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12.53. In triangle $ABC$, the angle bisectors $AD$ and $BE$ are drawn. Find the measure of angle $C$, given that $AD \cdot BC = BE \cdot AC$ and $AC \neq BC$. | 12.53. The quantities $A D \cdot B C \sin A D B$ and $B E \cdot A C \sin A E B$ are equal, as they are equal to twice the area of triangle $A B C$. Therefore, $\sin A D B = \sin A E B$. There are two possible cases.
1. $\angle A D B = \angle A E B$; in this case, points $A, E, D, B$ lie on the same circle, so $\angle ... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,592 |
12.54. Find the angle $B$ of triangle $A B C$, if the length of the height $C H$ is half the length of side $A B$, and $\angle B A C=75^{\circ}$. | 12.54. Let $B'$ be the point of intersection of the perpendicular bisector of segment $AC$ with line $AB$. Then $AB' = CB'$ and $\angle AB'C = 180^\circ - 2 \cdot 75^\circ = 30^\circ$. Therefore, $AB' = CB' = 2CH = AB$, i.e., $B' = B$ and $\angle B = 30^\circ$. | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,593 |
12.55*. In a right triangle $ABC$ with a right angle at $A$, a circle is constructed on the altitude $AD$ as a diameter, intersecting side $AB$ at point $K$ and side $AC$ at point $M$. Segments $AD$ and $KM$ intersect at point $L$. Find the acute angles of triangle $ABC$, given that $AK: AL = AL: AM$. | 12.55. It is clear that $A K D M$ is a rectangle and $L$ is the point of intersection of its diagonals. Since $A D \perp B C$ and $A M \perp B A$, then $\angle D A M = \angle A B C$. Similarly, $\angle K A D = \angle A C B$. Drop a perpendicular $A P$ from point $A$ to the line $K M$. Let's assume for definiteness that... | 15 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,594 |
12.56*. In triangle $ABC$, angle $C$ is twice as large as angle $A$ and $b=2a$. Find the angles of this triangle. | 12.56. Let $CD$ be the bisector. Then $BD = ac/(a+b)$. On the other hand, $\triangle BDC \sim \triangle BCA$, so $BD:BC = BC:BA$, i.e., $BD = a^2/c$. Therefore, $c^2 = a(a+b) = 3a^2$. The sides of triangle $ABC$ are $a, 2a$, and $\sqrt{3}a$, so its angles are 30, 90, and $60^{\circ}$. | 30,90,60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,595 |
12.57*. In triangle $A B C$, the bisector $B E$ is drawn and a point $K$ is taken on side $B C$ such that $\angle A K B=2 \angle A E B$. Find the measure of angle $A K E$, if $\angle A E B=\alpha$. | 12.57. Let $\angle ABC = 2x$, then the external angle $A$ of triangle $ABE$ is $\angle ABE + \angle AEB = x + \alpha$. Further, $\angle KAE - \angle BAK = (180^\circ - x - \alpha) - (180^\circ - 2x - 2\alpha) = x + \alpha$. Therefore, $AE$ is the bisector of the external angle $A$ of triangle $ABK$. Since $BE$ is the b... | 90-\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,596 |
12.60*. In an acute-angled triangle $A B C$, the segments $B O$ and $C O$, where $O-$ is the center of the circumscribed circle, are extended to intersect the sides $A C$ and $A B$ at points $D$ and $E$. It turns out that $\angle B D E=50^{\circ}$ and $\angle C E D=30^{\circ}$. Find the measures of the angles of triang... | 12.60. Since $\angle B D E=50^{\circ}$ and $\angle C E D=30^{\circ}$, then $\angle B O C=\angle E O D=$ $=180^{\circ}-50^{\circ}-30^{\circ}=100^{\circ}$. Let's assume that the diameters $B B^{\prime}$ and $C C^{\prime}$ of the circle are fixed, with $\angle B O C=100^{\circ}$, and point $A$ moves along the arc $B^{\pri... | \angleA=50,\angleB=70,\angleC=60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,599 |
12.61. Circle $S$ with center $O$ is inscribed on the base $BC$ of isosceles triangle $ABC$, touching the equal sides $AB$ and $AC$. Points $P$ and $Q$ are taken on sides $AB$ and $AC$ such that segment $PQ$ is tangent to circle $S$. Prove that then $4 PB \cdot CQ = BC^2$. | 12.61. Let $D, E$ and $F$ be the points of tangency of the circle with $BP, PQ$ and $QC$; $\angle BOD = 90^{\circ} - \angle B = 90^{\circ} - \angle C = \angle COF = \alpha, \angle DOP = \angle POE = \beta$ and $\angle EOQ = \angle QOF = \gamma$. Then $180^{\circ} = \angle BOC = 2\alpha + 2\beta + 2\gamma$, i.e., $\alph... | 4PB\cdotCQ=BC^2 | Geometry | proof | Yes | Yes | olympiads | false | 27,600 |
12.62*. Let $E$ be the midpoint of side $AB$ of square $ABCD$, and let points $F$ and $G$ be chosen on sides $BC$ and $CD$ such that $AG \| EF$. Prove that segment $FG$ is tangent to the circle inscribed in square $ABCD$. | 12.62. Let $P$ and $Q$ be the midpoints of sides $BC$ and $CD$ respectively. Points $P$ and $Q$ are the points of tangency of the inscribed circle with sides $BC$ and $CD$. Therefore, it is sufficient to check that $PF + GQ = FG$. Indeed, if $F'G'$ is a segment parallel to $FG$ and tangent to the inscribed circle, then... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,601 |
12.63*. A chord of a circle is at a distance $h$ from the center. In each of the segments cut off by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc, and the other two - on the chord or its extension (Fig. 12.3). What is the difference in the lengths of the sides of these s... | 12.63. Let's denote the vertices of the squares as shown in Fig. 12.7. Let \( O \) be the center of the circle, \( H \) be the midpoint of the given chord, and \( K \) be the midpoint of the segment \( A A_{1} \). Since \(\operatorname{tg} A H B = 2 = \operatorname{tg} A_{1} H D_{1}\), the point \( H \) lies on the lin... | \frac{8h}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,602 |
