geometry problems from Olympiad of Youth Mathematical School of St. Petersburg State University (Russia) with aops links in the names
collected inside aops: here
2005 - 2021
(only the problems found get posted)
Ordinary Type Problems
correspodence round
In a convex pentagon, the k diagonals have a length less than 1 cm, and the remaining 5-k of the diagonals - are more than 2 cm long. What can k be equal to? (Find all possible meanings of k and prove that there are no others).
One night, three flying saucers landed on the field, red, green and blue. Each of them densely covered a
certain triangular area with a powder of its color. The next morning, surprise villagers found that the
intersection of red and green has the shape of a triangle, the intersection of red and blue is quadrangular,
and the intersection of green and blue is pentagonal. Can the intersection of all three sections be
hexagonal?
Kostya drew a triangle and accurately measured the lengths of three sides and three medians in it. He got six different numbers. He told these numbers to Sasha, without specifying which of them are the lengths of the sides, and which are the lengths of the medians. Prove that Sasha can determine at least one of the numbers which is the length of the side.
How to cut the kite of the figure in 6 identical triangles? Remember to prove that the triangles are indeed equal.
In a right-angled triangle ABC, angle B is right and angle A is 30^o. Let BH be the altitude and AM the median. Point N lies on the segment AC, where AN = BM, and point L is symmetric to H wrt line AB. Prove that triangle LMN is equilateral.
Four points are marked inside the square and nine segments are drawn (see figure). The lengths of all line segments are 1. Line segments drawn from points to the sides of the square, perpendicular to them. Find the side length of the square.
In an acute-angled triangle ABC, the points B_1 and C_1 are the midpoints of the sides AC and AB, respectively, and B_2 and C_2 are the feet of the altitudes drawn on these sides. It is known that the angle between straight lines B_1C_2 and B_2C_1 is 60 degrees greater than angle A. What is it equal to?
In convex quadrilateral ABCD, angle A is 40^o, angle D is 45^o, and the bisector of angle B divides AD in half. Prove that AB> BC.
ABCDE and BEFGH are regular pentagons on a plane.
a) Find the value of the angle CFD.
b) Prove that eight more points can be added to the eight points mentioned in the condition so that the resulting 16 points serve as the vertices of six regular pentagons..
In triangle ABC, angle A is 50^o, BH is altitude, point M lies on BC such that BM = BH. The perpendicular bisector of the segment MC meets AC at the point K. It turns out that AC = 2 HK. Find the angles of triangle ABC.
In triangle ABC, angle B is right. M is the midpoint of the side BC and a point K was found on the hypotenuse such that AB = AK and \angle BKM = 45^o. In addition, on the sides AB and AC there were points N and L, respectively, such that BC = CL and \angle BLN = 45^o. In what ratio does point N divide side AB?
final round
Is it possible to mark points A, B, C, D, E on a straight line in such a way that AB = 14, BC = 16, CD = 12, DE = 8, AE = 16 cm?
There are three identical tiles in the shape of a right-angled triangle with angles of 30^o and 60^o. Vasya made one big triangle out of all of them. Petya claims that he can make a different triangle from the same tiles (also all three). Is Petya right?
In a convex hexagon ABCDEF exist a point M such that ABCM and DEFM are parallelograms . Prove that exists a point N such that BCDN and EFAN are also parallelograms.
In \vartriangle ABC, it is known that \angle A = 20^o, \angle B = 50^o, CH altitude of \vartriangle ABC, HE angle bisector \vartriangle AHC. Find \angle BEH.
In triangle ABC, it is known that \angle C = 3\angle A, and AB = 2 BC. Prove that \angle B = 60^o.
Inside (or on the border) of a square with side 2, you took 5 points. What is the largest number of triangles with vertires at these points may have an area greater than 1?
In the triangle ABC, AB = BC. On rays CA, AB and BC, points D, E and F are marked, respectively, so that AD = AC, BE = BA, CF = CB. Find \angle ADB + \angle BEC + \angle CFA.
In pentagon ABCDE, the angle A is 80^o, \angle B = 140^o, \angle C = 90^o, \angle D = 130^o. There is a point X on the AE side such that BA + AX = XE + ED = 1. Calculate the length of the line segment CX.
