geometry problems from Qualifying Round and Final Battle from All-Ukrainian Tournament of Young Mathematicians (TYM) named after M. Y. Yadrenko with aops links
collected inside aops:
qualifying round + final battle
I-XXIII
Qualifying Round
The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.
Given a natural number n> 3. On the plane are considered convex n - gons F_1 and F_2 such that on each side of F_1 lies one vertex of F_2 and no two vertices F_1 and F_2 coincide.
For each n, determine the limits of the ratio of the areas of the polygons F_1 and F_2.
For each n, determine the range of the areas of the polygons F_1 and F_2, if F_1 is a regular n-gon.
Determine the set of values of this value for other partial cases of the polygon F_1.
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.
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}.
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.
A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?
A right triangle when rotating around a large leg forms a cone with a volume of 100\pi. Calculate the length of the path that passes through each vertex of the triangle at rotation of 180^o around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to 17.
Inside a right cylinder with a radius of the base R are placed k (k\ge 3) of equal balls, each of which touches the side surface and the lower base of the cylinder and, in addition, exactly two other balls. After that, another equal ball is placed inside the cylinder so that it touches the upper base of the cylinder and all other balls. Find the volume V (R, k) of the cylinder.
Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?
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?
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.
A circle centered at point O is separated by points A_1,A_2,...,A_n on n equal parts (points are listed sequentially clockwise) and the rays OA_1,OA_2,...,OA_n are constructed. The angle A_2OA_3 is divided by rays into two equal angles at vertex O, the angle A_3OA_4 is divided into three equal angles, and so on, finally, the angle A_nOA_1 divided into n equal angles at vertex O. A point belonging to the ray other than OA_1, is connected by a segment with its orthogonal projection B_0 on the neighboring (clockwise) arrow) with ray OA_1, point B_1 is connected by a segment with its orthogonal projection on the next (clockwise) ray, etc. As a result of such process it turns out the broken line B_0B_1B_2B_3... infinitely "twists". Consider the question of giving the thus obtained broken numerical value of "length" L (n) and explore the value of L(n) depending on n.
Inside the regular n -gon M with side a there are n equal circles so that each touches two adjacent sides of the polygon M and two other circles. Inside the formed "star", which is bounded by arcs, these n equal circles are reconstructed so that each touches the two adjacent circles built in the previous step, and two more newly built circles. This process will take k steps. Find the area S_n (k) of the "star", which is formed in the center of the polygon M. Consider the spatial analogue of this problem.
Is it possible to "cut" the plane on convex pentagons, each of which can be inscribed in a circle? Consider the similar question of the existence of "cutting" the plane into convex pentagons, to each of which you can write a circle, and "cut" into convex pentagons, each of which is inscribed and circumscribed (such that it can be both inscribed in a circle and circumscribed around a circle).
Let \gamma_m (m \ge 3) be the number of faces of the convex polyhedra, which are m -gons. Describe numerical sets (\gamma_3, \gamma_4, \gamma_5, ...) (For each polyhedron, such a set is, of course, finite).
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.
Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed \frac16 of the area of the hexagon.
What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than \frac16 the area of the hexagon?
Given a convex body in space. Prove that four points can be marked on its surface so that the tangent plane to the surface at any of these four points is parallel to the plane passing through the other three.
Given a triangle ABC and points D, E, F, which are points of contact of the inscribed circle to the sides of the triangle.
i) Prove that \frac{2pr}{R} \le DE + EF + DF \le p
(p is the semiperimeter, r and R are respectively the radius of the inscribed and circumscribed circle of \vartriangle ABC).
ii). Find out when equality is achieved.
In the tetrahedron ABCD, the point E is the projection of the point D on the plane (ABC). Prove that the following statements are equivalent:
a) C = E or CE \parallel AB
b) For each point M belonging to the segment CD, the following equation is satisfied
S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}where S_{\vartriangle XYZ} means the area of triangle XYZ.
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.
Let the polyhedron F be inscribed in the spatial integer lattice, ie - all three coordinates each of its vertices are integers. Investigate how many such polyhedron F can have vertices, faces and edges, if there is no lattice node inside F. What dependencies (in particular - necessary, sufficient) between the number v (F) of the vertices of the polyhedron F inscribed in the spatial integer lattice, and the number of its nodes on the faces (s (F)), edges (e (F)) and inside F (i (F)) will you be able to install?
Investigate cases e(F) = 0, s (F) = 0, find the limits of change s (F) at given v (F).
Give non-trivial infinite series of sets (v (F), s (F), e (F), i (F)), which can be implemented (or, conversely, can not be implemented).
Let X be a point inside an equilateral triangle ABC such that BX+CX <3 AX. Prove that
3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5
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.
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.
Consider an arbitrary (optional convex) polygon. It's chord 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?