12.64*. Find the ratio of the sides of a triangle, one of whose medians is divided into three equal parts by the inscribed circle. | 12.64. Let the median $B M$ of triangle $A B C$ intersect the inscribed circle at points $K$ and $L$, such that $B K=K L=L M=x$. For definiteness, let the point of tangency of the inscribed circle with side $A C$ lie on segment $M C$. Then, since under symmetry with respect to the perpendicular bisector of segment $B M... | :b:=5:10:13 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,603 |
12.65*. A square is inscribed in a circle, and another square is inscribed in the segment cut off from the circle by one of the sides of the first square. Find the ratio of the lengths of the sides of these squares. | 12.65. Let $2a$ and $2b$ be the side lengths of the first and second squares, respectively. Then the distance from the center of the circle to the vertices of the second square, which lie on the circle, is $\sqrt{(a+2b)^{2}+b^{2}}$. On the other hand, this distance is $\sqrt{2}a$. Therefore, $(a+2b)^{2}+b^{2}=2a^{2}$, ... | 5b | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,604 |
12.66*. On the segment $A B$, a point $C$ is taken, and on the segments $A C$, $B C$, and $A B$ as diameters, semicircles are constructed lying on the same side of the line $A B$. Through the point $C$, a line perpendicular to $A B$ is drawn, and in the curvilinear triangles $A C D$ and $B C D$ inscribed circles $S_{1}... | 12.66. Let $P$ and $Q$ be the midpoints of segments $AC$ and $AB$, $R$ be the center of circle $S_{1}$; $a=AC/2$, $b=BC/2$, $x$ be the radius of circle $S_{1}$. It is easy to verify that $PR=a+x$, $QR=a+b-x$, and $PQ=b$. Let's draw the altitude $RH$ in triangle $PQR$. The distance from point $R$ to line $CD$ is $x$, so... | \frac{}{+b} | Geometry | proof | Yes | Yes | olympiads | false | 27,605 |
12.67*. The centers of circles with radii 1, 3, and 4 are located on the sides $A D$ and $B C$ of rectangle $A B C D$. These circles touch each other and the lines $A B$ and $C D$ as shown in Fig. 12.5. Prove that there exists a circle that touches all these circles and the line $A B$.
## $\S 9$. Various Problems | 12.67. Let $x$ be the radius of the circle $S$ touching the circles $S_{1}$ and $S_{2}$ and the ray $A B$, and $y$ be the radius of the circle $S^{\prime}$ touching the circles $S_{2}$ and $S_{3}$ and the ray $B A$. The position of the circle touching the circle $S_{1}$ and the ray $A B$ (respectively, $S_{3}$ and the ... | \frac{4}{3} | Geometry | proof | Yes | Yes | olympiads | false | 27,606 |
12.69. Find the height of a trapezoid with bases $A B$ and $C D$ equal to $a$ and $b(a<b)$, the angle between the diagonals is $90^{\circ}$, and the angle between the extensions of the lateral sides is $45^{\circ}$. | 12.69. Complete the triangle $ABC$ to form a parallelogram $ABCE$ (Fig. 12.8). Let $BC = x$ and $AD = y$. Then $(b-a) h = 2 S_{AED} = x y \sin 45^{\circ}$ and $(b-a)^{2} = x^{2} + y^{2} - 2 x y \cos 45^{\circ} = x^{2} + y^{2} - 2 x y \sin 45^{\circ}$. By the Pythagorean theorem, $a^{2} + b^{2} = (AO^{2} + BO^{2}) + (CO... | \frac{}{b-} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,608 |
12.70. The inscribed circle touches the side $B C$ of triangle $A B C$ at point $K$. Prove that the area of the triangle is $B K \cdot K C \operatorname{ctg}(\alpha / 2)$. | 12.70. Since $B K=(a+c-b) / 2$ and $K C=(a+$ $+b-c) / 2$ (see problem 3.2), then $B K \cdot K C=\left(a^{2}-(b-c)^{2}\right) / 4=$ $=S \operatorname{tg}(\alpha / 2)$ (see problem 12.12). | BK\cdotKC\operatorname{ctg}(\alpha/2) | Geometry | proof | Yes | Yes | olympiads | false | 27,609 |
12.71. Prove that if $\operatorname{ctg}(\alpha / 2)=(b+c) / a$, then the triangle is a right triangle. | 12.71. Since $(b+c) / a=\cos ((\beta-\gamma) / 2) / \sin (\alpha / 2)$ (Problem 12.4), then $\cos ((\beta-\gamma) / 2)=\cos (\alpha / 2)$, i.e., $\beta-\gamma=$
Fig. 12.8 $= \pm \alpha$. I... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,610 |
12.72. The continuations of the angle bisectors of triangle $ABC$ intersect the circumscribed circle at points $A_{1}, B_{1}$ and $C_{1}$. Prove that $S_{ABC} / S_{A_{1}B_{1}C_{1}} = 2r / R$, where $r$ and $R$ are the radii of the inscribed and circumscribed circles of triangle $ABC$. | 12.72. It is easy to check that $S_{A B C}=2 R^{2} \sin \alpha \sin \beta \sin \gamma$. Similarly, $S_{A_{1} B_{1} C_{1}}=2 R^{2} \sin ((\beta+\gamma) / 2) \sin ((\alpha+\gamma) / 2) \sin ((\alpha+$ $+\beta) / 2)=2 R^{2} \cos (\alpha / 2) \cos (\beta / 2) \cos (\gamma / 2)$. Therefore, $S_{A B C} / S_{A_{1} B_{1} C_{1}... | 2r/R | Geometry | proof | Yes | Yes | olympiads | false | 27,611 |
12.73. Prove that the sum of the cotangents of the angles of triangle $ABC$ is equal to the sum of the cotangents of the angles of the triangle formed by the medians of triangle $ABC$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result dir... | 12.73. The sum of the cotangents of the angles of a triangle is equal to $\left(a^{2}+b^{2}+c^{2}\right) / 4 S$ (Problem 12.44, a)). In addition, $m_{a}^{2}+m_{b}^{2}+m_{c}^{2}=3\left(a^{2}+b^{2}+c^{2}\right) / 4$ (Problem 12.11, b)) and the area of the triangle formed by the medians of triangle $A B C$ is $3 / 4$ the ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,612 | |
12.74*. Let $A_{4}$ be the orthocenter of triangle $A_{1} A_{2} A_{3}$. Prove that there exist numbers $\lambda_{1}, \ldots, \lambda_{4}$ such that $A_{i} A_{j}^{2}=\lambda_{i}+\lambda_{j}$, and if the triangle is not right-angled, then $\sum\left(1 / \lambda_{i}\right)=0$.
