1. In triangle 
point 
is the midpoint of side 
, and point 
is the foot of altitude 
. Prove that 
if and only if 
.
point is the midpoint of side , and point is the foot of altitude . Prove that if and only if . 2. A cyclic 
-gon is divided by non-intersecting (inside the -gon) diagonals to 
triangles. Each of these triangles is similar to at least one of the remaining ones. For what this is possible?
-gon is divided by non-intersecting (inside the -gon) diagonals to triangles. Each of these triangles is similar to at least one of the remaining ones. For what this is possible? 
3. A circle with center 
touches sides 
of triangle in points 
. Lines 
meet 
in points 
respectively. Prove that 
.
touches sides of triangle in points . Lines meet in points respectively. Prove that . 4. Given triangle . Point is the midpoint of side 
, and point 
is the projection of 
to the perpendicular bisector of segment 
. Line 
meets in point 
. Prove that triangle 
is isosceles.
. Point is the midpoint of side , and point is the projection of to the perpendicular bisector of segment . Line meets in point . Prove that triangle is isosceles. 5. On side of triangle an arbitrary point is selected . The tangent in to the circumcircle of triangle 
meets in point 
; point 
is defined similarly. Prove that 
.
of triangle an arbitrary point is selected . The tangent in to the circumcircle of triangle meets in point ; point is defined similarly. Prove that .6. Point of hypothenuse of a right-angled triangle is such that 
. Point 
on cathetus is such that 
; point 
is defined similarly. Find angle 
, where is the midpoint of 
.
of hypothenuse of a right-angled triangle is such that . Point on cathetus is such that ; point is defined similarly. Find angle , where is the midpoint of . 7. In a non-isosceles triangle the bisectors of angles 
and are inversely proportional to the respective sidelengths. Find angle 
.
the bisectors of angles and are inversely proportional to the respective sidelengths. Find angle . 8. Let 
be the median of right-angled triangle 
. The incircle of triangle 
touches sides 
in points 
; points 
are defined similarly. Prove that lines 
and 
meet on the bisector of angle .
be the median of right-angled triangle . The incircle of triangle touches sides in points ; points are defined similarly. Prove that lines and meet on the bisector of angle . 9. In triangle , given lines 
and 
containing the bisectors of angles and , and the foot 
of the bisector of angle . Restore triangle .
, given lines and containing the bisectors of angles and , and the foot of the bisector of angle . Restore triangle . 10. In a convex quadrilateral all sidelengths and all angles are pairwise different.
a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side?
b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?
11. Given triangle and point . Points 
are the projections of to 
. A line passing through and parallel to meets the circumcircle of triangle 
for the second time in point . Points 
are defined similarly. Prove that
and point . Points are the projections of to . A line passing through and parallel to meets the circumcircle of triangle for the second time in point . Points are defined similarly. Prove thata) lines 
concur;
concur;b) triangles and 
are similar.
and are similar. 12. Let 
be the circumcenter of an acute-angled triangle . A line passing through and parallel to meets and in points and respectively. The sum of distances from to and is equal to 
. Prove that 
.
be the circumcenter of an acute-angled triangle . A line passing through and parallel to meets and in points and respectively. The sum of distances from to and is equal to . Prove that . 
and the centroid of triangle are concyclic. 
. 15. Given triangle . Consider lines 
with the next property: the reflections of in the sidelines of the triangle concur. Prove that all these lines have a common point.
. Consider lines with the next property: the reflections of in the sidelines of the triangle concur. Prove that all these lines have a common point. 16. Given right-angled triangle with hypothenuse . Let be the midpoint of and be the center of circumcircle 
of triangle 
. Line meets for the second time in point 
. Segment 
meets the circumcircle of triangle in point 
. Prove that segments 
and 
meet on the circumcircle of triangle 
.
with hypothenuse . Let be the midpoint of and be the center of circumcircle of triangle . Line meets for the second time in point . Segment meets the circumcircle of triangle in point . Prove that segments and meet on the circumcircle of triangle . 17. A square 
is inscribed into a circle. Point lies on arc , 
meets 
in point , 
meets in point . Prove that the area of quadrilateral 
is equal to the half of the area of the square.
is inscribed into a circle. Point lies on arc , meets in point , meets in point . Prove that the area of quadrilateral is equal to the half of the area of the square. 18. A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to 
, one of two remaining angles is equal to 
, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.
, one of two remaining angles is equal to , one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers. 19. Two circles with radii 1 meet in points 
, and the distance between these points also is equal to 
. Point lies on the first circle, and lines 
are tangents to the second one. These tangents meet the first circle for the second time in points 
. Lines 
and 
meet in point 
. Find angle 
.
, and the distance between these points also is equal to . Point lies on the first circle, and lines are tangents to the second one. These tangents meet the first circle for the second time in points . Lines and meet in point . Find angle . 20. Point lies on side of triangle . Let 
and 
and 
be the incircles and the excircles (touching segment ) of triangles 
and 
Prove that the common external tangents to and 
and meet on .
lies on side of triangle . Let and and be the incircles and the excircles (touching segment ) of triangles and Prove that the common external tangents to and and meet on . 21. Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments.
22. A circle with center is inscribed into a segment of the disk, formed by an arc and a chord . Point is the midpoint of this arc , and point 
is the midpoint of the complementary arc. The tangents from touch in points and . The opposite sidelines and of quadrilateral meet in point 
, and the diagonals of meet in point 
. Prove that points 
and are collinear.
with center is inscribed into a segment of the disk, formed by an arc and a chord . Point is the midpoint of this arc , and point is the midpoint of the complementary arc. The tangents from touch in points and . The opposite sidelines and of quadrilateral meet in point , and the diagonals of meet in point . Prove that points and are collinear. 23. An arbitrary point is selected on each of twelve diagonals of the faces of a cube.The centroid of these twelve points is determined. Find the locus of all these centroids.
24. Given are 
points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?
points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops? Κάντε κλικ εδώ, για να τα κατεβάσετε.
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