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Lomonosov Tournament 1978 - 2022 (Russia) 95p

geometry problems from  Lomonosov Tournament  (Russian)
with aops link

started in 1978
since 2013 it has a 2nd, final round

collected inside aops round 1 here and finals here


2013-2022 Final Round

The extensions of the opposite sides $AB$ and $CD$ of the inscribed quadrilateral $ABCD$ intersect at point $P$, and the sides $BC$ and $AD$ intersect at point $Q$. The bisector of the angle $APD$ intersects the line $AD$ at point $K$. the bisector of the angle $AQB$ intersects the line $AB$ at point $L$. Prove that the circles circumscribed around the triangle $ALK$ and the quadrilateral $ABCD$, are tangent.
In triangle $ABC$, $\angle B=90^o$, point $M$ lies on the side $BC$  such that $BM: MC = 1: 2$, point $N$ is the midpoint of $AC$. Prove that the  $\angle AMB=  \angle CMN$

In the square $ABCD$, the square $KLMN$ was inscribed, as in the figure. We marked the centers of the inscribed circles of the resulting right-angled triangles. Prove that they are the vertices of a square equal to $KLMN$.

2016 Lomonosov Tournament Final p2
Let $ABC$ be triangle with angle $B$  obtuse. A point $D$ is marked on the plane such that $\angle BDC -  \angle ADB = 2 \angle BAC$,  $\angle ABD = 90^o$, with points $C$ and $D$ lying on one side of the line $AB$. Prove that $AD = CD$.

2017 Lomonosov Tournament Final p4
In a right triangle $ABC$, angle $B$ is a right , $AB> BC$. A circle $\Omega$ is circumscribed around the triangle $ABC$ . The tangent to $\Omega$ at point $B$ intersects the line $AC$ at point $Q$. Point $M$ is the midpoint of the arc $AB$ of circle $\Omega$ not containing point $C$. The line $QM$ intersects the circle $\Omega$ at points $M$ and $P$. The tangent to $\Omega$ at  $P$ intersects the line $BC$ at point $D$. Prove that the angle $QDC$ is right .

2018 Lomonosov Tournament Final p2
In an acute-angled scalene triangle $ABC, \angle A=60^o$ and $BD, CE$ are altitudes. Calculate $\frac{AB -  AC}{BE - CD}$

$2018$ points are marked on the plane so that any line contains at most two marked points. Prove that there are $1009$ pairwise intersecting lines such that each marked point lies on one of these lines.

Two non intersecting circles of radius $1$ and $3$ are inscribed in the angle $\angle POQ$, where $P$ is the point of contact of the side of the corner with the first circle, and $Q$ is the point of contact of the other side of the angle with the second circle. The common internal tangent of circles $AB$ intersects the ray $OP$ at point $A$, and the segment $OQ$ at point $B$. Find $AB$ if $OP = 3$.

A set of unit circles is drawn on the plane so that each line on the plane intersects at least one circle. Is there a line that intersects infinitely many circles?

A triangle is drawn on the coordinate plane with vertices at points with integer coordinates. It is known that inside this triangle there is exactly one point with integer coordinates. What is the largest number of points with integer coordinates (excluding vertices) can lie on the sides of this triangle?

2020 Lomonosov Tournament Final p2
A circle $\omega$ with center $O$ is inscribed in the rhombus $ABCD$. The points $P$ and $Q$ are chosen on the sides $BC$ and $CD$, respectively, so that $PQ$ is tangent to $\omega$ at the point $L$. We denote the point of tangency $\omega$ with the side $CD$ by $K$. Prove that the area of the triangle $PQD$ is equal to the area of the quadrilateral $OLQK$.
Let $O$ be the center of the circumscribed circle of an acute-angled triangle ABC. Points $P$ and $Q$ are marked on sides $AB$ and $AC$, respectively. It turned out that the circumcircle of the triangle $APO$ touches the straight line $BO$, the circumcircle $AQO$ touches the straight line $CO$, and the perimeter of the triangle $APQ$ is equal to $AB + AC$. Find the angle $\angle BAC$.

