one point with rational coordinates lies?

To protect valuable equipment from bad weather, it is required to make a tent in the form of a pyramid, at the base of which a rectangle should lie, and one of the side ribs should be perpendicular to the base. Find the largest possible volume of the tent, provided that none of the edges of the pyramid should be longer than $2$ meters.

Point $D$ is taken on the  side $AB$ of the triangle $ABC$. In the triangle $ADC$, bisectors $AP$ and $CQ$ are drawn. A point $R$ is taken on the side $AC$ of triangle $ADC$  so that $PR \perp CQ$. It is known that the bisector of angle $D$ of triangle $BCD$ is perpendicular to the segment $PB, AB = 18, AP = 12$. Calculate $AR$.

2010 High Standards Olympiad 11.10
All edges of a regular quadrangular pyramid $SABCD$ are equal to $b$. The height of the regular quadrangular prism $ABCDA_1B_1C_1D_1$ is equal to the height of the indicated pyramid, the base $ABCD$ they have in common. 
i) Through points $A$ and $C_1$ the plane $\alpha$ is drawn parallel to the straight line $BD$. Find the area of the polygon formed at the intersection of the plane $\alpha$ and the pyramid SABCD . 
ii) Plane $\beta$ is drawn through points $B$ and $D_1$. Find the smallest possible area of the polygon formed at the intersection of the plane $\beta$ and the pyramid $SABCD$.

original wording
Все ребра правильной четырехугольной пирамиды SABCD равны $b$. Высота правильной четырехугольной призмы ABCDA_1B_1C_1D_1 равна высоте указанной пирамиды, основание ABCD у них общее. 
(1) Через точки А и С_1 проведена плоскость $\alpha$ параллельно прямой BD. Найдите величину площади многоугольника, образовавшегося при пересечении плоскости $\alpha$ i и пирамиды SABCD. 
(2) Через точки В и D_1 проведена плоскості, $\beta$. Найдите минимально возможную величину площади многоугольника, образовавшегося при пересечении плоскости  $\beta$ и пирамиды SABCD.

In a convex quadrilateral $ABCD$, $AB = 1, BC = 2, CD = 4, DA = 3$. Extensions of sides $AB$ and $CD$ beyond points $B$ and $C$, respectively, intersect at point $E$. Extension of sides $AD$ and $BC$ beyond points $A$ and $B$, respectively, intersect at point $F$. Find $AF - BF + BE - CE$.

In the plane given are a disjoint square with side $2$ and a circle of radius $3$. Find the maximum distance between the midpoints of segments $AB$ and $CD$ such that points $A$ and $C$ lie on the square, and points $B$ and $D$ lie on the  circle.

The centers of three spheres with radii $1, 2, 3$ form an isosceles triangle with side $100500$. Find the locus of the intersection points of the medians of triangles $ABC$ such that point $A$ lies on the first sphere, point $B$ lies on the second sphere, and point $C$ lies on the third sphere.

2012 High Standards Olympiad 9.4
Each of the four circles passes through the three vertices of a given parallelogram. Find the area of the quadrangle formed by the centers of the circles when parallelogram has an angle of $45^o$ and an area of $2$

Each of the four circles passes through the three vertices of a given parallelogram. Find the area of the quadrangle formed by the centers of the circles when parallelogram has $2/ 3$ aspect ratio of sides  and an area of $1$.

The tangential quadrilateral $ABCD$ is divided by the diagonal $AC$  into two similar, but not equal, triangles. What can be the length of the $AC$ diagonal if the lengths of the sides $AB$ and $CB$ are $5$ and $10$, respectively?

In the acute-angled triangle $ABC$, the heights $AA_1, BB_1, CC_1$ are drawn. On the side $AB$, the point $P$ is chosen so that the circle circumscribed around the triangle $PA_1 B_1$ is tangent to side $AB$ Find $PC_1$ if $PA = 30$ and $PB = 10$.