A quadrilateral whose perimeter is equal to P is inscribed in a circle of radius R and is circumscribed around a circle of radius r. Check whether the inequality P\le \frac{r+\sqrt{r^2+4R^2}}{2} holds.
Try to find the corresponding inequalities for the n-gon (n \ge 5) inscribed in a circle of radius R and circumscribed around a circle of radius r.
The convex polygon A_1A_2...A_n is given in the plane. Denote by T_k (k \le n) the convex k-gon of the largest area, with vertices at the points A_1, A_2, ..., A_n and by T_k(A+1) the convex k-gon of the largest area with vertices at the points A_1, A_2, ..., A_n in which one of the vertices is in A_1. Set the relationship between the order of arrangement in the sequence A_1, A_2, ..., A_n vertices:
1) T_3 and T_3 (A_2)
2) T_k and T_k (A_1)
3) T_k and T_{k+1}
Let a, b, and c be the lengths of the sides of an arbitrary triangle, and let \alpha,\beta, and \gamma be the radian measures of its corresponding angles. Prove that \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.Suggest spatial analogues of this inequality.
Investigate the properties of the tetrahedron ABCD for which there is equality
\frac{AD}{ \sin \alpha}=\frac{BD}{\sin \beta}=\frac{CD}{ \sin \gamma}where \alpha, \beta, \gamma are the values of the dihedral angles at the edges AD, BD and CD, respectively.
Find the largest value of the expression \frac{p}{R}\left( 1- \frac{r}{3R}\right) , where p,R, r is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.
The set M of points of a plane is called obtuse if for any three points A, B, C of this set the triangle ABC is obtuse. Investigate the existence of such a point X \notin M for an obtuse set M such that the set M\cup \{X\} is also obtuse.
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.
Propose an algorithm for cutting an arbitrary convex quadrilateral with as few lines as possible into parts from which a triangle with equal area to this quadrilateral can be formed.
Let n\ge 2 be a natural number. What is the smallest number of m_n and what is the largest number of M_n parts that can n straight lines divide the plane? For which natural k, m_n \le k \le M_n, there is a partition of k parts?
Inside the convex polygon A_1A_2...A_n , there is a point M such that \sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}. Prove that nP\ge 4d, where P is the perimeter of the polygon, and d=\sum_{k=1}^n |\overrightarrow {A_kM}| . Investigate the question of the achievement of equality in this inequality.
A paper square is bent along the line \ell, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.
About 20 years ago, scientists discovered that carbon atoms are behind under certain conditions form molecules of 60 or more atoms located in vertices of a polyhedron with pentagonal and hexagonal faces. Let the polyhedron has n vertices and only pentagonal and hexagonal faces. Find the number of these faces depending on the number of vertices n.
Chords AB and CD, which do not intersect, are drawn in a circle. On the chord AB or on its extension is taken the point E. Using a compass and construct the point F on the arc AB , such that \frac{PE}{EQ} = \frac{m}{n}, where m,n are given natural numbers, P is the point of intersection of the chord AB with the chord FC, Q is the point of intersection of the chord AB with the chord FD. Consider cases where E\in PQ and E \notin PQ.
Prove that there exists a point K in the plane of \vartriangle ABC such thatAK^2 - BC^2 = BK^2 - AC^2 = CK^2 - AB^2.Let Q, N, T be the points of intersection of the medians of the triangles BKC, CKA, AKB, respectively. Prove that the segments AQ, BN and CT are equal and have a common point.
On the plane there are two cylindrical towers with radii of bases r and R. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.
Let I be the point of intersection of the angle bisectors of the \vartriangle ABC, W_1,W_2,W_3 be point of intersection of lines AI, BI, CI with the circle circumscribed around the triangle, r and R be the radii of inscribed and circumscribed circles respectively. Prove the inequalityIW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr}.
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.
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.$
Given a triangle ABC, inside which the point M is marked. On the sides BC,CA and AB the following points A_1,B_1 and C_1 are chosen, respectively, that MA_1 \parallel CA, MB_1 \parallel AB, MC_1 \parallel BC. Let S be the area of triangle ABC, Q_M be the area of the triangle A_1 B_1 C_1.
a) Prove that if the triangle ABC is acute, and M is the point of intersection of its altitudes , then 3Q_M \le S. Is there such a number k> 0 that for any acute-angled triangle ABC and the point M of intersection of its altitudes, such thatthe inequality Q_M> k S holds?
b) For cases where the point M is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest k_1> 0 and the smallest k_2> 0 such that for an arbitrary triangle ABC, holds the inequality k_1S \le Q_M\le k_2S (for the center of the circumscribed circle, only acute-angled triangles ABC are considered).