ABCDEFGHIJK and GIMNOPQRSTU are two regular 11-gons (the vertex order is counterclockwise). Prove that AP = BQ = IC.
The perimeters of triangles ABC and DEF are 239 and 533, respectively. Can triangles ABD, BCE, CAF be equilateral?
All angles of an convex pentagon with equal sides are different. Prove that the largest and the smallest of them are adjacent.
Point M is the midpoint of the side AC of the triangle ABC. Point N lies on the segment AM such that the angle MBN is equal to the angle CBM . On the extension of the segment BN beyond the point N, point K is selected such that the angle BMK is right. Prove that BC = AK + BK.
I is the center of the inscribed circle \omega of the triangle ABC. The circumscribed circle of the triangle AIC intersects \omega at points P and Q. Prove that if PQ and AC are parallel, then triangle ABC is isosceles.
A line intersects the sides AB and BC of the square ABCD at the points X and Y, and the extensions of the sides AD and CD at the points Z and T. Prove that the triangles CXZ and AYT have the same area.
An isosceles triangle ABC with base AC is given. The angle BAC is 37 degrees. X is the point of intersection of the altitude from the vertice A with a straight line passing through B parallel to the base, M is a point on the line AC such that BM = MX. Find the measure of the angle MXB.
On the base AE of trapezoid ABCE, point D is selected such that S_{ABCD} = S_{CDE}. It is known that ABCD is a parallelogram, and its diagonals meet at point O. On the segment DE, point T is selected. Prove that if OT \parallel BE, then OD \parallel CT.
In a right-angled triangle ABC, point M is the midpoint of the hypotenuse AB. On the ray CM mark the points X and Y. It is known that lines XA and YA form the same angles with line AC. Prove that lines XB and YB form the same angles with line BC.
On the diagonal BD of the isosceles trapezoid ABCD, there is a point E such that the base BC and the segment CE are the legs of a right-angled isosceles triangle. Prove that lines AE and CD are perpendicular.
Point D is marked on the lateral side AB of an isosceles triangle ABC, point E is marked on the lateral side AC, and point F is marked on the extension of the base BC beyond point B, such that CD = DF. Point P is selected on line DE, and point Q is selected on segment BD so that PF\parallel AC and PQ\parallel CD. Prove that DE = QF.
In a triangle ABC, the angle A is right. Let BM be the median of the triangle, D the midpoint of BM. It turned out that \angle ABD = \angle ACD. And what are these angles equal to?
Through the center O of a circle circumscribed about a regular pentagon ABCDE, are drawn straight lines parallel to sides AB and BC. Their intersection points with BC and CD are denoted by F and G respectively. What is the ratio of the areas of the OFCG to the pentagon ABCDE?
Points X and Y are marked in triangle ABC, so that the rays AX, CY intersect on the extension of segments AX and CY and are perpendicular to straight lines BY and BX, respectively. Sum of distances from X and Y to line AC is less than BH. Prove that AX + CY <AC
Research Type Problems
Correspodence Round
The axis of symmetry of a set on a plane is a straight line such that for any point from this set, a point symmetric to it with respect to this line also lies in this set.
1. The angle between two straight lines is eight degrees. Is there a polygon in which each of these lines is an axis of symmetry?
2. AD is a non-convex quadrilateral. It is known that each of the triangles ABC, ABD, ACD, BCD has an axis of symmetry. Prove that one of them is the axis of symmetry of the whole quadrilateral.
3. Is the statement of the previous item true if ABCD is a convex quadrilateral?
4. The set of points on the plane has exactly 100 axes of symmetry. What is the smallest number of points this set can consist of?
A trapezoid is a (flat) quadrilateral in which two sides are parallel and the other two are not (ie, a parallelogram is not a trapezoid).
1. Is there a 6-hedron in which all faces are trapeziums?
2. Is there a polyhedron ABCDEFGH, whose faces ABFE, BCGF, CDHG and DAEH are isosceles trapezoids, and AE = BF = CG \ne DH?
3. The 6-hedron ABCDEFGH has all faces, trapeziums. Is it true that among them there are two parallel ones?
4. 6-hedron ABCDEFGH has all faces , isosceles trapezoids. Is it true that it is inscribed?