## §10. The Method of Coordinates | 12.74. One of the points $A_{i}$ lies inside the triangle formed by the other three points, so we can assume that the triangle $A_{1} A_{2} A_{3}$ is acute (or right). The numbers $\lambda_{1}, \lambda_{2}$, and $\lambda_{3}$ are easily found from the corresponding system of equations; as a result, we get $\lambda_{1}=... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,613 |
12.75. The coordinates of the vertices of a triangle are rational. Prove that the coordinates of the center of its circumscribed circle are also rational. | 12.75. Let $\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)$ and $\left(a_{3}, b_{3}\right)$ be the coordinates of the vertices of a triangle. The coordinates of the center of its circumscribed circle are given by the system of equations
$$
\begin{aligned}
& \left(x-a_{1}\right)^{2}+\left(y-b_{1}\right)^{2}=\left(... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,614 |
12.76. Diameters $A B$ and $C D$ of circle $S$ are perpendicular. Chord $E A$ intersects diameter $C D$ at point $K$, and chord $E C$ intersects diameter $A B$ at point $L$. Prove that if $C K: K D=2: 1$, then $A L: L B=3: 1$. | 12.76. Let's take points $K$ and $L$ on segments $A B$ and $C D$, respectively, dividing them in the given ratios. It is sufficient to prove that the point of intersection of lines $A K$ and $C L$ lies on the circle $S$. Introduce a coordinate system with the origin at the center $O$ of the circle $S$ and axes $O x$ an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,615 |
12.77. In triangle $ABC$, angle $C$ is right. Prove that under a homothety with center $C$ and coefficient 2, the inscribed circle transforms into a circle that is tangent to the circumscribed circle. | 12.77. Let $d$ be the distance from the center of the circumscribed circle to the image of the center of the inscribed circle under the considered homothety. It is sufficient to check that $R=d+2 r$. Let $(0,0),(2 a, 0)$ and $(0,2 b)$ be the coordinates of the vertices of the given triangle. Then $(a, b)$ are the coord... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,616 |
13.1. a) Prove that a triangle can be formed from the medians of a triangle.
b) A triangle $A_{1} B_{1} C_{1}$ is formed from the medians of triangle $A B C$, and a triangle $A_{2} B_{2} C_{2}$ is formed from the medians of triangle $A_{1} B_{1} C_{1}$. Prove that triangles $A B C$ and $A_{2} B_{2} C_{2}$ are similar,... | 13.1. a) Let $\boldsymbol{a}=\overrightarrow{B C}, \boldsymbol{b}=\overrightarrow{C A}$ and $\boldsymbol{c}=\overrightarrow{A B} ; A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ be the medians of triangle $A B C$. Then $\overrightarrow{A A^{\prime}}=(\boldsymbol{c}-\boldsymbol{b}) / 2, \overrightarrow{B B^{\prime}}=(\b... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,618 |
13.2. The sides of triangle $T$ are parallel to the medians of triangle $T_{1}$. Prove that the medians of triangle $T$ are parallel to the sides of triangle $T_{1}$. | 13.2. Let $\boldsymbol{a}, \boldsymbol{b}$ and $\boldsymbol{c}$ be the vectors of the sides of triangle $T$. Then $(\boldsymbol{b}-\boldsymbol{a}) / 2$, $(\boldsymbol{a}-\boldsymbol{c}) / 2$ and $(\boldsymbol{c}-\boldsymbol{b}) / 2$ are the vectors of its medians. We can consider that $\boldsymbol{a}, \boldsymbol{b}$ a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,619 |
13.3. $M_{1}, M_{2}, \ldots, M_{6}$ are the midpoints of the sides of a convex hexagon $A_{1} A_{2} \ldots A_{6}$. Prove that there exists a triangle whose sides are equal and parallel to the segments $M_{1} M_{2}, M_{3} M_{4}, M_{5} M_{6}$. | 13.3. It is clear that $2 \overrightarrow{M_{1} M_{2}}=\overrightarrow{A_{1} A_{2}}+\overrightarrow{A_{2} A_{3}}=\overrightarrow{A_{1} A_{3}}, 2 \overrightarrow{M_{3} M_{4}}=\overrightarrow{A_{3} A_{5}}$ and $2 \overrightarrow{M_{5} M_{6}}=$ $=\overrightarrow{A_{5} A_{1}}$. Therefore, $\overrightarrow{M_{1} M_{2}}+\ove... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,620 |
13.5. The sum of four unit vectors is zero. Prove that they can be divided into two pairs of opposite vectors. | 13.5. From the given vectors, a convex quadrilateral can be formed. All sides of this quadrilateral are equal to 1, so it is a rhombus; pairs of its opposite sides provide the required partition. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,622 |
13.6. Let $E$ and $F$ be the midpoints of sides $AB$ and $CD$ of quadrilateral $ABCD$, and let $K, L, M$, and $N$ be the midpoints of segments $AF, CE, BF$, and $DE$. Prove that $KLMN$ is a parallelogram. | 13.6. Let $\boldsymbol{a}=\overrightarrow{A E}, \boldsymbol{b}=\overrightarrow{D F}$ and $\boldsymbol{v}=\overrightarrow{A D}$. Then $2 \overrightarrow{A K}=\boldsymbol{b}+\boldsymbol{v}$ and $2 \overrightarrow{A L}=\boldsymbol{a}+\boldsymbol{v}+2 \boldsymbol{b}$, so $\overrightarrow{K L}=\overrightarrow{A L}-\overrigh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,623 |
13.7*. Given $n$ pairwise non-collinear vectors ( $n \geqslant 3$ ), the sum of which is zero. Prove that there exists a convex $n$-gon, the set of vectors of whose sides coincides with the given set of vectors. | 13.7. Let's place these vectors from one point and, moving clockwise, number them in order: $\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{n}$. Consider the closed broken line $A_{1} \ldots A_{n}$, for which $\overrightarrow{A_{i} A_{i+1}}=\boldsymbol{a}_{i}$. We will prove that $A_{1} \ldots A_{n}$ is a convex polygon. ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,624 |
13.9*. Given four pairwise non-parallel vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$, and $\boldsymbol{d}$, the sum of which is zero. Prove that
$$
|\boldsymbol{a}|+|\boldsymbol{b}|+|\boldsymbol{c}|+|\boldsymbol{d}|>|\boldsymbol{a}+\boldsymbol{b}|+|\boldsymbol{a}+\boldsymbol{c}|+|\boldsymbol{a}+\boldsymbol... | 13.9. According to problem 13.8, b) from the given vectors, a self-intersecting four-link broken line can be formed; it can be represented as two diagonals and two opposite sides of a convex quadrilateral. There are two cases: vector $\boldsymbol{a}$ can be either a side or a diagonal of this quadrilateral. In both cas... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,626 |
13.10*. In a convex pentagon $A B C D E$, side $B C$ is parallel to diagonal $A D$, $C D\|B E$, $D E\| A C$, and $A E \| B D$. Prove that $A B \| C E$.