The following method of approximate measurement is known for distances. Suppose, for example, that the observer is on the river bank at point C in order to measure its width. To do this, he fixes point A on the opposite bank so that the angle between the shoreline and the line CA is close to the line. Then the observer pulls forward the right hand with the raised thumb, closes left eye and aligns the raised finger with point A. Next, opens the left eye, closes right and estimates the distance between the point on the opposite bank to which the finger points, and point A. Multiply this distance by 10 and get the approximate value of the distance to point A, ie the width of the river. Justify this method of measuring distance.
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.
On the plane is drawn a triangle ABC and a circle \omega passing through the vertex C, the midpoints of the sides AC and BC and the point of intersection of the medians of the triangle ABC. The point K lies on the circle \omega such that \angle AKB=90^o. Using only with a ruler, draw a tangent to the circle \omega at point K.
The problem deals with the shortest paths between points A (0, 0) and B (n, n), composed of segments of lines x = k, 0 \le k \le n and y = s, 0 \le s \le n. Each such path is broken. It can contain from two (such broken two) to 2n (such broken also two) links. To get from A to B along a broken line consisting of 2n links, change the direction of traffic at each intersection. Let's divide all possible breaks (paths from A to B) into classes, depending on how many rectilinear segments they are composed of.
A) How many broken lines have exactly m, 3 \le m \le 6, links?
Let m be a natural number, 1 \le m \le n.
B) How many are broken, which have exactly 2m - 1 links?
C) How many are broken, which have exactly 2m links?
Having numbers that determine the quantitative composition of all classes of broken lines, make an identity for binomial coefficients.
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.
Points A, B, C, D lie on the sphere of radius 1. It is known that AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}. Prove that ABCD is a regular tetrahedron.
Eight circles of radius r located in a right triangle ABC (angle C is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle ABC.
The circle \omega_0 touches the line at point A. Let R be a given positive number. We consider various circles \omega of radius R that touch a line \ell and have two different points in common with the circle \omega_0. Let D be the touchpoint of the circle \omega_0 with the line \ell, and the points of intersection of the circles \omega and \omega_0 are denoted by B and C (Assume that the distance from point B to the line \ell is greater than the distance from point C to this line). Find the locuss of the centers of the circumscribed circles of all such triangles ABD.
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.
Given a quadrilateral ABCD, inscribed in a circle \omega such that AB=AD and CB=CD . Take the point P \in \omega. Let the vertices of the quadrilateral Q_1Q_2Q_3Q_4 be symmetric to the point P wrt the lines AB, BC, CD, and DA, respectively.
a) Prove that the points symmetric to the point P wrt lines Q_1Q_22, Q_2Q_3, Q_3Q_4 and Q_4Q_1, lie on one line.
b) Prove that when the point P moves in a circle \omega, then all such lines pass through one common point.
Investigate the largest number of sides of polygons that can be obtained in the cross-section of a plane with:
a) regular polyhedra;
b) convex octahedra that are not regular (here we consider a convex octahedron to be any convex polyhedron with 8 triangular faces, 12 edges and 6 vertices, from each of which 4 edges emerge).
Is there such a convex n-hedron that any polygon obtained in its cross-sectional plane has no more than n/10 sides?
The triangle ABC is drawn on the board such that AB + AC = 2BC. The bisectors AL_1, BL_2, CL_3 were drawn in this triangle, after which everything except the points L_1, L_2, L_3 was erased. Use a compass and a ruler to reconstruct triangle ABC.
Let E be an arbitrary point on the side BC of the square ABCD. Prove that the inscribed circles of triangles ABE, CDE, ADE have a common tangent.
Is it possible to cut a regular tetrahedron into several regular tetrahedra?
Given a circle \omega, on which marks the points A,B,C. Let BF and CE be the altitudes of the triangle ABC, M be the midpoint of the side AC. Find a the locus of the intersection points of the lines BF and EM for all positions of point A , as A moves on \omega.
Given a triangle PQR, the inscribed circle \omega which touches the sides QR, RP and PQ at points A, B and C, respectively, and AB^2 + AC^2 = 2BC^2. Prove that the point of intersection of the segments PA, QB and RC, the center of the circle \omega, the point of intersection of the medians of the triangle ABC, the point A and the midpoints of the segments AC and AB lie on one circle.
Inside the acute-angled triangle ABC, mark the point O so that \angle AOB=90^o, a point M on the side BC such that \angle COM=90^o, and a point N on the segment BO such that \angle OMN = 90^o. Let P be the point of intersection of the lines AM and CN, and let Q be a point on the side AB that such \angle POQ = 90^o. Prove that the lines AN, CO and MQ intersect at one point.
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 .
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.
In the triangle ABC on the ray BA mark the point K so that \angle BCA= \angle KCA , and on the median BM mark the point T so that \angle CTK=90^o . Prove that \angle MTC=\angle MCB .
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?