1. Is it possible that the distances from one point inside the \vartriangle ABC to the straight lines AB, BC and AC, respectively, are 7, 5 and 9 cm, and the distances from another point inside the same triangle to the same straight lines are 8, 9 and 11 cm, respectively?
2. Draw three straight lines and mark three points on the drawing, so that the distances from the first point to the straight lines are 0, 0 and 12 cm, from the second point - 0, 15 and 24 cm, respectively, from the third point - 20, 0 and 24 cm respectively. Try to describe the construction of the drawing using a ruler with divisions, a compass and a protractor.
3. Several straight lines are drawn on the plane and one point is marked. Prove that there is a point on the plane that is at a greater distance to each of the drawn lines than the marked point.
4. Three straight lines are drawn on the plane. It is known that for any numbers a, b and c, if on the plane there is a point with distances to these straight lines: a - to the first, b - to the second, and c - to the third, then there is also a point, the distance from which to these straight lines: b - to the first, c - to the second, and a - to the third. Pick the shape formed by these lines. Give all (up to similarity and congruence) options and justify why there are no others.
Consider a convex quadrilateral ABCD and a point E on the side BC such that the areas satisfy the condition: S_{AED} = S_{ABE} + S_{ECD}.
1. Give an example of a quadrilateral for which there is more than one such point E.
2. It is known that the \angle B and \angle C are right, and the corners \angle A and \angle D are not right. Prove that BE = EC.
3. It is known that E lies on the bisector of the angles \angle A and \angle D. Prove that AD = AB + CD.
4. It is known that BE = EC, O is the point of intersection of the diagonals. Prove S_{AOD} = S_{BOC}.
1. Given a triangle. Prove that you can construct a triangle whose side lengths are the sines of the angles of the given triangle.
2. Prove that among the pairwise products of the tangents of the angles of an acute-angled triangle, one number is at least 3, the other is at least 2, and the remaining number is at least 1.
3. Given an acute-angled triangle, the cosine of the smaller angle is not more than 2/3. Prove that there is a triangle whose side lengths are equal to the cotangents of the angles of this triangle.
4. It is known that there is a triangle, the lengths of the sides of which are equal to the tangents of the angles of this triangle. In addition, there is a triangle whose side lengths are equal to the cotangents of the angles of this triangle. Prove that all angles of this triangle lie in the interval (\pi/4, \pi/2))
For each three vertices of the n-gon (where n> 4), the center of the circle passing through them is marked.
1. Any center coincides with one of two given points, A and B. Prove that the original n-gon is cyclic.
2. It is known that any center coincides with one of six given points. Prove that the original n-gon is cyclic.
3. For each natural n, give an example of an n-gon with exactly \frac{n^2-3n+4}{2} different centers.
4. Is there a natural n, such that some n-gon has more than one but fewer than \frac{n^2-3n+4}{2} different centers?
Consider four points on the plane: A, B, C and D, and the following three points: P , the point of intersection of the lines AB and CD, Q , the point of intersection of the lines AD and BC, and R, the point of intersection of the lines AC and BD. Let's call the resulting points accompanying to points A, B, C and D, and the quadrilateral APQR also accompanying to ABCD.
1. Let the points A, B, Q and R be given on the plane, and none of these four points lie on one straight line. Prove that the remaining points C, D and P can be uniquely reconstructed.
2. Prove that for any APQR on the plane there is at least one nonconvex quadrilateral ABCD for which APQR is accompanying .
3. Let APQR be given on the plane, and RM be is its median. Find on this median infinitely many such positions of point A that you can build a convex quadrilateral ABCD, for which APQR will be accompanying and infinitely many positions of the vertex A, for which you can construct a non-convex accompanying quadrilateral.
4. Let the position of points P, Q, R and point A inside APQR be given on the plane. Prove that there is at most one quadrilateral ABCD for which APQR is accompanying .
On the third day, Chuk and Gek studied geometry and after lessons started cutting pieces into pieces. They cut the triangle into 9 triangles as follows (see fig): three straight lines intersecting at one point and passing so that any side is intersected by exactly two of them, and three more segments, connecting the points of intersection of lines with the sides of the original triangle.