## §2. Scalar Product. Relations | 13.10. Let diagonal $B E$ intersect diagonals $A D$ and $A C$ at points $F$ and $G$. The sides of triangles $A F E$ and $B C D$ are parallel, so they are similar and $A F: F E = B C: C D$. Therefore, $A D: B E = (A F + B C):(E F + C D) = B C: C D$. Similarly, $A E: B D = D E: A C$. From the similarity of triangles $B E... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,627 |
13.11. Prove that if the diagonals of a quadrilateral $A B C D$ are perpendicular, then the diagonals of any other quadrilateral with the same side lengths are also perpendicular. | 13.11. Let $\boldsymbol{a}=\overrightarrow{A B}, \boldsymbol{b}=\overrightarrow{B C}, \boldsymbol{c}=\overrightarrow{C D}$ and $\boldsymbol{d}=\overrightarrow{D A}$. It is sufficient to check that $A C \perp B D$ if and only if $a^{2}+c^{2}=b^{2}+d^{2}$. Clearly, $d^{2}=$ $=|\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,628 |
13.12. a) Let $A, B, C$ and $D$ be arbitrary points on a plane. Prove that $(\overrightarrow{A B}, \overrightarrow{C D})+(\overrightarrow{B C}, \overrightarrow{A D})+(\overrightarrow{C A}, \overrightarrow{B D})=0$.
b) Prove that the altitudes of a triangle intersect at a single point. | 13.12. a) Express all vectors involved in the given formula through $\overrightarrow{A B}, \overrightarrow{B C}$, and $\overrightarrow{C D}$, i.e., write $\overrightarrow{A D}=\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C D}, \overrightarrow{C A}=-\overrightarrow{A B}-\overrightarrow{B C}$, and $\overrigh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,629 |
13.13. Let $O$ be the center of the circumscribed circle of triangle $ABC$, and let point $H$ have the property that $\overrightarrow{O H}=\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}$. Prove that $H$ is the point of intersection of the altitudes of triangle $ABC$. | 13.13. We will prove that $A H \perp B C$. $\overrightarrow{A H}=\overrightarrow{A O}+\overrightarrow{O H}=\overrightarrow{A O}+\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\overrightarrow{O B}+$ $+\overrightarrow{O C}$ and $\overrightarrow{B C}=\overrightarrow{B O}+\overrightarrow{O C}=-\overrightarr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,630 |
13.14. Let $\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{n}$ be the vectors of the sides of an $n$-gon, $\varphi_{i j}=\angle\left(\boldsymbol{a}_{i}, \boldsymbol{a}_{j}\right)$. Prove that $a_{1}^{2}=a_{2}^{2}+\ldots+a_{n}^{2}+2 \sum_{i>j>1} a_{i} a_{j} \cos \varphi_{i j}$, where $a_{i}=\left|\boldsymbol{a}_{i}\right|$... | 13.14. Let $\alpha_{i}=\angle\left(\boldsymbol{a}_{i}, \boldsymbol{a}_{1}\right)$. Considering projections onto the line parallel to $\boldsymbol{a}_{1}$ and the line perpendicular to $\boldsymbol{a}_{1}$, we get $a_{1}=\sum_{i>1} a_{i} \cos \alpha_{i}$ and $0=$ $=\sum_{i>1} a_{i} \sin \alpha_{i}$ respectively. Squarin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,631 |
13.15. Given a quadrilateral $A B C D$. Let $u=A D^{2}, v=B D^{2}, w=$ $=C D^{2}, U=B D^{2}+C D^{2}-B C^{2}, V=A D^{2}+C D^{2}-A C^{2}, W=A D^{2}+$ $+B D^{2}-A B^{2}$. Prove that $u U^{2}+v V^{2}+w W^{2}=U V W+4 u v w$ (Gauss). | 13.15. Let $\boldsymbol{a}=\overrightarrow{A D}, \boldsymbol{b}=\overrightarrow{B D}$ and $\boldsymbol{c}=\overrightarrow{C D}$. Since $B C^{2}=|\boldsymbol{b}-\boldsymbol{c}|^{2}=B D^{2}+$ $+C D^{2}-2(\boldsymbol{b}, \boldsymbol{c})$, then $U=2(\boldsymbol{b}, \boldsymbol{c})$. Similarly, $V=2(\boldsymbol{a}, \boldsym... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,632 |
13.16*. Points $A, B, C$ and $D$ are such that for any point $M$, the pairs $(\overrightarrow{M A}, \overrightarrow{M B})$ and $(\overrightarrow{M C}, \overrightarrow{M D})$ are different. Prove that $\overrightarrow{A C}=\overrightarrow{D B}$. | 13.16. Fix an arbitrary point $O$. Let $\boldsymbol{m}=\overrightarrow{O M}, \boldsymbol{a}=\overrightarrow{O A}, \ldots, \boldsymbol{d}=\overrightarrow{O D}$. Then $(\overrightarrow{M A}, \overrightarrow{M B})-(\overrightarrow{M C}, \overrightarrow{M D})=(\boldsymbol{a}-\boldsymbol{m}, \boldsymbol{b}-\boldsymbol{m})-(... | \overrightarrow{AC}=\overrightarrow{DB} | Geometry | proof | Yes | Yes | olympiads | false | 27,633 |
13.17*. Prove that in a convex $k$-gon, the sum of the distances from any interior point to the sides is constant if and only if the sum of the vectors of the unit external normals is zero. | 13.17. Let $\boldsymbol{n}_{1}, \ldots, \boldsymbol{n}_{k}$ be the unit outward normals to the sides, and $M_{1}, \ldots, M_{k}$ be arbitrary points on these sides. For any point $X$ inside the polygon, the distance to the $i$-th side is $(\overrightarrow{X M_{i}}, \boldsymbol{n}_{i})$. Therefore, the sums of distances... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,634 |
13.18*. In a convex quadrilateral, the sum of the distances from a vertex to the sides is the same for all vertices. Prove that this quadrilateral is a parallelogram.