From the railway station - point S - come two tracks - rays, along which two long-distance trains are moving at constant speeds; the rays do not lie on one line. One by one from the tracks the first train moves in the direction of station S, and on the other , the second train departs from this station. We will consider the movement of trains only during the period of time during which they are not go beyond the tracks-rays. At each fixed point in time we will consider convex quadrilaterals, the vertices of which are the ends of trains-segments. Find out which one is needed and sufficient condition so that all points of intersection the diagonals of such quadrilaterals will lie on some parabola.
Akopyan A.V., Zaslavsky A.A. Geometric properties of curves of the second order. - M .: MTsNMO, 2011
a) Prove that when there is no face of a convex polyhedron triangle, there are at least eight of its vertices, from which it follows exactly three edges . (There are exactly eight such vertices in a cube).
b) Prove that when from each vertex of a convex polyhedron it turns out at least four edges, there are at least eight of its faces, each of which is a triangle. (The octahedron has exactly eight such faces).
In \vartriangle ABC on the sides BC, CA, AB mark feet of altitudes H_1, H_2, H_3 and the midpoint of sides M_1, M_3, M_3. Let H be orthocenter \vartriangle ABC. Suppose that X_2, X_3 are points symmetric to H_1 wrt BH_2 and CH_3. Lines M_3X_2 and M_2X_3 intersect at point X. Similarly, Y_3,Y_1 are points symmetric to H_2 wrt C_3H and AH_1.Lines M_1Y_3 and M_3Y_1 intersect at point Y. Finally, Z_1,Z_2 are points symmetric to H_3 wrt AH_1 and BH_2. Lines M_1Z_2 and M_2Z_1 intersect at the point Z Prove that H is the incenter \vartriangle XYZ .
The inscribed circle \omega of triangle ABC with center I touches the sides AB, BC, CA at points C_1, A_1, B_1. The circle circumsrcibed around \vartriangle AB_1C_1 intersects the circumscribed circle of ABC for second time at the point K. Let M be the midpoint BC, L be the midpoint of B_1C_1. The circle circumsrcibed around \vartriangle KA_1M cuts intersects \omega for second time at the point T. Prove that the circumscribed circles of triangles KLT and LIM are tangent.
Let k, m and n be given natural numbers, and | n - m | <k. Grasshopper wants from the origin - point (0, 0) - to get to the point (m, n), moving jumps, each of which takes place either 1 up or 1 right. Find the number of all such trajectories of a conic that hasno common points with lines y = x + k and y = x - k.
There are 6 points on the plane, none of which lie on one line. A straight line was drawn every two of these points. A point on a plane other than given, we call it triple, if exactly three of the conducted ones pass through it direct. Find the largest possible number of triple points.
What is the largest number of points of the coordinate plane can be noted in the ring \{(x,y)|1 \le x^2 + y^2 \le 2\} so that the distance between any two of they were not less than 1?
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
Is there such a convex polyhedron, all faces of which are triangles and the surface of which is painted blue, which can be cut into tetrahedra so that all vertices of tetrahedra are vertices of a polyhedron, any two tetrahedra with a common vertex had either a common edge or a common face, and each of the formed tetrahedra had exactly one face blue?
What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?
Let CH be the altitude of the triangle ABC drawn on the board, in which \angle C = 90^o, CA \ne CB. The mathematics teacher drew the perpendicular bisectors of segments CA and CB, which cut the line CH at points K and M, respectively, and then erased the drawing, leaving only the points C, K and M on the board. Restore triangle ABC, using only a compass and a ruler.
Let A_1A_2... A_{2n + 1} be a convex polygon, a_1 = A_1A_2, a_2 = A_2A_3, ..., a_{2n} = A_{2n}A_{2n + 1}, a_{2n + 1} = A_{2n + 1}A_1. Denote by: \alpha_i = \angle A_i, 1 \le i \le 2n + 1, \alpha_{k + 2n + 1} = \alpha_k, k \ge 1, \beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}, 1 \le i \le 2n + 1. Prove what if
\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}then a circle can be circumscribed around this polygon.
Does the inverse statement hold a place?
An acute-angled triangle ABC is given, through the vertices B and C of which a circle \Omega, A \notin \Omega, is drawn. We consider all points P \in \Omega, that do not lie on none of the lines AB and AC and for which the common tangents of the circumscribed circles of triangles APB and APC are not parallel. Let X_P be the point of intersection of such two common tangents.
a) Prove that the locus of points X_P lies to some two lines.
b) Prove that if the circle \Omega passes through the orthocenter of the triangle ABC, then one of these lines is the line BC.
The inscribed circle \omega of the triangle ABC touches its sides BC, CA, and AB at the points D, E, and F, respectively. Let the points X, Y, and Z of the circle \omega be diametrically opposite to the points D, E, and F, respectively. Line AX, BY and CZ intersect the sides BC, CA and AB at the points D', E' and F', respectively. On the segments AD', BE' and CF' noted the points X', Y' and Z', respectively, so that D'X'= AX, E'Y' = BY, F'Z' = CZ. Prove that the points X', Y' and Z' coincide.