1. First, they took an isosceles right-angled triangle as their starting point. What is the largest number of equilateral triangles (among 9 small ones) they can get?
2. Then the guys took an equilateral original triangle. Can all 9 triangles be right?
3. The original triangle is equilateral, but of the three lines intersecting inside it, no two are perpendicular. What is the largest number of right-angled triangles can it work now?
A cardboard square was laid on a sheet of paper. The bully pierced the square with a needle (by attaching it to the paper). After that, he turned the square around the needle, drawing on paper the path of each of the vertices of the square, and then threw out the square itself.
1. How many circles can be drawn?
2. Is it true that it is always possible to reconstruct from a picture which two of the four circles correspond to the vertices of the square located diagonally?
3. One of the four circles was erased. What is the maximum number of options for drawing the fourth circle so that the resulting circles can turn out by rotating the vertices of the square?
Final Round
Let's call the circle passing through the midpoints of the sides of the triangle the median circle of this triangle.
1. Prove that the median circle of a triangle touches the circumscribed circle if and only if the triangle is right-angled.
2. Two circles are given with the ratio of radii 1: 2, and one is strictly inside the other. Prove that for any point on the larger circle there is exactly one triangle, in which this point will be one of the vertices: the larger circle is circumscribed, and the smaller one is the median.
3. Given two intersecting circles with the ratio of the radii 1: 2. Find the locus of the points which are one of the vertices of a triangle for which the larger circle is the circumcircle, and the smaller one is the median.
4. The radius of the circumscribed circle about a triangle is equal to 1, the distance between the centers of the circumscribed and the middle one is \frac{\sqrt3+1}{2}, one of the angles is 150^o. Find the area of that triangle.
Recall that the similarity of a figure with a coefficient k> 0 is a transformation such that any two points X and Y of the figure are associated with points X' and Y' such that X'Y'= k \cdot XY. A figure \Phi ' is called similar to a figure \Phi with coefficient k if there is a similarity with coefficient k that transforms \Phi into \Phi '.
1. The right-angled triangle A'B'C is inscribed in a similar triangle ABC, the names of the vertices are corresponding, while the vertex A' lies on BC, B' on AC, and C' on AB. Find all possible values for the coefficient of similarity.
2. Triangle A'B'C' is inscribed in triangle ABC, similar to it with coefficient k <1, the names of the vertices correspond responsible. In this case, vertex A' lies on BC, B' on AC, C' on AB, \angle BC'A'= \theta, \angle CAB = \alpha. Prove that k\cos \frac{\alpha-\theta}{2}=\frac12.
3. The heights of the tetrahedron intersect at one point, and the tetrahedron with vertices at the bases of the heights is similar to the original one. Prove that the original tetrahedron is regular.
4. The tetrahedron with vertices at the bases of heights dropped from the vertices of the original tetrahedron turned out to be correspondingly similar to the initial one. Prove that the square of the similarity coefficient is 1 -\sin^2\alpha \sin^2 \beta, where \alpha is the dihedral angle at some edge, and \beta is the angle between this edge and the opposite one, as between crossing lines.
There is a square with a side length of 1 km. Two runners run along its diagonals at a speed of 10 km / h (each on its own diagonal, when it reaches the end of the diagonal, turns around and runs in the opposite direction at the same speed).
1. Prove that at some point the distance between the runners will be at least \sqrt{2}/2 km.
2. Prove that someday the distance between the runners will be no more than 1/2 km.
3. Find how long the minimum of these distances can be.
4. Prove that if at some moment the distance was less than a m, then at some other moment it will be more (1000 - a) m.
Points A and B on the plane are connected by a broken line so that the following conditions are met:
(I) moving from A to B along a given broken line, we keep moving away from A,
(II) moving from A to B along this broken line, we are constantly approaching B.
In addition, the length of the segment AB is 1 cm.
1. Prove that any segment of such a broken line, except the initial and final, lies between the bases of the perpendiculars dropped from points A and B to the line containing this segment.
2. Let A and B be connected by a two-link broken line so that conditions (I) and (II) are satisfied. Find the maximum possible length of such a broken line.