## §3. Inequalities | 13.18. Let $l$ be an arbitrary line, $\boldsymbol{n}$ be a unit vector perpendicular to the line $l$. If points $A$ and $B$ lie in the same half-plane defined by the line $l$ as the vector $\boldsymbol{n}$, then $\rho(B, l)-\rho(A, l)=(\overrightarrow{A B}, \boldsymbol{n})$, where $\rho(X, l)$ is the distance from poin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,635 |
13.19. Given points $A, B, C$ and $D$. Prove that $A B^{2}+B C^{2}+C D^{2}+$ $+D A^{2} \geqslant A C^{2}+B D^{2}$, and equality is achieved if and only if $A B C D$ is a parallelogram. | 13.19. Let $\boldsymbol{a}=\overrightarrow{A B}, \boldsymbol{b}=\overrightarrow{B C}$ and $\boldsymbol{c}=\overrightarrow{C D}$. Then $\overrightarrow{A D}=\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}, \overrightarrow{A C}=\boldsymbol{a}+\boldsymbol{b}$ and $\overrightarrow{B D}=\boldsymbol{b}+\boldsymbol{c}$. It is al... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,636 |
13.20. Prove that from five vectors, it is always possible to choose two such that the length of their sum does not exceed the length of the sum of the remaining three vectors. | 13.20. Consider five vectors $\boldsymbol{a}_{1}, \boldsymbol{a}_{2}, \boldsymbol{a}_{3}, \boldsymbol{a}_{4}, \boldsymbol{a}_{5}$ and assume that the length of the sum of any two of them is greater than the length of the sum of the remaining three. Since $\mid \boldsymbol{a}_{1}+$ $+\boldsymbol{a}_{2}|>| \boldsymbol{a}... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 27,637 |
13.21. Ten vectors are such that the length of the sum of any nine of them is less than the length of the sum of all ten vectors. Prove that there exists an axis, the projection of each of the ten vectors onto which is positive. | 13.21. Let's denote these vectors as $\boldsymbol{e}_{1}, \ldots, \boldsymbol{e}_{10}$. Suppose $\overrightarrow{A B}=\boldsymbol{e}_{1}+\ldots+\boldsymbol{e}_{10}$. We will prove that the ray $A B$ defines the desired axis. It is clear that $\left|\overrightarrow{A B}-\boldsymbol{e}_{i}\right|^{2}=A B^{2}-2\left(\over... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,638 |
13.22. Points $A_{1}, \ldots, A_{n}$ lie on a circle with center $O$, and $\overrightarrow{O A_{1}}+\ldots+\overrightarrow{O A_{n}}=\overrightarrow{0}$. Prove that for any point $X$ the inequality $X A_{1}+\ldots+X A_{n} \geqslant n R$ holds, where $R$ is the radius of the circle. | 13.22. Let $a_{i}=\overrightarrow{O A_{i}}$ and $\boldsymbol{x}=\overrightarrow{O X}$. Then $\left|\boldsymbol{a}_{i}\right|=R$ and $\overrightarrow{X A_{i}}=\boldsymbol{a}_{i}-\boldsymbol{x}$. Therefore, $\sum X A_{i}=\sum\left|\boldsymbol{a}_{i}-\boldsymbol{x}\right|=\sum\left|\boldsymbol{a}_{i}-\boldsymbol{x}\right|... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,639 |
13.23. Given eight real numbers $a, b, c, d, e, f, g, h$. Prove that at least one of the six numbers $a c+b d, a e+b f, a g+b h, c e+d f, c g+d h, e g+f h$ is non-negative. | 13.23. Consider four vectors $(a, b),(c, d),(e, f)$ and $(g, h)$ on a plane. One of the angles between these vectors does not exceed $360^{\circ} / 4=90^{\circ}$. If the angle between vectors does not exceed $90^{\circ}$, then their dot product is non-negative.