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.
Using only a compass and a ruler, reconstruct triangle ABC given the following three points: point M the intersection of its medians, point I is the center of its inscribed circle and the point Q_a is touch point of the inscribed circle to side BC.
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.
In an isosceles trapezoid ABCD with bases AD and BC, diagonals intersect at point P, and lines AB and CD intersect at point Q. O_1 and O_2 are the centers of the circles circumscribed around the triangles ABP and CDP, r is the radius of these circles. Construct the trapezoid ABCD given the segments O_1O_2, PQ and radius r.
Points P, Q, R were marked on the sides BC, CA, AB, respectively. Let a be tangent at point A to the circumcircle of triangle AQR, b be tangent at point B to the circumcircle of the triangle BPR, c be tangent at point C to the circumscribed circle triangle CPQ. Let X be the point of intersection of the lines b and c, Y be the point the intersection of lines c and a, Z is the point of intersection of lines a and b. Prove that the lines AX, BY, CZ intersect at one point if and only if the lines AP, BQ, CR intersect at one point.
The altitude AH, BT, and CR are drawn in the non isosceles triangle ABC. On the side BC mark the point P; points X and Y are projections of P on AB and AC. Two common external tangents to the circumscribed circles of triangles XBH and HCY intersect at point Q. The lines RT and BC intersect at point K.
a). Prove that the point Q lies on a fixed line independent of choice P.
b). Prove that KQ = QH.
Specify at least one right triangle ABC with integer sides, inside which you can specify a point M such that the lengths of the segments MA, MB, MC are integers. Are there many such triangles, none of which are are similar?
The Fibonacci sequence is given by equalitiesF_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N.
a) Prove that for every m \ge 0, the area of the triangle A_1A_2A_3 with vertices A_1(F_{m+1},F_{m+2}), A_2 (F_{m+3},F_{m+4}), A_3 (F_{m+5},F_{m+6}) is equal to 0.5.
b) Prove that for every m \ge 0 the quadrangle A_1A_2A_4 with vertices A_1(F_{m+1},F_{m+2}), A_2 (F_{m+3},F_{m+4}), A_3 (F_{m+5},F_{m+6}), A_4 (F_{m+7},F_{m+8}) is a trapezoid, whose area is equal to 2.5.
c) Prove that the area of the polygon A_1A_2...A_n , n \ge3 with vertices does not depend on the choice of numbers m \ge 0, and find this area.
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.
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.
In the acute triangle ABC, the altitude AH is drawn. Using segments AB,BH,CH and AC as diameters circles \omega_1, \omega_2, \omega_3 and \omega_4 are constructed respectively. Besides the point H, the circles \omega_1 and \omega_3 intersect at the point P, and the circles \omega_2 and \omega_4 interext at point Q. The lines BQ and CP intersect at point N. Prove that this point lies on the midline of triangle ABC, which is parallel to BC.
Inside the circle of diameter 1 are several segments, whose total length equal to 30. The lengths of the segments and their number can be any, segments can intersect or touch a circle. Can that happen if no line intersects more than:
a) 17 segments?
b) 25 segments?
Hannusya, Petrus and Mykolka drew independently one isosceles triangle ABC, all angles of which are measured as a integer number of degrees. It turned out that the bases AC of these triangles are equals and for each of them on the ray BC there is a point E such that BE=AC, and the angle AEC is also measured by an integer number of degrees. Is it in necessary that:
a) all three drawn triangles are equal to each other?
b) among them there are at least two equal triangles?
On the base BC of the isosceles triangle ABC chose a point D and in each of the triangles ABD and ACD inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers
the apex A is located . Use a compass and ruler to restore the triangle ABC , if we know that :
a) AD is angle bisector,
b) AD is median.
At the altitude AH_1 of an acute non-isosceles triangle ABC chose a point X , from which draw the perpendiculars XN and XM on the sides AB and AC respectively. It turned out that H_1A is the angle bisector MH_1N. Prove that X is the point of intersection of the altitudes of the triangle ABC.
Let \omega_a, \omega_b, \omega_c be the exscribed circles tangent to the sides a, b, c of a triangle ABC, respectively, I_a, I_b, I_c be the centers of these circles, respectively, T_a, T_b, T_c be the points of contact of these circles to the line BC, respectively. The lines T_bI_c and T_cI_b intersect at the point Q. Prove that the center of the circle inscribed in triangle ABC lies on the line T_aQ.
There are n different points on the plane. Divide this set of points into two non-empty subsets \{A_1,A_2,...,A_k\} and \{B_1,B_2,...,B_{n-k}\}. Let's call this partition balanced, if for this partition , exists a point M on the plane, such thatMA_1+MA_2+ ...+MA_k=MB_1+MB_2+...+MB_{n-k}.
a) Prove that for any set of n \ge 2 points there is at least one balanced partition.
b) Prove that for any set with an even number n \ge 2 points there is at least one balanced division into sets of
\frac{n}{2} points in each.
c) Prove that for every n \ge 2 there is a set of n points for which the number of balanced partitions is not less than 2^{n-2}.
