League of Winners Tournament 2014, 2016-17 (Russia) 23p

 geometry problems from League of Winners Tournament of Mathematical Battles (Russia), with aops links in the names

Лига Победителей

collected inside aops here

juniors inside aops here

2014, 2016-2017

it lasted only these 3 years

2014 1st League of Winners Tournament Juniors 1.9

$n$ points are marked on the plane -- all vertices of some convex polygon and some points inside it. For every two of these points on the segment connecting them, as on a diameter, a circle is constructed. Prove that every point inside the polygon is covered by at least $n-1$ circles.

MathOverflow

2014 1st League of Winners Tournament Juniors 2.9

Given a square $ABCD$. It contains two conguent convex shapes, one of which contains $A$ and the other contains $C$. These figures have exactly one common point $P$. Find all possible positions of the point $P$.

D. Belov

2014 1st League of Winners Tournament Juniors 3.2

Point $H$ is the orthocenter of acute-angled triangle $ABC$. Circles $\omega_1$ and $\omega_2$ are circumscribed around triangles $AHB$ and $AHC$. The circle $\Gamma$ passes through the points $B$ and $C$, and also intersects the segments $BH$ and $CH$ for second time at points $B '$ and $C'$. Finally, the circle $\gamma$ touches $\omega_1$ and $\omega_2$ externally, and also touches the arc $BC$ of the circle $\Gamma$ internally , lying inside $ABC$, at the point $T$. Prove that $T$ is the midpoint of arc $B'C '$ of the circle $\Gamma$.

I. Bogdanov

2014 1st League of Winners Tournament Juniors 3.10

The points $A_1$, $\dots$, $A_ {2n}$ and $M$ lie on the circle. Points $A_1$, $\dots$, $A_ {2n}$ are divided into pairs and a straight line is drawn through each pair. Prove that the product of the distances from $M$ to all $n$ drawn lines does not depend on the division of points into pairs.

folklore, generalized by P. Kozhevnikov

2014 1st League of Winners Tournament Juniors 4.7

Let $L$ and $K$ be the feet of the internal and external bisectors of the angle $A$ acute-angled triangle $ABC$. Let $P$ be the intersection point tangent to its circumcircle at points $B$ and $C$. Perpendicular from point $L$ on line $BC$ intersects line $AP$ at point $Q$. Prove that $Q$ lies on the midline of the triangle $LKP$.

F. Ivlev

2014 1st League of Winners Tournament Juniors 4.10

The sum of the diagonals of a convex quadrilateral is $2$, and the sum of two opposite sides is equal to $a$. Find the largest possible area of this quadrangle.

I. Bogdanov

2014 1st League of Winners Tournament Seniors 1.1

$n$ lines are chosen on the plane, no two of which are parallel, no three intersect at the same point, and no four touch the same circle. For each three of them we inscribe a circle into the triangle they form. What can be the number of such circles, intersected by exactly $k$ chosen lines?

                                                                                                                                            D.Q. Naiman

2014 1st League of Winners Tournament Seniors 1.7

In the trapezoid $ABCD$, the base of $AB$ is less than the base of $CD$. Point $K$ is chosen so that ${AK \parallel BC}$ and $BK \parallel AD$. Points $P$ and $Q$ are selected on rays $AK$ and $BK$, respectively, so that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$. The circumscribed circles of triangles $APD$ and $BCQ$ intersect at points $U$ and $V$. Prove that one of the points $U$ and $V$ lies on the line $PQ$.

F. Ivlev

2014 1st League of Winners Tournament Seniors 2.5

Circle $\omega$ inscribed in quadrilateral $ABCD$ touches sides $AB$ and $BC$ at points $E$ and $F$, respectively. Diagonal $AC$ intersects the circle $\omega$ at the points $P$ and $Q$ (the point $Q$ lies on segment $AP$). Prove that lines $BD$, $EP$ and $FQ$ intersect at one point.

Kozepiskolai Matematikai Lapok, 2013:09

2014 1st League of Winners Tournament Seniors 3.3

A convex polyhedron $K$ is given in coordinate space. Is it true, that one can necessarily choose such a (possibly degenerate) a parallelepiped $P$ lying in $K$ such that the number of integer points inside $P$ and inside $K$ differ by no more than a million times?