2013 High Standards Olympiad 10.3
Find the lateral side of the isosceles triangle with base $1$, in which the bisector, median and the segment of the opposite side enclosed between them also form an isosceles triangle.

2013 High Standards Olympiad 11.6
Given two tall cylindrical cups of radii $r$ and $R, r <R$, the wide one was placed on a horizontal table, and the narrow one was placed on it in every possible way so that it rests on the wide edge with two points of its edge and one point on the side surface (see figure). Describe the geometrical locus of the points of space at which the top point of the edge of the narrow glass in contact with the wide can be at the same time.
2014 High Standards Olympiad 8.6
Lines containing the heights of the non-isosceles triangle $ABC$ intersect at point $H$. $I$ is the center of the inscribed circle of triangle $ABC$, and $O$ is the center of the circumscribed circle of the triangle $BHC$. It is known that point $I$ lies on the segment $OA$. Find the angle $BAC$.

2014 High Standards Olympiad 9.2
Point $B$ is the midpoint of the segment $AC$. Square $ABDE$ and equilateral triangle  $BCF$ are located in the same half-plane wrt the line $AC$. Find  the measuere of the acute angle between the lines $CD$ and $AF$.

2014 High Standards Olympiad 10.4
Point $O$ is the center of the circumscribed circle of an acute-angled triangle $ABC$. The straight line $AO$ intersects the $BC$ side at point $P$. The points $E$ and $F$ on the sides $AB$ and $AC$ are respectively selected so that a circle can be circumscribed around the quadrilateral $AEPF$. Prove that the length of the projection of the segment $EF$ on the side $BC$ does not depend on the choice of points $E$ and $F$.

2014 High Standards Olympiad 11.2
Through the vertices of a regular hexagon, $6$ different parallel lines are drawn. Could it be that all pairwise distances between these lines are integers?

2014 High Standards Olympiad 11.4
The polyhedron is inscribed in a sphere of radius $K$, and its volume is numerically equal to its surface area.
a. Prove that $K> 3$.
b. Could $K$ be greater than $1000$?

A triangle $ABC$ is given, in which $AB = BC$ and  $\angle ABC = 90^o$. The altitude BH is drawn. On side CA, point P is chosen such that $AP = AB$. On side $CB$, point $Q$ is chosen such that $BQ = BH$. Prove that lines $PQ$ and $AB$ are parallel.

Given a triangle $ABC, \angle B = 90^o$. On the sides $AC, BC$, points $E ,D$ are selected, respectively, such that $AE = EC$,  $\angle ADB = \angle EDC$. Find the ratio  $CD: BD$.

Three points $A, B, C$ are given, forming a triangle with angles of $30^o, 45^o, 105^o$. Two of these points are selected, and the perpendicular bisector is drawn to the segment connecting them, after which the third point is reflected relative to this perpendicular bisector . We get the fourth point $B$. With the resulting set of $4$ points, the same procedure is carried out - two points are selected, the perpendicular bisector is drawn and all points are reflected relative to it. What is the greatest number of different points that can be obtained as a result of repeated repetition of this procedure?

Given triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of sides $BC, AC, AB$, respectively. Prove that the three lines passing through these points and parallel to the bisectors of the opposite angles intersect at one point.

A circle is circumscribed around triangle $ABC$ with an angle $\angle B = 60^o$. Tangents to the circle  at points $A$ and $C$ intersect at point $B_1$. On rays $AB$ and $CB$, points $A_0$ and $C_0$ were noted respectively, such that $AA_0 = AC = CC_0$. Prove that the points $A_0, C_0, B_1$ lie on the same line.

2017 High Standards Olympiad 8.5 9.5
In the triangle $ABC, \angle B = 90^o, \angle A = 30^o$. Inscribed circle touches side $AB$ at point $P$, and side $AC$ at point $Q$, $M$ is the midpoint of the side $AC$. Prove that $PM = PQ$.