Partitions in which two sets simply change with each other, we consider the same.
Petrik has the same parallelograms with angles 45^o and 135^o and lengths of sides 1 and 2.
a) Prove that in a rectangular box of size 2 \times n, n \ge 2 , he will not be able to place more than 2n-2 parallelograms.
b) Prove that in a square box of size 4\times 4 he will not be able to place 13 parallelograms. Give an example of placing 12 parallelograms in a box.
c) Give an example of placement of 4n-4 parallelograms in a rectangular box of size 4 \times n, n \ge 4.
Parallelograms can be inverted.
n points are marked on the board points that are vertices of the regular n -gon. One of the points is a chip. Two players take turns moving it to the other marked point and at the same time draw a segment that connects them. If two points already connected by a segment, such a move is prohibited. A player who can't make a move, lose. Which of the players can guarantee victory?
Cut an isosceles trapezoid into three similar trapezoids in all possible ways. For an isosceles trapezoid with bases a and b and lateral side c=1, find out the necessary and sufficient conditions for the implementation of each method
cutting.
A parallelogram is given on the plane. Using only one-sided ruler, we want to draw n lines on the plane so that along the formed parallelogram lines could be cut into 5 equal polygons. At this we allow some polygons to be "assembled" from several parts. Will we succeed if we do this for a given parallelogram if:
a) n =12 ?
b) n=11 ?
c) n=10 ?
In triangle ABC, point I is the center, point I_a is the center of the excircle tangent to the side BC. From the vertex A inside the angle BAC draw rays AX and AY. The ray AX intersects the lines BI, CI, BI_a, CI_a at points X_1, ..., X_4, respectively, and the ray AY intersects the same lines at points Y_1, ..., Y_4 respectively. It turned out that the points X_1,X_2,Y_1,Y_2 lie on the same circle. Prove the equality\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.
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.
On the side CD of the square ABCD, the point F is chosen and the equal squares DGFE and AKEH are constructed (E and H lie inside the square). Let M be the midpoint of DF, J is the incenter of the triangle CFH. Prove that:
a) the points D, K, H, J, F lie on the same circle;
b) the circles inscribed in triangles CFH and GMF have the same radii.
In the triangle ABC on the side BC, the points D and E are chosen so that the angle BAD is equal to the angle EAC. Let I and J be the centers of the inscribed circles of triangles ABD and AEC respectively, F be the point of intersection of BI and EJ, G be the point of intersection of DI and CJ. Prove that the points I, J, F, G lie on one circle, the center of which belongs to the line I_bI_c, where I_b and I_c are the centers of the exscribed circles of the triangle ABC, which touch respectively sides AC and AB.
There are a million points on the plane, none of which lie on one lines . Nine players take turns connecting these points with segments, each pencil of its color. It is allowed to connect any two points in one move, which have not yet been connected. The winner is the one who gets the triangle first vertices at given points, all sides of which have the same color. Can such a game end in a draw?
Final Battle
From the point O on the plane several vectors are drawn, the sum of the lengths of which is equal to 4. Prove that you can choose one or more of these vectors so that the length of their sum is greater than 1.
The apple pie has the shape of a regular n-gon inscribed in a circle of radius 1. From the midpoint of each side in any direction of the inner part of the pie make a straight cut of length 1. Prove that at least one piece will be cut from the pie.
Prove that in a convex quadrilateral with area S and perimeter P it is possible to place a circle of radius \frac{S}{P}.
Let A_1A_2...A_n is a regular polygon inscribed in a circle of radius R with center at the point O ,X is an arbitrary point of a circle of radius r with center O. Prove that the value of the sum \sum_{i=1}^{n}XA_i^{2m} (for a fixed m \in N) does not depend on the choice of points X. Express the value of the specified sum in terms of R, r, m and n.
The lengths of the two legs of one right triangle coincide with the lengths of the leg and the hypotenuse of the other right triangle, respectively. Can the lengths of all sides of both triangles be expressed as integers at the same time?
Is it possible to represent a five-pointed star (see figure) so that the lengths of the segments AB, BC, CD, DE, EF, FG, GH, HI, IJ, JA are ten consecutive natural numbers?
Given a prime number p> 3. Consider on the coordinate plane Oxy the set M of such points (x, y) with integer coordinates that 0 \le x <p, 0 \le y <p. Prove that we can note p different points of the set M so that any 4 of them are not vertices of the parallelogram, and any 3 of them do not lie on the same line.
Triangle ABC is isosceles. The circle \omega_1 passes through points B and C, and intersects the segments AB and AC at the inner points M and N, respectively. Let O_1 be the center of the circle \omega_1, X and A be the points of intersection of the circumscribed circles of triangles ABC and AMN. Prove that \angle AXO_1 = 90^o.