I.Bogdanov, MathOverflow

2014 1st League of Winners Tournament Seniors 3.5

A convex polygon $K$ is drawn on checkered paper. Prove that it contains a parallelogram (possibly degenerate) covering at least $1/1000$ of all nodes in $K$.

I.Bogdanov, MathOverflow

2016 2nd League of Winners Tournament Juniors 1.1

Angle bisectors of $ACB$, $ADB$, $CBD$ and $CAD$ are drawn in the inscribed quadrilateral $ABCD$. They intersect the sides of the quadrilateral at points $X$, $Y$, $Z$ and $T$. Prove that the points $X$, $Y$, $Z$ and $T$ lie on the same circle.

Kozepiskolai Matematikai Lapok, 2016 No 1, B.4765

2016 2nd League of Winners Tournament Juniors 1.9

In the convex polygon $P$, a chord is chosen (any segment connecting two points on the perimeter) of the smallest length, dividing the area in half. Could any of the angles formed this chord with the sides to which it is adjacent, be less than $45 ^ \circ$?

I. Bogdanov

2016 2nd League of Winners Tournament Juniors 2.5

$n\geq 3$ points in general position are given in the plane. For each calculated the sum of the distances to the others, and all these sums turned out to be equal. Prove that the points are vertices of a convex polygon.

MathOverflow

2016 2nd League of Winners Tournament Juniors 3.8

Is there a finite set of points on the plane such that each point has nearest at least $4$ ?

MathOverflow

2016 2nd League of Winners Tournament Juniors 4.1

The altitudes $BP$ and $CQ$ are drawn in an acute-angled triangle $ABC$. Point $M$ is the midpoint of $BC$. The circumscribed circle of the triangle $CMP$ touches the side $AB$. Prove that the circumcircle of triangle $BMQ$ touches the straight line $AC$.

2016 2nd League of Winners Tournament Seniors 3.6

In an acute-angled triangle $ABC$ any two sides differ at least $d$. Prove that the sum of the areas of the triangles $AIO$, $BIO$ and $CIO$ at least $rd$, where $I$ and $O$ are the centers of the inscribed and the circumcircle, and $r$ is the radius of the inscribed circle.

(very deep processing of the Czech and Slovak Olympiad 2015 III-A p5)

2017 3rd League of Winners Tournament Juniors 2.3

Point $D$ is selected inside triangle $ABC$. It turned out that circles $S_1$ and $S_2$ inscribed in triangles $ABD$ and $CBD$ respectively, touch each other. Denote by $K$ the intersection point of common external tangents to $S_1$ and $S_2$. Prove that $K$ lies on the line $AC$.

S. Berlov

2017 3rd League of Winners Tournament Juniors 3.9

Let $O$ be the center of the circumscribed circle of an acute-angled triangle $ABC$. Let the circle with the center $O_1$ circumscribed around the triangle $OAB$, intersects $BC$ for second time at the point $D$, and the circle with the center $O_2$, circumscribed around the triangle $OAC$ intersects $BC$ for second time at the point $E$. The perpendicular bisector of the segment $BC$ meets $AC$ at the point $F$. Prove that the center of the circumcircle of triangle $ADE$ lies on $AC$ if and only if the points $O_1$, $O_2$ and $F$ are collinear.

2017 3rd League of Winners Tournament Seniors 1.8

In a triangle, denote by $f$ and $g$ the lengths of segments dividing the angle between sides with lengths $a$ and $b$ into $3$ equal parts. Prove that ${1\over a}+{1\over b}<{1\over f}+{1\over g}$.

D. Shiryaev, based on Kozepiskolai Matematikai Lapok, 2017, No 3, B.4865

2017 3rd League of Winners Tournament Seniors 2.10

For each parallelogram $OABC$ of area $1$, where $O$ is the beginning coordinates, and the coordinates of the remaining vertices are non-negative integers numbers, calculate the value $OA + OC-OB$. Prove that the sum of such values for different parallelograms are not more than $2$.

N. Kalinin, M. Shkolnikov

2017 3rd League of Winners Tournament Seniors 3.9

Is there a set of $10$ circles on the plane, for which will be at least $20$ points, through each of which passes at least three circles?

Inspired by MathOverflow

sourceσ: http://www.guas.info/competit/league.html ,  https://2cyr.com/decode/?lang=ru