Let $O$ be the center of the circumscribed circle, $G$ the intersection point of the medians of an acute triangle $ABC$. A line perpendicular to $OG$ passing through point $G$ intersects segment $BC$ at point $K$. The tangent to the circumscribed circle of triangle$ ABC$ at point A intersects line $KG$ at point $ L$. Find the measure of the angle $\angle  ACB$ if $\angle LOK = 155^o$ , and $ \angle ABC = 53^o$.


1978 - 2021 [1st round]

A sphere of unit radius is tangent to all edges of some triangular prism. What is the volume of this prism?

Nikita drew and filled in a convex pentagon with perimeter $20$ and area $21$. Tanya painted over all points that are at a distance of no more than $1$ from painted over by Nikita (see fig.). By how much has the shaded area increased?


It is known that if a regular $N$-gon inside a circle extends all sides to the intersection with this circle, then $2N$ segments added to the sides can be divided into two groups with the same sum of lengths. Is a similar statement true for inside the sphere
a) an arbitrary cube?
b) an arbitrary regular tetrahedron?

(Each edge is extended in both directions until it intersects with the sphere. A line segment is added to each edge on both sides. It is required to paint each of them is either red or blue, so that the sum of the lengths of the red the segments were equal to the sum of the lengths of the blue ones.)

Explain your answers to the previous questions.
King Arthur wants to order the blacksmith a new knight's shield according to his sketch. King took a compass and drew three arcs with a radius $1$ yard as shown in the picture. What is the area of the shield?
The height of each step of the “stairs” (see figure) is $1$, and the width of each step increases from one to $2019$. Is it true that the segment from the lower left point of the stairs to the upper right point of the stairs doesn’t cross the stairs?
You need to divide the curvilinear triangle in the picture into $2$ parts of equal area, having drawn one line with a compass. This can be done by selecting one of the marked points as the center and drawing an arc through another marked point. Find a way to do this and prove that it is a suitable solution.
Is there a triangular pyramid among six edges of which
a) two edges have the length of less than $1$ cm, and the other four edges more than $1$ km?
b) four edges have the length of less than 1 cm, and the other two edges more than $1$ km?

Lyosha drew a geometric picture outlining his plastic right triangle for $4$ times, placing the short leg (cathetus) to the hypotenuse and superimposing the vertex of the acute angle with the vertex of the right angle (see the pic.). It turns out that the “closing” fifth triangle is isosceles (see the pic., the marked (!) sides are equal). Find the size of the angles of Lyosha’s triangle.

(Kazitsina T.V.)

It’s easy to paste over the surface of a cube with $6$ rhombuses, i.e. with $6$ squares which match the faces. Is it possible to paste over the surface of the cube (without any gaps or overlaps) with less than $6$ rhombuses (not necessarily congruent)?

(Shapovalov A.V.)

In a convex quadrilateral two opposite sides are equal and perpendicular, and the two others are equal to $a$ and $b$. Find the area of the quadrilateral.
(Bakaev E.V.)

Cut the regular tetrahedron into equal polyhedrons with six faces.

(Merzon G.)

Six equilateral triangles are located, as in the figure. Prove that the sum of the areas of the shaded triangles is equal to the sum of the areas of the filled triangles.
(Bakaev E.V.)

A square frame was laid on the ground, and a vertical pole was installed in the center of the square. When the fabric was pulled on top of this structure, a small tent was formed. If you put two of the same frames side by side, put a vertical pole of the same length in the center of each and pull the fabric on top, you get a big tent. A small tent took $4$ square meters of fabric. And how much fabric is needed for a large tent?

(M. Raskin)

Let us consider polyhedrons that possess the following property: For any two vertices of such polyhedron it is possible to find a third vertex such that these three vertices together form an equilateral triangle. The regular tetrahedron has this property. Are there any other polyhedrons like this?

(Bakaev E.V.)

The four segments marked on the sides of the square are identical. (See the picture.) Prove that the two marked angles are of the same size.
(Bakaev E.V.)

The “Young Geometer” building kit contains several 2D polygons. Alexander, a geometry student, used the kit to build a 3D convex polyhedron. Next, Alexander disassembled the polyhedron and divided the polygons into two groups. Can it be possible that all polygons of each group can be assembled to a convex polyhedron so that each polygon from a given group is a face of the corresponding polyhedron and each of its faces is a polygon from this group?