3. Let A and B be connected by a three-link broken line so that conditions (I) and (II) are satisfied. Find the maximum possible length of such a broken line.
4. Is it true that polygonal lines satisfying conditions (I) and (II) (and connecting these points A and B) can have arbitrarily large length?
1. You are given a circle of radius 1 with center O and diameter AB. Find the largest possible area of a triangle APB if the length of the segment OP is 5.
2. The diagonal of the quadrilateral passes through the center of its inscribed circle with a radius of 1. Find the smallest possible perimeter of this quadrilateral.
3. What can be the smallest perimeter of a triangle, the radius of the inscribed circle is 1, and the length of at least one of the sides is equal to \ell?
4. The radius of the inscribed circle of a quadrilateral is 1, and the length of one of its diagonals is \ell > 3. Find the smallest possible value of the perimeter of this quadrilateral.
1. There are 4 circles S_1, S_2, S_3, S_4, which are externally touch each other consecutively in a cycle, i.e., pairs of touching- circles: S_1 and S_2, S_2 and S_3, S_3 and S_4, S_4 and S_1. Prove that if the centers of the circles form a convex quadrilateral, then it is tangential.
2. Lines \ell_1 and \ell_2 touch four circles described in the previous paragraph. Line \ell_1 separates circles S_1 and S_2 from S_3 and S_4, i.e. circles S_1 and S_2 are on the same side of the straight line \ell_1, and S_3 and S_4 - on the other. Similarly, the line \ell_2 separates S_2 and S_3 from S_1 and S_4. Prove that the quadrilateral formed by the centers of the circles is a rhombus.
3. There are 8 spheres that touch the "dice": S_1, S_2, S_3 and S_4 (consecutively in a cycle) and S_5, S_6, S_7 and S_8 (also consecutively in a cycle), as well as S_1 and S_5, S_2 and S_6, S_3 and S_7, S_4 and S_8. There are also three planes, each touching all eight spheres, where the first separates the spheres S_1 - S_4 from the spheres S_5 - S_8, the second - spheres S_1, S_2, S_5 and S_6 from S_3, S_4, S_7 and S_8, and the third - S_1, S_4, S_5 and S_8 from S_2, S_3, S_6 and S_7. Prove that the centers of the spheres S_1, S_2, S_3 and S_4 lie in the same plane.
4. The centers of the eight spheres described in the previous paragraph are the vertices of a hexagon with quadrangular faces. Prove that if this hexagon is inscribed, then it is a cube.
1. Three points move in a straight line at constant speeds so that the sum of the squares of all three pairwise distances between them remains constant. Prove that the speeds of all three points are equal.
2. Three points move in a circle with constant velocity so that the sum of the squares of all three pairwise distances between remains constant (in this problem, the distance between two points on the circle is the smaller of the lengths of the two arcs between them). Prove that all three points are moving at the same speed.
3. Three points move in a circle with constant velocity so that the sum of the squares of all three distances between them is remains constant (in this and the next problem, the distance is is the length of the line segment connecting these points). Moreover, it is known that that not all points move at the same speed. Prove that there are two points that always remain diametrically opposite to each other.
4.Four points move in a circle with constant speeds so that the sum of the squares of the distances between them remains is constant. Prove that among the speeds of these four points no more than two different.
Grade 9: 1,3,4, Grades 10-11: 2,3,4
1. Point B is selected on segment AC, then on segments AB and BC are constructed equilateral triangles ABK and BCL, lying in one half-plane relative to the straight line AC. Points P, Q,R divide the segments AB, BC, KL in a ratio of 1: 2, respectively. Prove that points P, Q, R form an equilateral triangle.
2. On the sides AB and BC of triangle ABC, isosceles triangles ABK and BCL with the same angles at top. Points P, Q, R divide segments AB, BC, KL, respectively in the same ratio k. Prove that triangle PQR is similar to triangle ABK.
3. Two similar triangles A_1B_1C_1 and A_2B_2C_2 are given, and their orientation coincides (the order of traversing the corresponding the vertices are the same). Points A_3, B_3, C_3 are the midpoints of segments A_1A_2, B_1B_2, C_1C_2 respectively. Prove that triangle A_3, B_3, C_3 is similar to the first two.