2017 High Standards Olympiad 9.3
Petya has a ruler $10$ cm long (using it it is impossible to draw segments longer than $10$ cm), and compass with a maximum solution of $6$ cm (using it, it is impossible to draw circles of radius greater than $6$ cm).There are no divisions on the ruler, (that is, they cannot measure distances). Two dots are drawn on a piece of paper. It is known that the distance between them is $17$ cm. Show how Petya can connect these points with a segment using only that ruler and compasses that he has.

2017 High Standards Olympiad 10.4
Inside the convex quadrilateral $ABCD$ are four circles of the same radius so that they have a common point and each of them is inscribed in one of quadrangle angles. Prove that quadrilateral $ABCD$ is cyclic.

2017 High Standards Olympiad 10.6
The heights $AA_1, BB_1, CC_1$ of the acute-angled triangle $ABC$ intersect at point $H$. Let $M$ be the midpoint of $BC, K$ the midpoint of $B_1C_1$. Prove that the circle passing through $K, H$ and $M$ is tangent to $AA_1$.

2017 High Standards Olympiad 11.3
A convex polyhedron has $8$ vertices and $6$ quadrangular faces. Can the projection of this polyhedron onto a certain plane turn out to be a regular octagon?

2018 High Standards Olympiad 7.4 8.4
Let a quadrangle  $ACDE$ be given such that the vertices $D$ and $E$ lie the same side wrt line $AC$. Let point $B$ be taken on the side $AC$,  so that the triangle $BCD$ is isosceles with base $BC$, i.e. $BD = CD$. Let the angles of the $BDC, ABE, ADE$ be equal to $80$ degrees. Find the angle $\angle EAD$.

2018 High Standards Olympiad 9.4 10.3. 11.3
The triangle $ABC$, with $AB> AC$, is inscribed in a circle centered at the point $O$. The heights $AA'$ and $BB'$ are drawn, and $BB'$ reintersects the circumscribed circle of triangle $ABC$ at point $N$. Let $M$ be the midpoint of the segment $AB$. Prove that if $\angle OBN = \angle N BC$, then the lines $AA', ON$, and $M B'$ intersect at one point.

2019 High Standards Olympiad 7.4
In the triangle $ABC$, in which all three sides are pairwise distinct, bisectors of angles $A$ and $B$ are
drawn, dividing it into a quadrangle and three triangles, two of which are isosceles. Find the angles
of the original triangle.

 2019 High Standards Olympiad 8.3.9.3
In a right triangle $ABC$, angle $B$ is right. On the leg $AB$, the point $M$ is chosen so that $AM = BC$, and on point $BC$, point $N$ is chosen so that $CN = M B$. Find the acute angle between the lines $AN$ and $CM$.

2019 High Standards Olympiad 10.6
In the acute-angled triangle $ABC$, the heights $AD, BE, CF$ are drawn, $H$ is the orthocenter. Circle with center at point $O$ passes through points $H$ and $A$, intersecting sides $AB$ and $AC$ at points $Q$ and $P$, respectively (point $O$ does not lie on sides $AB$ and $AC$). The circumscribed circle around the triangle $QOP$ touches side $BC$ at point $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}$

2019 High Standards Olympiad 11.5
From the vertices $A,B,C$ of a triangle $ABC$ pass three parallel lines $a,b,c$ respectively, not parallel to the sides of the triangle. Let $A_0,B_0,C_0$ be the midpoints $BC,CA,AC$ respectively. Let $A_1,B_1,C_1$ be the intersection points of the pairs $a$ and $B_0C_0, b$ and $C_0A_0, c$ and $A_0B_0$, respectively. Prove that lines $A_0A_1,B_0B_1$ and $C_0C_1$ are concurrent.