The snowflake on the figure below has rotational symmetry: if rotated $60^o$ around point $O$, the snowflake will match itself. It also has reflectional symmetry with respect to the line $OX$. Find $OX:XY$ , the ratio of the lengths of the segments $OX$ and $XY$ . (Dotted lines connect points that belong to the same lines.)
(M. Raskin)

An isosceles triangle with a $120^o$ angle is made out of $3$ layers of folded paper. When it was unfolded, we got a rectangular piece. Draw such a rectangle and plot the folding lines on it.

(Merzon G.)

A side of a rectangle of area $14$ divides a side of a square at the ratio $1:3$ (see the figure). Find the area of the square.
(Golenishcheva-Kutuzova T.I.)

A convex quadrilateral is drawn on a blackboard. Three boys made one claim each: Alexey said, “This quadrilateral can be cut by its diagonal into two acute triangles”. Boris replied: “This quadrilateral can be cut by its diagonal into two right triangles”. And Charlie concluded: “This quadrilateral can be cut by its diagonal into two obtuse triangles”. It turned out that just one of them was wrong. Name the boy who certainly was right, and prove that he was.

(Frenkin B.R.)

A diagonal of a $1\times 4$ rectangle is drawn on checkered paper. Show how, using only the ruler without divisions, divide this segment into three equal parts.

Two round coins were put on the left side of the scale, and another one on the right, so that the scale was in balance. And which of the bowls will outweigh if each of the coins is replaced with a ball of the same radius? (All balls and coins are made entirely of the same material, all coins have the same thickness.)

(Halperin G.A.)

Two balls of radii $3$ and $5$ were put on the left side of the scale, and one ball of radius $8$ on the right one. Which of the bowls will outweigh? (All balls are made entirely of the same material.)

Draw a polygon and a point on its border so that any line passing through this point divides the area of this polygon in half.

The Egyptians calculated the area of a convex quadrangle using the formula $(a+c)(b+d)/4$ , where $a,b,c,d$ are the lengths of the sides in the round-trip order. Find all the quadrangles for which this formula is true.

(Sergeev P.V.)

The figure shows the figure of $ABCD$ . The sides $AB,CD$ and $AD$ of this figure are segments (moreover, $AB\parallel CD$ and $AD\perp CD$), $BC$ is an arc of a circle, and any tangent to this arc cuts off a trapezoid or rectangle from the figure. Explain how to draw a tangent to the arc $BC$ so that the cut-off shape has the largest area.
Little Petya sawed all the legs of a square stool and lost four pieces of sawn off. It turned out that the lengths of all the pieces are different, and that after that the stool is on the floor, albeit obliquely, but still touching the floor with all four ends of the legs. Grandfather decided to fix the stool, but found only three pieces with lengths of $8,9$ and $10$ cm. How long can a fourth piece be?

There are three triangles: acute-angled, right, and obtuse. Sasha took one triangle for himself, and Borya took the two remaining ones. It turned out that Borya can attach (without imposing) one of his triangles to another, and get a triangle equal to Sashin. Which of these triangles did Sasha take?

The sum of the three positive angles is $90^o$ . Can the sum of the cosines of two of them be equal to the cosine of the third?

 On the board was a pentagon inscribed in a circle. Masha measured his angles and it turned out that they are $80^o$,$90^o$,$100^o$,$130^o$ and $140^o$ (in that order). Was Masha wrong?

Is there a polyhedron whose all faces are isosceles right triangles?

Given a triangle with sides $AB=2$,$BC=3$,$AC=4$. It's inscribed circle touches $BC$ at point $M$. Connect point $M$ with the point $A$ . Circles are inscribed in the triangles $AMB$ and $AMC$ . Find the distance between their points of contact with the straight line $AM$ .

Is there a tetrahedron whose all faces are isosceles triangles, and no two of them are congruent?

Four vertices of the square are marked. Mark four more points so that on all the perpendiculars bisector of the segments with ends at the marked points, there are two marked points.

The polyhedron is inscribed in a sphere. Could this polyhedron be non-convex? (A polyhedron is inscribed in a sphere if all the ends of its edges lie on the sphere.)