4. Let (in the notation of the previous section) the lines containing the altitudes A_1H_1 and A_2H_2 of the first two triangles intersect at the point H, and the points H_1 and H_2 lie, respectively on the segments A_1H and A_2H. It turned out that these triangles are visible from points H at the same angles. Prove that A_3H \perp B_3C_3.
Let XY and XZ be chords of circles U_1 and U_2, and line XY touches U_2, line XZ touches U_1, and T is the second pointintersection of these circles.
1. Prove that if XY = XZ then TY = TZ.
2. Prove that point X is equidistant from lines TY and TZ.
3. Let points I and J be the centers of the incircles \vartriangle XTY and \vartriangle XTZ, respectively. Prove that triangles TIJ and XYT are similar (for some vertex order).
4. Let point O be the center of the circumscribed circle of the triangle \vartriangle XIJ. Prove that IOJT is cyclic.
5. Let W be the intersection point of the segments XT and IJ. Prove those that if the sides \vartriangle XTY are rational, then the length of the segment WX is so is rational.
Grades 9-10: 1-3,5 Grade 11: 2-5
Three circles were drawn on the plane, any two of which intersect at two points. Then all the intersection points were marked, and the circles themselves were erased.
1. Let 6 points be marked. Can they be the vertices of a regular hexagon?
2. Let 4 points be marked, and they lie at the vertices of a rhombus with angles 60^o and 120^o. The radius of the smaller circle is 1. What are the radii of the other two?
3. Let 6 points be marked. Can different sets of circles correspond to them?
4. Suppose there were not three, but 100 circles, and 9900 points were marked. Prove that the original circles can be recovered uniquely.
1.In triangle ABC, point M lies on side AC, and point N lies on side BC. Can line segments AN and BM split triangle ABC into four parts with equal areas?
2. In triangle ABC, points N and M lie on side BC, and points K and L lie on side AC. Can 9 parts, into which triangle ABC is divided by segments AN, AM, BK, BL, have equal areas?
3. In triangle ABC, point M lies on side AB, point K on BC, and point L on AC. How many parts did the segments AK, BL, CM divide the triangle ABC into if the areas of all the parts obtained are equal?
4. In quadrilateral ABCD, point K lies on side BC, L on side CD, M on AD, N on AB. Can all parts, into which ABCD divided the segments AK, BL, CM, DN, have equal areas?
A stencil is cut out of cardboard in the form of the requlat n-gon with a side of 1 cm. We will call it an n-stencil. The following geometric constructions are allowed:
a) attach the stencil to a sheet of paper so that several vertices of the stencil coincide with the points previously marked on the paper;
b) completely or partially circle the stencil attached to the sheet, marking all the vertices obtained in this case;
c) mark the intersection points of the segments already drawn;
d) attach the stencil to a sheet of paper so that one of the sides stencil lay on the previously drawn segment.
1. Prove that using operations a) –– d) and a 10-stencil, you can construct the center of a regular 10-gon with a side of 1 cm.
2. Prove that using operations a) –– c) and a 12-stencil, you can construct the center of a regular 12-gon with a side of 1 cm.
3. Prove that using operations a) –– b) and a 10-stencil, you can construct the center of a regular 10-gon with a side of 1 cm.
4. Prove that using operations a) –– b) and a 14-stencil you can construct the center of a regular 14-gon with a side of 1 cm.
5. Prove that using operations a) –– b) and a 2014-stencil, you can construct the center of a regular 2014-gon with a side of 1 cm.
6. Prove that using operations a) –– d) and an n-stencil you can construct the center of a regular n-gon with a side of 1 cm for any n> 4. In addition to operations a) –– d), it is allowed to connect by a segment any two points, the distance between which is not more than 1 cm.
Grade 10: 1-3,5 Grade 11: 1-2,4,6
A triangle ABC is considered, on the sides of which points are selected X, Y and Z (one point on each side). Moreover, it is known that the centers of the inscribed circles triangles ABC and XYZ coincide.
1. Let points X, Y and Z be the midpoints of the sides of triangle ABC. Prove that triangles ABC and XYZ are equilateral.
2. Let the radius of the incircle of \vartriangle XYZ be half the radius of the incircle of \vartriangle ABC. Prove that if triangle XYZ is equilateral, then ABC is equilateral.