These six numbers are the dot products of all pairs of ou... | proof | Algebra | proof | Yes | Yes | olympiads | false | 27,640 |
13.24*. On a circle of radius 1 with center $O$, there are $2 n+1$ points $P_{1}, \ldots, P_{2 n+1}$, lying on one side of some diameter. Prove that $\left|\overrightarrow{O P}_{1}+\ldots+\overrightarrow{O P}_{2 n+1}\right| \geqslant 1$. | 13.24. We will prove this statement by induction. For $n=0$, the statement is obviously true. Assume that the statement is proven for $2 n+1$ vectors. Consider in the system of $2 n+3$ vectors the two extreme vectors (i.e., the two vectors between which the angle is maximal). For definiteness, let these be the vectors ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,641 |
13.25*. Let $a_{1}, a_{2}, \ldots, a_{n}$ be vectors whose lengths do not exceed 1. Prove that in the sum $\boldsymbol{c}= \pm \boldsymbol{a}_{1} \pm \boldsymbol{a}_{2} \ldots \pm \boldsymbol{a}_{n}$, the signs can be chosen such that $|\boldsymbol{c}| \leqslant \sqrt{2}$. | 13.25. First, let's prove that if $\boldsymbol{a}, \boldsymbol{b}$, and $\boldsymbol{c}$ are vectors with lengths not exceeding 1, then at least one of the vectors $\boldsymbol{a} \pm \boldsymbol{b}, \boldsymbol{a} \pm \boldsymbol{c}, \boldsymbol{b} \pm \boldsymbol{c}$ has a length not exceeding 1. Indeed, two of the v... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,642 |
13.26*. From point $O$, $n$ unit vectors emanate, and in any half-plane bounded by a line passing through point $O$, there are at least $k$ vectors (it is assumed that the boundary line is included in the half-plane). Prove that the length of the sum of these vectors does not exceed $n-2k$.
## §4. Sums of Vectors | 13.26. We can assume that the sum of the given vectors \(a\) is not zero, as otherwise the statement of the problem is obvious. Let's introduce a coordinate system, directing the \(O y\) axis along the vector \(\boldsymbol{a}\). Number the vectors in the lower half-plane in order - clockwise: \(\boldsymbol{e}_{1}, \bol... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,643 |
13.27. Prove that the point $X$ lies on the line $A B$ if and only if $\overrightarrow{O X}=t \overrightarrow{O A}+(1-t) \overrightarrow{O B}$ for some $t$ and any point $O$. | 13.27. Point $X$ lies on the line $A B$ if and only if $\overrightarrow{A X}=\lambda \overrightarrow{A B}$, i.e. $\overrightarrow{O X}=\overrightarrow{O A}+\overrightarrow{A X}=(1-\lambda) \overrightarrow{O A}+\lambda \overrightarrow{O B}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,644 |
13.28. Given several points and for some pairs $(A, B)$ of these points, vectors $\overrightarrow{A B}$ are taken, and in each point, as many vectors start as end. Prove that the sum of all selected vectors is $\overrightarrow{0}$. | 13.28. Let's take an arbitrary point $O$ and write all the selected vectors in the form $\overrightarrow{A_{i} A_{j}}=\overrightarrow{O A_{j}}-\overrightarrow{O A_{i}}$. By the condition of the problem, each vector $\overrightarrow{O A_{i}}$ will enter the sum of all selected vectors with a plus sign as many times as i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,645 |
13.29. Inside triangle $ABC$, a point $O$ is taken. Prove that
$$
S_{BOC} \cdot \overrightarrow{OA} + S_{AOC} \cdot \overrightarrow{OB} + S_{AOB} \cdot \overrightarrow{OC} = \overrightarrow{0}
$$ | 13.29. Let $e_{1}, e_{2}$ and $e_{3}$ be unit vectors in the direction of vectors $\overrightarrow{O A}, \overrightarrow{O B}$ and $\overrightarrow{O C} ; \alpha=\angle B O C, \beta=\angle C O A$ and $\gamma=\angle A O B$. We need to prove that $\boldsymbol{e}_{1} \sin \alpha+\boldsymbol{e}_{2} \sin \beta+\boldsymbol{e... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,646 |
13.30. Points $A$ and $B$ move along two fixed rays with a common origin $O$ such that the quantity $\frac{p}{O A}+\frac{q}{O B}$ remains constant. Prove that the line $A B$ passes through a fixed point. | 13.30. Let $\boldsymbol{a}$ and $\boldsymbol{b}$ be unit vectors in the directions of rays $O A$ and $O B$, $\lambda=O A$ and $\mu=O B$. The line $A B$ consists of all points $X$ such that $\overrightarrow{O X}=t \overrightarrow{O A}+$ $+(1-t) \overrightarrow{O B}=t \lambda \boldsymbol{a}+(1-t) \mu \boldsymbol{b}$. It ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,647 |
13.31. Through the point $M$ of intersection of the medians of triangle $ABC$, a line is drawn intersecting the lines $BC$, $CA$, and $AB$ at points $A_{1}$, $B_{1}$, and $C_{1}$. Prove that $\left(1 / \overline{M A_{1}}\right)+\left(1 / \overline{M B_{1}}\right)+\left(1 / \overline{M C_{1}}\right)=0$ (segments $M A_{1... | 13.31. Let $\boldsymbol{a}=\overrightarrow{M A}, \boldsymbol{b}=\overrightarrow{M B}$ and $\boldsymbol{c}=\overrightarrow{M C}$. Then $\boldsymbol{e}=\overrightarrow{M C_{1}}=p \boldsymbol{a}+(1-p) \boldsymbol{b}$ and $\overrightarrow{M A_{1}}=q \boldsymbol{c}+(1-q) \boldsymbol{b}=-q \boldsymbol{a}+(1-2 q) \boldsymbol{... | 1/\alpha+1/\beta+1=0 | Geometry | proof | Yes | Yes | olympiads | false | 27,648 |
13.32. On the sides $BC$, $CA$, and $AB$ of triangle $ABC$, points $A_1$, $B_1$, and $C_1$ are taken. Segments $BB_1$ and $CC_1$, $CC_1$ and $AA_1$, $AA_1$ and $BB_1$ intersect at points $A_2$, $B_2$, and $C_2$ respectively. Prove that if $\overrightarrow{A A_2} + \overrightarrow{B B_2} + \overrightarrow{C C_2} = \over... | 13.32. Adding the equations $\overrightarrow{A A_{2}}+\overrightarrow{B B_{2}}+\overrightarrow{C C_{2}}=\overrightarrow{0}$ and $\overrightarrow{A_{2} B_{2}}+\overrightarrow{B_{2} C_{2}}+\overrightarrow{C_{2} A_{2}}=\overrightarrow{0}$, we get $\overrightarrow{A B_{2}}+\overrightarrow{B C_{2}}+\overrightarrow{C A_{2}}=... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,649 |
13.33. Quadrilateral \(ABCD\) is inscribed. Let \(H_a\) be the orthocenter of triangle \(BCD\), and \(M_a\) be the midpoint of segment \(AH_a\); points \(M_b\), \(M_c\), and \(M_d\) are defined similarly. Prove that points \(M_a\), \(M_b\), \(M_c\), and \(M_d\) coincide. | 13.33. Let $O$ be the center of the circumscribed circle of a given quadrilateral, $\boldsymbol{a}=\overrightarrow{O A}, \boldsymbol{b}=\overrightarrow{O B}, \boldsymbol{c}=\overrightarrow{O C}$ and $\boldsymbol{d}=\overrightarrow{O D}$. If $H_{a}$ is the orthocenter of triangle $B C D$, then $\overrightarrow{O_{a}}=\b... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,650 |
13.34*. Quadrilateral $ABCD$ is inscribed in a circle of radius $R$.