2020 High Standards Olympiad  7.4
In the convex quadrilateral $ABCD, AB = BC = CD$, and each of the diagonals is equal to a side. Find the angles of the quadrangle

2020 High Standards Olympiad  8.4
On the side $BC$ of the parallelogram $ABCD$, the point $M$ is chosen so that  each of the triangles  $ABM, AMD, CDM$ turned out to be isosceles. Find the angles of the parallelogram.

2020 High Standards Olympiad  9.2 10.2 11.2
The tangential quadrangle $ABCD$ is given for which the radii of the inscribed circles of the triangles $ABC$ and $ADC$ are equal. Find the angle between the diagonals $AC$ and $BD$.

2020 High Standards Olympiad 11.4
Points $P$ and $Q$ lie respectively on the sides $BC$ and $CD$ of the square $ABCD$. Lines $AP$ and $AQ$ intersect $BD$ at points $M$ and $N$, respectively, and the lines $PN$ and QM intersect at $H$. Prove that $AH \perp PQ$ if and only if the points $P, Q, M, N$ lie on one circle.

The area of the parallelogram $ABCD$ is 2 and its angle $A$ is $45^o$. Around triangles $ABC, BCD, CDA$, and $DAB$, circles were drawn with centers $K, L, M$, and $N$, respectively. Find the area of the quadrilateral $KLMN$.

Given a rectangle with side lengths $5$ and $6$. Divide it into seven non-overlapping rectangles with integer sides parallel to the sides of the original rectangle so that the areas of these seven rectangles are pairwise different.

Around the triangle $ABC$ with an angle $\angle B = 60^o$, a circle is circumscribed, The tangents to the circle drawn at points $A$ and $C$ intersect at point $B_1$. Points $A_0$ and $C_0$ are marked on the rays $AB$ and $CB$, respectively, such that $AA_0 = AC = CC_0$. Prove that points $A_0, C_0, B_1$ are collinear,

Through the vertices of triangle $ABC$, three parallel straight lines $a,b,c$ respectively, are drawn not parallel to the sides of the triangle. Let $A_0, B_0, C_0$ be the midpoints of $BC, CA, AB$. Let $A_1, B_1, C_1$ be the points of intersection of $a$ and $B_0C_0, b$ and $C_0A_0, c$ and $A_0B_0$. Prove that the lines $A_0A_1,B_0B_1$ and $C_0C_1$ intersect at one point.

Given an isosceles right triangle $ABC$ with right angle $B$. On side $AC$, a point $K$ is chosen such that the angle $CBK$ is $15^o$. A point $M$ is marked on the ray $BK$ such that that the angle $ACM$ is $90^o$. Prove that $AC=BM$.

Point $K$ is marked in parallelogram $ABCD$ such that $AB=BK=KC$. Prove that the center of the parallelogram is equidistant from the midpoints of all sides of triangle $AKD$.

In triangle $ABC$, points $A_1$, $B_1$, $C_1$ are the midpoints of sides $BC$, $AC$, $AB$, respectively. Points $A_2$, $B_2$, $C_2$ are the midpoints of broken lines $BAC$, $ABC$, $ACB$, respectively (point is called the midpoint of a broken line if it belongs to a broken line and divides it into two broken lines equal length). Prove that lines $A_1A_2$, $B_1B_2$, $C_1C_2$ pass through one point.

$M$ is the midpoint of side $BC$ of triangle $ABC$. Tangents drawn from $M$ to the inscribed circles of triangle $ABC$ are tangent to this circle at points $P, Q$. The tangents from $M$ to the excircle $ABC$, the one tangent to side $BC$, are tangent of this circle at the points $R, S$. The lines $PQ$, $RS$ intersect at the point $X$. It turned out that that $AH = AM$. Find the angle $\angle BAC$.

The hypotenuse $AB$ of right triangle $ABC$ is tangent to the inscribed and corresponding excircles at points $T_1$, $T_2$, respectively. The circle passing through the midpoints of the sides, touches the same circles at the points $S_1$, $S_2$, respectively. Prove that $\angle S_1CT_1=\angle S_2CT_2$.




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