3. Let the radius of the incircle of \vartriangle XYZ be half the radius of the incircle of \vartriangle ABC. Prove that \vartriangle XYZ is equilateral.
4. Let the ratio of the radius of the incircle of \vartriangle ABC to the radius of the incircle of \vartriangle XYZ be equal to t. Prove thatt^2 \le \frac{xy}{ (p-x) (p-y)},where x = YZ, y = XZ, p is the semiperimeter of \vartriangle XYZ.
The perpendicular bisector of the side AC of the triangle ABC intersects the BC segment and the ray AB at points D and E, respectively. Points M and N are the midpoints of the segments AC and DE, respectively.
1. It turned out that \angle BAC = 2\angle BCA. Prove that DM \le DB.
2. Prove that \angle ABC + \angle MBN = 180^o if AB = BE.
3. The point J\ne D on the segment BD is chosen such that DE = JE. It is known that \angle ABM = 90^o. Find the ratio of the areas of triangles ADJ and BEJ.
4. For what values of \alpha can the equality \angle MBN = \alpha = \angle ABC be satisfied?
In all problems O denotes the center of the circumcircle of triangle ABC, and I the center of its incircle.
1. Consider an acute-angled triangle ABC and its orthocenter H. It turns out that points B, O, H and C lie on the same circle. Prove that point I lies on the same circle.
2. Points X and Y are the midpoints of arcs AC and AB of the circumscribed circle of triangle ABC, respectively. The segment XY and the side of the triangle AC meet at the point Z. Prove that|IZ|> \frac{|AC|-|IC|}{2}.3. Given a non-isosceles triangle ABC with angle \angle A = 60^o. Prove that the intersection point of lines OI and BC is equidistant from points A and I.
4. An arbitrary acute-angled triangle ABC is given. X is some point inside the triangle. The circumcircles of triangles AOX, BOX and COX intersect the circumcircle of triangle ABC at points A_1, B_1 and C_1, respectively. Prove that X is the center of the incircle ABC if and only if X is the orthocenter A_1B_1C_1.
Let I be the center of the inscribed circle \omega of the triangle ABC. The circumscribed circle of the triangle AIC intersects \omega at points P and Q so that P and A lie on one side of the straight line BI, and Q and C on the other. We denote by M the midpoint of the smaller arc AB of the circumscribed circle of the triangle ABC, and by N the midpoint of the smaller arc BC.
1. Prove that if PQ\parallel AC, then triangle ABC is isosceles.
2. Given a triangle DEF. The circle passing through the vertices E and F intersects the sides DE and DF at points X and Y, respectively. The bisector of angle \angle DEY intersects DF at point Y ', and the bisector of angle \angle DFX intersects DE at point X'. Prove that XY\parallel X'Y '.
3. Prove that MN> PQ
4. Let L be the point of intersection of lines AP and CM, S be the point of intersection of lines AN and CQ. Prove that LS\parallel PQ.
5. Prove that MN\parallel PQ.
6. Let T be the point of intersection of lines AP and CQ, and K be the point of intersection of lines MP and NQ. Prove that T, K and I are collinear.
Grade 9: 1,3-5 Grade 10: 1-2,5-6
An arbitrary triangle ABC with orthocenter H is given. The inner and external bisectors of angle B intersect the line AC at points L and K, respectively. Two circles are considered: \omega_1 the circumcircle of the triangle AHC, \omega_2 having segment KL as it's diameter.
1. Let point T be such that TL is the angle bisector of the triangle ATC . Prove that TK is the external bisector of the same triangle.
2. Let X be a point of intersection of the circles \omega_1 and \omega_2 such that X and B lie on opposite sides wrt line AC. Prove that the point X lies at the alitude BH of the triangle ABC.
3. Let Y be a point of intersection of the circles \omega_1 and \omega_2 such that the points Y and B lie on the same side wrt line AC. Prove that point Y lies on the median BM.
4. Prove that the tangent to the circle \omega_1 and \omega_2 at the point of intersection with the median BM intersects the line AC at the midpoint of the segment KL.
