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 that$$AK^2 - BC^2 = BK^2 - AC^2 = CK^2 - AB^2.$$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 inequality$$IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr}.$$
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, 1$2$ 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 E$M$ 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 equalities$$F_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 that$$MA_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 $R$.
source: http://tym.in.ua/
No comments:
Post a Comment