Found problems: 83
Ukrainian TYM Qualifying - geometry, 2013.17
Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .
Ukrainian TYM Qualifying - geometry, 2016.1
The points $K$ and $N$ lie on the hypotenuse $AB$ of a right triangle $ABC$. Prove that orthocenters the triangles $BCK$ and $ACN$ coincide if and only if $\frac{BN}{AK}=\tan^2 A.$
Ukrainian TYM Qualifying - geometry, 2010.6
Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.
Ukrainian TYM Qualifying - geometry, IX.12
Let $AB,AC$ and $AD$ be the edges of a cube, $AB=\alpha$. Point $E$ was marked on the ray $AC$ so that $AE=\lambda \alpha$, and point $F$ was marked on the ray $AD$ so that $AF=\mu \alpha$ ($\mu> 0, \lambda >0$). Find (characterize) pairs of numbers $\lambda$ and $\mu$ such that the cross-sectional area of a cube by any plane parallel to the plane $BCD$ is equal to the cross-sectional area of the tetrahedron $ABEF$ by the same plane.
Ukrainian TYM Qualifying - geometry, VI.2
Let $A_1,B_1,C_1$ be the midpoints of the sides of the $BC,AC, AB$ of an equilateral triangle $ABC$. Around the triangle $A_1B_1C_1$ is a circle $\gamma$, to which the tangents $B_2C_2$, $A_2C_2$, $A_2B_2$ are drawn, respectively, parallel to the sides $BC, AC, AB$. These tangents have no points in common with the interior of triangle $ABC$. Find out the mutual location of the points of intersection of the lines $AA_2$ and $BB_2$, $AA_2$ and $CC_2$, $BB_2$ and $CC_2$ and the circumscribed circle $\gamma$. Try to consider the case of arbitrary points $A_1,B_1,C_1$ located on the sides of the triangle $ABC$.
Ukrainian TYM Qualifying - geometry, 2010.15
On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.
Ukrainian TYM Qualifying - geometry, XII.2
The figure shows a triangle, a circle circumscribed around it and the center of its inscribed circle. Using only one ruler (one-sided, without divisions), construct the center of the circumscribed circle.
Ukrainian TYM Qualifying - geometry, 2018.16
Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.
Ukrainian TYM Qualifying - geometry, 2011.11
Let $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, which intersect its angle bisector $AL$ at two different points $P$ and $Q$, respectively. Denote by $F$ such a point that $PF\parallel AB$ and $QF\parallel AC$, and by $T$ the intersection point of the tangents drawn at points $B$ and $C$ to the circumscribed circle of the triangle $ABC$. Prove that the points $A, F$ and $T$ lie on the same line.
Ukrainian TYM Qualifying - geometry, 2020.11
In the acute-angled triangle $ABC$, the segment $AP$ was drawn and the center was marked $O$ of the circumscribed circle. The circumcircle of triangle $ABP$ intersects the line $AC$ for the second time at point $X$, the circumcircle of the triangle $ACP$ intersects the line $AB$ for the second time at the point $Y$. Prove that the lines $XY$ and $PO$ are perpendicular if and only if $P$ is the foor of the bisector of the triangle $ABC$.
Ukrainian TYM Qualifying - geometry, II.18
Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.
Ukrainian TYM Qualifying - geometry, IV.7
Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.
Ukrainian TYM Qualifying - geometry, II.16
Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?
Ukrainian TYM Qualifying - geometry, 2016.15
A non isosceles triangle $ABC$ is given, in which $\angle A = 120^o$. Let $AL$ be its angle bisector, $AK$ be it's median, drawn from vertex $A$, point $O$ be the center of the circumcircle of this triangle, $F$ be the point of intersection of the lines $OL$ and $AK$. Prove that $\angle BFC = 60^o$.
Ukrainian TYM Qualifying - geometry, VI.9
Consider an arbitrary (optional convex) polygon. It's [i]chord [/i] is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than $\frac13$ the area of the polygon?
Ukrainian TYM Qualifying - geometry, VI.1
Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.
Ukrainian TYM Qualifying - geometry, I.8
One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of this triangle to the area of the quadrilateral formed by three such lines and one of the sides of the triangle.
Ukrainian TYM Qualifying - geometry, 2014.9
Construct a point $Q$ in triangle $ABC$ such that at least two of the segments $CQ, BQ, AQ$, divide the inscribed circle in half. For which triangles is this possible?
Ukrainian TYM Qualifying - geometry, V.3
Fix the triangle $ABC$ on the plane.
1. Denote by $S_L,S_M$ and $S_K$ the areas of triangles whose vertices are, respectively, the bases of bisectors, medians and points of tangency of the inscribed circle of a given triangle $ABC$. Prove that $S_K\le S_L\le S_M$.
2. For the point $X$, which is inside the triangle $ABC$, consider the triangle $T_X$, the vertices of which are the points of intersection of the lines $AX, BX, CX$ with the lines $BC, AC, AB$, respectively.
2.1. Find the position of the point $X$ for which the area of the triangle $T_x$ is the largest possible.
2.2. Suggest an effective criterion for comparing the areas of triangles $T_x$ for different positions of the point $X$.
2.3. Find the positions of the point $X$ for which the perimeter of the triangle $T_x$ is the smallest possible and the largest possible.
2.4. Propose an effective criterion for comparing the perimeters of triangles $T_x$ for different positions of point $X$.
2.5. Suggest and solve similar problems with respect to the extreme values of other parameters (for example, the radius of the circumscribed circle, the length of the greatest height) of triangles $T_x$.
3. For the point $Y$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $\Delta_Y$, the vertices of which are the points of intersection $AY, BX, CX$ with the circle $\omega$. Suggest and solve similar problems for triangles $\Delta_Y$ for different positions of point $Y$.
4. Suggest and solve similar problems for convex polygons.
5. For the point $Z$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $F_Z$, the vertices of which are orthogonal projections of the point $Z$ on the lines $BC$, $AC$ and $AB$. Suggest and solve similar problems for triangles $F_Z$ for different positions of the point $Z$.
Ukrainian TYM Qualifying - geometry, 2014.1
In the triangle $ABC$, one of the angles of which is equal to $48^o$, side lengths satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$. Express in degrees the measures of the other two angles of this triangle.
Ukrainian TYM Qualifying - geometry, 2018.17
Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.
Ukrainian TYM Qualifying - geometry, XII.15
Given a triangular pyramid $SABC$, in which $\angle BSC = \alpha$, $\angle CSA =\beta$, $\angle ASB = \gamma$, and the dihedral angles at the edges $SA$ and $SB$ have the value of $\phi$ and $\delta$, respectively. Prove that $\gamma > \alpha \cdot \cos \delta +\beta \cdot \cos \phi.$$
Ukrainian TYM Qualifying - geometry, 2015.18
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
Ukrainian TYM Qualifying - geometry, I.9
Prove that for any interior point of a triangle the sum of squares of distances from it to the sides of a triangle is not less than $\frac{4S^2}{9R^2}$.
[hide=about S,R]they are not defined, but I suppose they mean it's area and circumradii respectively[/hide]
Ukrainian TYM Qualifying - geometry, I.10
Given a circle of radius $R$. Find the ratio of the largest area of the circumscribed quadrilateral to the smallest area of the inscribed one.