Let ABCD be an arbitrary convex quadrilateral, X be an arbitrary point of a plane that does not lie on the lines AB, BC, CD, DA. Denote M_1,M_2, M_3,M_4 the points of intersection of medians of triangles ABX,CDX,BCX,DAX respectively. Prove that the area of the quadrilateral M_1M_2M_3M_4 does not depend on the position of point X. Formulate and prove the spatial analogue of the problem.
The diagonal of a right parallelepiped is 1, k_1 and k_2 are the ratio of the lengths of its two edges to the third. Calculate the surface area of this parallelepiped. Let k_1 = ak_2 (a \in R). At what value of k_2 is the value of S (k_1, k_2) maximum?
Prove that in any triangle (p-a) (p-b) + (p-c) (p-a)> \sqrt3 S, where a, b, c are sides, p is half perimeter, S is area of a triangle.
In triangle ABC, \angle ABC = 100^o, \angle ACB = 65^o, point M belongs to side AB, point N belongs to side AC, and \angle MCB = 55^o, \angle NBC = 80^o. Find \angle NMC.
10 points are selected on the circle. What is the largest number of segments with ends at these
points that can be made so that none of the three segments do not form a triangle?
The volume of a parallelepiped is 216 cm^3, and the surface area (total) is 216 cm^2. Prove that the parallelepiped is a cube.
The altitude drawn from one of the vertices of the acute-angled triangle ABC intersects the opposite side at the point H. From the point H, the perpendiculars NE and HF on the other two sides are drawn. Prove that the length of the segment EF does not depend on the choice of the vertex.
Is it always possible that 12 points marked on a plane, none of which lie on the same line, can be divided into 3 groups of 4 points each, so that each of the four points of the group is the vertices of a convex quadrilateral and the interior of any two of did these quadrilaterals have no common points?
Circles \omega_1 and \omega_2 with center O_1 and ,respectively, intersect at points A and B. Line O_1B intersects \omega_1 at point Q (Q \ne B) and line O_2B intersects \omega_1 at point P (P\ne B). A line m passing through point B intersects \omega_1 and \omega_2 at points M and N, respectively (M \ne B, N \ne B). Prove that AP + AQ= MN.
Let H be the point of intersection of the altitudes of an acute-angled triangle ABC. Prove that the midpoints of the segments AB and CH and the point of intersection of the bisectors of the angles CAH and CBH lie on the same line.
Points C_1,A_1 and B_1 are chosen on the sides AB, BC and CA of the triangle ABC, respectively, different from the vertices. It is known that \angle A_1C_1B_1=\angle B_1A_1C=\angle C_1B_1A=\phi. Is the triangle ABC necessarily equilateral?
Let the quadrilateral ABCD be inscribed in a circle. Through points A and D are drawn lines \ell_A and \ell_D perpendicular on the line AD respectively; Through points B and C are drawn lines \ell_B and \ell_C perpendicular on the line BC, respectively. Let \ell_A cap \ell_C=M, \ell_B cap \ell_D=N, E be the point of intersection of the lines AD and BC. Prove that \angle DEN = \angle CEM.
Let AK, BL and CM be the angle bisectors of triangle ABC. Prove that\frac{BC}{AK}\cdot \cos \frac{\angle BAC}{2}+ \frac{CA}{BL}\cdot \cos \frac{\angle CBA}{2}+ \frac{AB}{CM}\cdot \cos \frac{\angle ACB}{2} \ge 3
Inside the convex quadrilateral ABCD , mark a point O such that \angle AOP = \angle COQ, where P is the point of intersection of the rays BA and CD, and O is the point of intersection of the rays AD and BC. Prove that the bisectors of the angles AOC and BOD are perpendicular.
Let ABC be an acute-angled triangle with orthocenter H and incenter I, and AC\ne BC. The lines CH and CI intersect the circumscribed circle of triangle ABC for second time at points D and L, respectively. Prove that \angle CIH = 90^o if and only if \angle IDL = 90^o.
Prove that the triangle ABC is right-angled if and only if
\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}-\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}=\frac{1}{2}
There are three points N, Q and K on the plane, and the point Q lies between the points N and K, and 1 <\frac{NQ}{QK}<3. Prove that there are exactly two such triplets of points A, B and C, that \angle ABC = 120^o, point Q is the center of the inscribed circle of triangle ABC, CN is the angle bisector of the triangle ABC, and K is its point of intersection with the segment connecting the ends M and P of the bisectors AP and BM of the triangle ABC.
The exscribed circle \omega_C of the triangle ABC touches the side AB and the extensions of the sides BC and CA at the points M, N, and P, respectively, and the exscribed circle \omega_B touches the side AC and the extensions of the sides AB and BC at the points S, Q, and R, respectively. Let X = MN \cap RS, Y = PN \cap RQ. Prove that the points X, Y and A lie on the same line.