A square with side $1$ is given. Each side is divided into three equal parts. Segments are drawn through the division points (see fig.). Find the area of the hatched square.
Given a straight line and a point outside it. How to use a compass and a ruler to construct a straight line parallel to a given line and passing through a given point, while drawing as few lines as possible (circles and lines), so that the last line drawn is the line you are looking for? How many lines did you manage to achieve?

Three equal triangles were cut into opposite medians (see fig. ). Is it possible to create one triangle from the resulting six triangles?
The base of the Cheops pyramid is a square, and its side faces are equal isosceles triangles. Pinocchio climbed up and measured the angle of the face at the top. It turned out $100^o$ . Could this be so?

Dunno thinks that only an equilateral triangle can be cut into three congruent triangles. Is he right?

Perpendiculars to the sides are drawn from point $M$ inside the quadrilateral $ABCD$ . The feet of the perpendiculars lie inside the sides. Denote these points: the one that lies on the side $AB$ passes through $X$ , the one that lies on the side $BC$ passes through $Y$ , the one that lies on the side of the $CD$ passes through $Z$ , the one that lies on the side of $DA$ passes through $T$ . It is known that $AX\ge XB$, $BY\ge YC$,$CZ\ge ZD$, $DT\ge TA$ . Prove that around the quadrilateral $ABCD$ one can circumscribe a circle.

Triangle $ABC$ is inscribed in a circle. Point $D$ is the midpoint of the arc $AC$ , points $K$ and L are selected on the sides $AB$ and $CB$, respectively, so that $KL$ is parallel to $AC$ . Let $K'$ and $L'$ be the intersection points of the lines $DK$ and $DL$, respectively, with the circle. Prove that a circle can be circumscribed around the quadrangle $KLL'K'$.

Six identical parallelograms of area $1$ pasted a cube with edge $1$. Is it possible to say that all parallelograms are squares? Can we say that they are all rectangles?

$n$ paper circles of radius $1$ are laid on a plane so that their borders pass through one point, and this point is inside the entire area of the plane covered with circles. This area is a polygon with curved sides. Find its perimeter.

(Kozhevnikov P.A.)

 In triangle $ABC$, points $A',B',C'$ lie on sides $BC$,$CA$ ,$AB$, respectively. It is known that $\angle AC'B'=\angle B'A'C$, $\angle CB'A'=\angle A'C'B$, $\angle BA'C'=\angle C'B'A$ . Prove that the points $A',B',C'$ are the midpoints of the sides of the triangle $ABC$ .

(Arbitrary VV)

Is it possible to cut an equilateral triangle into five pairwise different isosceles triangles?

In triangle $ABC$, angle $A$ is $120^o$, point $D$ lies on the bisector of angle $A$ , and $AD=AB+AC$ . Prove that the triangle $DBC$ is equilateral.

Find the sum of the angles $MAN,MBN,MCN,MDN$, and $MEN$ drawn on grid paper as shown.
Several (finitely many) different chords were drawn in the circle so that each of them passes through the midpoint of some other chord. Prove that all of these chords are diameters of a circle.

The altitude of length $AB$ of the right trapezoid $ABCD$ is equal to the sum of the lengths of the bases $AD$ and $BC$ . In what ratio does the bisector of angle B divide the side $CD$?

$M_a,M_b,M_c$ are midpoints of the sides, $H_a,H_b,H_c$ are feet of altitudes of triangle $ABC$, of area $S$ . Prove that from the segments $M_aH_b,M_bH_c,M_cH_a$ you can construct a triangle, and find its area.

The rectangle $ABCD$ ( $AB=a$,$BC=b$ ) was folded so that a pentagon of area $S$ was formed ( $C$ lay in $A$ ). Prove that $S<3/4ab$ .

Two circles are given on the plane, one inside the other. Construct a point $O$ such that one circle is obtained from another by homothety with respect to the point $O$ (in other words, so that stretching the plane from the point $O$ with a certain coefficient transfers one circle to another).

The vertices $A,B,C$ of the triangle are connected to the points $A_1,B_1,C_1$ lying on opposite sides (not at the vertices). Can the midpoints of segments $AA_1,BB_1,CC_1$ lie on one straight line?