a) Let $S_{a}$ be the circle of radius $R$ centered at the orthocenter of triangle $BCD$; circles $S_{b}, S_{c}$, and $S_{d}$ are defined similarly. Prove that these four circles intersect at one point.
b) Prove that the nine-point circles of triang... | 13.34. Let $O$ be the center of the circumscribed circle of a given quadrilateral, $\boldsymbol{a}=\overrightarrow{O A}, \boldsymbol{b}=\overrightarrow{O B}, \boldsymbol{c}=\overrightarrow{O C}$, and $\boldsymbol{d}=\overrightarrow{O D}$. If $H_{d}$ is the orthocenter of triangle $A B C$, then $\overrightarrow{O H_{d}}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,651 |
13.35. Point $X$ lies inside triangle $ABC$, $\alpha=S_{BXC}$, $\beta=S_{CXA}$, and $\gamma=S_{AXB}$. Let $A_1, B_1$, and $C_1$ be the projections of points $A, B$, and $C$ onto an arbitrary line $l$. Prove that the length of the vector $\alpha \overrightarrow{AA_1} + \beta \overrightarrow{BB_1} + \gamma \overrightarro... | 13.35. Let $X_{1}$ be the projection of point $X$ onto line $l$. The vector $\alpha \overrightarrow{A A_{1}}+\beta \overrightarrow{B B_{1}}+ \gamma \overrightarrow{C C_{1}}$ is the projection of the vector $\alpha \overrightarrow{A X_{1}}+\beta \overrightarrow{B X_{1}}+\gamma \overrightarrow{C X_{1}}$ onto a line perpe... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,652 |
13.38*. Let $O$ and $R$ be the center and radius of the circumcircle of triangle $ABC$, $Z$ and $r$ be the center and radius of its incircle; $K$ be the point of intersection of the medians of the triangle with vertices at the points of tangency of the incircle with the sides of triangle $ABC$. Prove that the point $Z$... | 13.38. Let the inscribed circle touch the sides $AB, BC$, and $CA$ at points $U, V$, and $W$. We need to prove that $\overrightarrow{O Z}=\frac{3 R}{r} \overrightarrow{Z K}$, i.e., $\overrightarrow{O Z}=\frac{R}{r}(\overrightarrow{Z U}+\overrightarrow{Z V}+\overrightarrow{Z W})$. We will prove, for example, that the pr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,653 |
13.39*. Given two sets of vectors $\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{n}$ and $\boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{m}$, such that the sum of the lengths of the projections of the vectors of the first set onto any line is not greater than the sum of the lengths of the projections of the vectors of the s... | 13.39. Let's introduce a coordinate system $O x y$. Let $l_{\varphi}$ be a line passing through point $O$ and forming an angle $\varphi (0 < \varphi < \pi)$ with the $O x$ axis, i.e., if point $A$ lies on $l_{\varphi}$ and the second coordinate of point $A$ is positive, then $\angle A O X = \varphi; l_{0} = l_{\pi} = O... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,654 |
13.40*. Prove that if one convex polygon lies inside another, then the perimeter of the inner polygon does not exceed the perimeter of the outer one. | 13.40. The sum of the lengths of the projections of the sides of a convex polygon onto any straight line is equal to twice the length of the projection of the polygon onto this line. Therefore, the sum of the lengths of the projections of the side vectors onto any straight line for the inner polygon is no greater than ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,655 |
13.41*. The sum of the lengths of several vectors on a plane is equal to $L$. Prove that from these vectors, one can choose some number of vectors (possibly only one) such that the length of their sum is not less than $L / \pi$.
| 13.41. If the sum of the lengths of the vectors is $L$, then according to the remark to problem 13.39, the average value of the sum of the lengths of the projections of these vectors is $2 L / \pi$.
The function $f$ on the interval $[a, b]$ cannot be everywhere less than its average value $c$, because otherwise
$$
c=... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 27,656 |
13.42*. Prove that if the lengths of all sides and diagonals of a convex polygon are less than $d$, then its perimeter is less than $\pi d$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 13.42. Let the projection of the polygon onto the line $l$ be denoted by $A B$. It is clear that points $A$ and $B$ are the projections of some vertices $A_{1}$ and $B_{1}$ of the polygon. Therefore, $A_{1} B_{1} \geqslant A B$, i.e., the length of the projection of the polygon does not exceed $A_{1} B_{1}$, and $A_{1}... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,657 | |
13.43*. On a plane, four vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$, and $\boldsymbol{d}$ are given, the sum of which is zero. Prove that
$$
|\boldsymbol{a}|+|\boldsymbol{b}|+|\boldsymbol{c}|+|\boldsymbol{d}| \geqslant|\boldsymbol{a}+\boldsymbol{d}|+|\boldsymbol{b}+\boldsymbol{d}|+|\boldsymbol{c}+\boldsy... | 13.43. According to problem 13.39, the inequality $|\boldsymbol{a}|+|\boldsymbol{b}|+|\boldsymbol{c}|+|\boldsymbol{d}| \geqslant|\boldsymbol{a}+\boldsymbol{d}|+|\boldsymbol{b}+ \boldsymbol{d}|+| \boldsymbol{c}+\boldsymbol{d}|$ is sufficient to prove for projections of vectors onto a line, i.e., we can assume that $\bol... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,658 |
13.44*. Inside a convex $n$-gon $A_{1} A_{2} \ldots A_{n}$, a point $O$ is taken such that $\overrightarrow{O A_{1}}+\ldots+\overrightarrow{O A_{n}}=\overrightarrow{0}$. Let $d=O A_{1}+\ldots+O A_{n}$. Prove that the perimeter of the polygon is not less than $4 d / n$ for even $n$ and not less than $4 d n /\left(n^{2}-... | 13.44. According to problem 13.39, it is sufficient to prove the inequality for projections of vectors onto any line. Let the projections of vectors $\overrightarrow{O A_{1}}, \ldots, \overrightarrow{O A_{n}}$ onto the line $l$ be (taking into account the sign) $a_{1}, \ldots, a_{n}$. We will divide the numbers $a_{1},... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,659 |
13.45*. The length of the projection of a closed convex curve onto any line is 1. Prove that its length is $\pi$.