A convex quadrilateral ABCD is defined on the coordinate plane xOy, all vertices of which lie on the graph of the function y = \frac{1}{x}, and the abscissas of points A and D are negative, the abscissas of points B and C are positive and x_B < x_C. The segment AC passes through the origin. Prove that \angle BAD = \angle BCD.
Let AA_1, BB_1 and CC_1 be the altitudes of the acute triangle ABC. Prove that the feet of the perpendiculars drawn from point C_1 on the lines AC, BC, BB_1 and AA_1 lie on one line.
Find the right triangle of the smallest perimeter, such that all lengths of the sides and the length of the altitude drawn to the hypotenuse are integers.
Can each of the distances from the center of the inscribed circle of a triangle to its vertices be less than the diameter of this circle?
On the parabola y = x^2 we take the point A (2,4). On this parabola, find all points B for which AB cannot be the hypotenuse of the triangle inscribed in it.
Let ABCD be a convex quadrilateral, m,n be even natural numbers. The sides of AB,CD are divided into m equal parts, the sides BC, AD are divided into n equal parts. Respective endpoints of opposite sides are connected. The formed cells are painted in yellow and blue colors in a checkerboard pattern. Prove that the sums of the areas of the yellow and blue cells are equal.
A right parallelepiped is given in a right Cartesian system. The coordinates of its four vertices, which do not lie in the same plane, are integers. Are the coordinates of the other vertices necessarily integers?
Suppose that the squares ABFP and CBED are constructed on the sides AB and BC of the triangle ABC. Let M and N be the midpoints of the segments PD and AE, respectively,. Prove that M, N, B are the vertices of an isosceles right triangle (if they do not coincide).
Construct an acute-angled triangle ABC given three segments: angle bisector AL = \ell_a, altitude AK = h_a and AH, where H is the point of intersection of the heights of this triangle.
XII didn't take place
Let h_a and h_b be the altitudes of triangle ABC drawn from its vertices A and B, respectively. Let r be the radius of the inscribed circle. Prove that if \sin \frac{\angle C}{2} \ge \frac{3}{4}, then h_a + h_b\ge 7r.
Two circles \omega_1 and \omega_2, which do not have common points, are internally tangent to the circle \omega. The two inner common tangents of circles \omega_1 and \omega_2 intersect the circle \omega at four points. Let M and N be any two of them lying on the same arc of the circle \omega with ends at its points of contact with the circles \omega_1 and \omega_2. Prove that the line MN is parallel to any of the external common tangents to the circles \omega_1 and \omega_2.
Given a convex quadrilateral ABCD, in which BA = BC and DA = DC. It is known that on the diagonal AC there is a point K such that KA = KB, and around the quadrilateral BCDK we can circumscribe a circle. Prove that the triangle BCD is isosceles.
On the sides AB, BC and CA of the triangle ABC mark points K, M and N, respectively, different from the vertices, such that KM\parallel AC, KN \parallel BC. Let the segments AM and NK intersect at the point E, and let the segments BN and MK intersect at the point F. Prove that EF\le \frac14 AB.
Prove that any isosceles trapezoid with perpendicular diagonals can be cut into four similar convex quadrilaterals in at least three different ways.
In a triangle ABC (AB> AC) an inscribed circle with center at point I touches the side BC at point D. The bisector of the angle BAC intersects the circumscribed circle of triangle ABC for the second time at point M, and the line MD intersects the same circle at point P. Prove that \angle API = 90^o.
Given a triangle ABC. Construct points X and Y on its sides AB and AC respectively, such that BX+CY=BC and around the quadrilateral BXYC one can circumscribe a circle.
The inscribed circle \omega from triangle ABC touches its side BC at point D. The bisector of the angle ADB intersects the circle \omega for the second time at the point N. The bisector of the angle ADC intersects the circle \omega for the second time at the point M. Prove that the lines BM, CN and AD intersect at one point.
soon 2014-2016
The segment AB is given, the point M is it's midpoint. Find the locus of points X such that \angle MXB = \angle XAB + \angle XBA.
Using a compass and a ruler, restore the triangle given two midpoints and a point inside the triangle, from which each side is visible under the same angle.
In the non-obtuse triangle ABC with \angle A = 60^o, the median BM is drawn, the altitude CH and the angle bisector AL so that they form a triangle similar to triangle ABC. Find the angles B and C of this triangle.
Given an acute triangle ABC, with AH its height, AL angle bisector, point T the point of tangency of the exscribed circle to the side BC, \omega is inscribed circle of triangle ABC. Let the rays AH, AL and AT intersect for the first time \omega at points P, Q and R, respectively. Using a compass and a ruler, restore the triangle ABC at the points P, Q and R.
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