In the triangle $ABC$, angle $A$ greater than the angle $B$ . Prove that the length of side $BC$ is greater than half the length of side $AB$ .

In the triangle $ABC$ on the side $AB$, a point $D$ is chosen such that $\frac{AD}{DC}=\frac{AB}{BC}$. Prove that angle $C$ is obtuse.

Circle $\omega_2$ is drawn through the center of circle $\omega_1$, $A$ and $B$ are the intersection points of the circles. The tangent to circle $\omega_2$ at point $B$ intersects circle $\omega_1$ at point $C$. Prove that $AB=BC$.

The vertices of the equilateral triangle $MNP$ are located on the sides $AB,CD$ and $EF$ of the regular hexagon $ABCDEF$. Prove that the triangle $MNP$ and the hexagon $ABCDEF$ have a common center.

(Sedrakyan N.)

Is it possible to draw $12$ circles on a plane so that each touches exactly five others?

Three pairwise intersecting circles are given on the plane. A line is drawn through the intersection points of each two of them. Prove that these three lines intersect at one point or are parallel.

Let $a,b,c$ be the lengths of the sides of the triangle, $A,B,C$ are the values of opposite angles.
Prove that $a A+b B+cC  \ge aB+b C+cA$.

Restore
a) a triangle,
b) a pentagon
given the midpoints of its sides.

Two circles and a point are given. Draw a segment whose ends lie on given circles, and the midpoint is at a given point.

In a triangle, two altitudes are no less than the sides on which they are drawn. Find the angles of the triangle.

In the circle marked the point. Cut the circle into
a) three parts
b) two parts
so that it is possible to make a new circle from them, with the marked point in the center.

A convex pentagon is given. Each diagonal cuts off a triangle from it. Prove that the sum of the areas of the triangles is greater than the area of the pentagon.

Prove that the sum of the diagonals of a convex quadrilateral is less than its perimeter, but larger than the half-perimeter.

Brother and sister divide a triangular cake like this: he points a point on the cake, and she draws a straight line passing through this point and selects a piece for herself. Everyone wants to get a piece as much as possible. Where should the brother put an end to? What part of the cake will each receive in this case?

On the plane, four points are given that do not lie on one straight line. Prove that there is a triangle not acute with vertices at these points.

Let $a,b,c,$d be the sides of the quadrilateral (in any order),let $S$ be its area. Prove that $S\le 1/2(ab+cd)$.

Through this point on the plane, all sorts of straight lines are drawn that intersect the given circle. Find the geometric locus of the midpoints of the resulting chords.

$a_1,a_2,a_3,a_4,a_5,a_6$ are consecutive sides of a hexagon, all angles of which are equal. Prove that $a_1-a_4=a_3-a_6=a_5-a_2$ .

The point inside the square was connected with the vertices. Four triangles were obtained, one of which is isosceles with angles at the base (side of the square) of $15^o$. Prove that the opposite triangle is equilateral.

Find the sum of the angles at the vertices of a self-intersecting five-pointed star.

The vertices of the parallelogram $A_1B_1C_1D_1$ lie on the sides of the parallelogram $ABCD$ (point $A_1$ lies on side $AB$ , point $B_1$ on side $BC$ , etc.). Prove that the centers of both parallelograms coincide.

Four houses are located on a circle. Where do you need to dig a well so that the sum of the distances from the houses to the well is the smallest?

Is it possible to draw five rays from one point on the plane so that among the angles they form are exactly four acute? The angles are considered not only between adjacent, but also between any two rays.

Two circles intersect the line. Prove that $\angle ABC=\angle DEM$ .
The point $M$ inside the convex quadrilateral $ABCD$ is such that the areas of the triangles $ABM,BCM,CDM$, and $DAM$ are equal. Is it true that $ABCD$ is a parallelogram, and the point $M$ is the intersection point of its diagonals?

Can a quadrilateral with sides $10,11,12,13$ be a trapezoid ?

a) Show that any triangle can be cut into several parts, from which a rectangle can be folded.
b) Show that any rectangle can be cut into several parts from which a square can be folded.
c) Is it true that any polygon can be cut into several parts from which a square can be folded?

Is there a convex $1978$-gon in which all angles are expressed by an integer number of degrees?



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