A circle \omega with a center at point I is inscribed in the triangle ABC and touches its sides AB and AC at points D and E, respectively. The angle bisectors of the triangle ADE intersect at point J. The segments BJ and CJ intersect the segment DE at points P and Q, respectively.
1. Prove that PJ> PD.
2. It is known that EJ = DE. Find the angle BAC.
3. Prove that the perimeter of the triangle BJC is greater than the perimeter of the perimeter BDEC.
4. Points M and N are the midpoints of DJ and JE respectively. Prove that PM = QN.
Georgy Konstantinovich has a garden, through which Erich sometimes runs. Erich runs in a straight line, but each time a new one. Georgy Konstantinovich wants to buy and place anti-tank hedgehogs inside or on the border of the garden (in the form of several segments) so that Erich is guaranteed to rest against them. The length of a hedgehog is the sum of the lengths of its constituent segments.
1. Let the garden have the shape of an equilateral triangle with side 1. Prove that hedgehogs with total length \sqrt3 are enough for Georgy Konstantinovich.
2a. Let the garden have the shape of a square with side 1. Prove that hedgehog with total length 2.65 are enough for Georgiy Konstantinovich.
2b. Let the garden have the shape of a square with side 1. Prove that hedgehog with total length 2.64 are enough for Georgiy Konstantinovich.
3a. Let the garden have the shape of an equilateral triangle with side 1. Prove that Georgy Konstantinovich will have to buy hedgehogs of total length at least \frac{3\sqrt3}{4}.
3b. Let the garden have the shape of an equilateral triangle with side 1. Prove that Georgy Konstantinovich will have to buy hedgehogs of total length at least 1.29.
4. Let the garden have the shape of a square with side 1. Prove that Georgy Konstantinovich will have to buy hedgehogs of total length at least 2.
Grade 9: 1,2a,3a,4 , Grade 10: 1,2a,3b,4 Grade 11: 1,2b,3b,4
Two circles inscribed in an angle with a vertex R meet at points A and B. A line is drawn through A that intersects the smaller circle at point C, and the larger one at point D. It turned out that AB = AC = AD.
1. Prove that the tangents to the circles at point A are perpendicular.
2. Let C and D coincide with the points of tangency of the circles and the angle. Prove that the \angle R is right.
3. Let C and D coincide with the points of tangency of the circles and the angle. Find the angle ADR.
4. Prove that if \angle R is right, then C and D coincide with the tangency points of the circles and the angle.
5. Let \angle R = 135^o. The perpendicular from A to the nearest side of the angle intersects the smaller circle at point P, the perpendicular from A to the second side intersects BP at point Q. Finally, let O_1 and O_2 be the centers of the original circles, O be the center of the circle circumscribed around \vartriangle ABQ. Prove that BO is the bisector of the angle O_1BO_2.
6. What values can the angle \angle RAO_1 take, where O_1 is the center of the smaller circle?
Grade 9: 1-4 , Grade 10: 2-5 Grade 11: 2-4,6
There is a square with a side of 2. Vasya painted a finite number in it polygons so that there are no filled points at a distance of 1 (the border is considered unpainted). Let A be the filled set, S (A) is its area.
1. Give an example of A such that S (A)\ge 1.
2. Prove that S (A) \le 2.
3. Give an example of A with S (A)\ge 2.5 -\sqrt2
4. Estimate S (A) from above as 1.65.
5. Give an example of A such that S (A) \le 1.14.
Grades 9: 1-4 Grade 11: 1,2,4,5
An equilateral triangle ABC is given. Points X and Y on sides AB and BC, respectively. Equilateral triangles AXD and BY E are built in the outside wrt the original triangle ABC. Point F is such that triangle DEF is equilateral, and points F and C lie on the same side wrt DE.
1. It is known that AX = BY, and G is the intersection point of the side AB and the line passing through F parallel to BC. Prove that GF = AB.
2. It is known that AB is parallel to FC. Prove that AX = BY.
3. Let the initial triangle ABC be fixed and its side equal to a, and the rest of the points are not fixed. Let F' be a point different from F such that the triangle DEF' is equilateral. Find locus of points F' and its length.
4. Let G be the point of intersection of the side AB and the line passing through F parallel to BC. Prove that BG = AX.
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