The length of the projection of a closed convex curve onto any line is 1. Prove that its length is $\pi$. | 13.45. The length of a curve is the limit of the perimeters of inscribed polygons. Consider an inscribed polygon with perimeter \( P \) and the length of its projection on a line \( l \), equal to \( d_{l} \). Let \( 1-\epsilon < d_{l} < 1 \) for all lines \( l \). The polygon can be chosen such that \( \epsilon \) can... | \pi | Geometry | proof | Yes | Yes | olympiads | false | 27,660 |
13.46*. Given several convex polygons, and it is impossible to draw a straight line that does not intersect any polygon and has at least one polygon on each side of it. Prove that these polygons can be enclosed in a polygon whose perimeter does not exceed the sum of their perimeters.
## §7. Pseudoscalar Product
The p... | 13.46. We will prove that the perimeter of the convex hull of all vertices of the given polygons does not exceed the sum of their perimeters. For this, it is sufficient to note that, by the condition, the projections of the given polygons onto any line cover the projection of the convex hull. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,661 |
13.47. Prove that:
a) $(\lambda \boldsymbol{a}) \vee \boldsymbol{b}=\lambda(\boldsymbol{a} \vee \boldsymbol{b})$
b) $\boldsymbol{a} \vee(\boldsymbol{b}+\boldsymbol{c})=\boldsymbol{a} \vee \boldsymbol{b}+\boldsymbol{a} \vee \boldsymbol{c}$. | 13.47. a) If $\lambda0$ the proof is obvious.
b) Let $\boldsymbol{a}=\overrightarrow{O A}, \boldsymbol{b}=\overrightarrow{O B}$ and $\boldsymbol{c}=\overrightarrow{O C}$. Introduce a coordinate system, directing the axis $O y$ along the ray $O A$. Let $A=\left(0, y_{1}\right), B=\left(x_{2}, y_{2}\right)$ and $C=\left... | proof | Algebra | proof | Yes | Yes | olympiads | false | 27,662 |
13.49. a) Prove that $S(A, B, C) = -S(B, A, C) = S(B, C, A)$.
b) Prove that for any points $A, B, C$, and $D$, the equality $S(A, B, C) = S(D, A, B) + S(D, B, C) + S(D, C, A)$ holds. | 13.49. a) It is clear that $\overrightarrow{A B} \vee \overrightarrow{A C}=\overrightarrow{A B} \vee(\overrightarrow{A B}+\overrightarrow{B C})=-\overrightarrow{B A} \vee \overrightarrow{B C}=\overrightarrow{B C} \vee \overrightarrow{B A}$.
b) To prove it, it is sufficient to use the equality $\overrightarrow{A B} \ve... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,664 |
13.50. Three runners $A, B$ and $C$ are running on parallel tracks at constant speeds. At the initial moment, the area of triangle $A B C$ is 2, and after 5 seconds, it is 3. What could it be after another 5 seconds? | 13.50. Let at the initial moment, i.e., at $t=0, \overrightarrow{A B}=\boldsymbol{v}$ and $\overrightarrow{A C}=\boldsymbol{w}$. Then at moment $t$ we get $\overrightarrow{A B}=\boldsymbol{v}+t(\boldsymbol{b}-\boldsymbol{a})$ and $\overrightarrow{A C}=\boldsymbol{w}+t(\boldsymbol{c}-\boldsymbol{a})$, where $\boldsymbol... | 48 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,665 |
13.51. Three pedestrians are walking along three straight roads at constant speeds. At the initial moment, they were not on the same straight line. Prove that they can be on the same straight line no more than twice. | 13.51. Let $\boldsymbol{v}(t)$ and $\boldsymbol{w}(t)$ be the vectors connecting the first pedestrian to the second and third at time $t$. Clearly, $\boldsymbol{v}(t)=t \boldsymbol{a}+\boldsymbol{b}$ and $\boldsymbol{w}(t)=t \boldsymbol{c}+\boldsymbol{d}$. The pedestrians are on the same line if and only if $\boldsymbo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,666 |
13.52. Solve problem 4.29, b using the pseudoscalar product.
4.29, b.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The note above is not part of the translation but is provided to clarify the instruction. It shoul... | 13.52. Let $\overrightarrow{O C}=\boldsymbol{a}, \overrightarrow{O B}=\lambda \boldsymbol{a}, \overrightarrow{O D}=\boldsymbol{b}$ and $\overrightarrow{O A}=\mu \boldsymbol{b}$. Then $\pm 2 S_{O P Q}=$ $=\overrightarrow{O P} \vee \overrightarrow{O Q}=((\boldsymbol{a}+\mu \boldsymbol{b}) / 2) \vee((\lambda \boldsymbol{a... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,667 | |
13.53*. Points $P_{1}, P_{2}$ and $P_{3}$, not lying on the same line, are located inside a convex $2 n$-gon $A_{1} \ldots A_{2 n}$. Prove that if the sum of the areas of triangles $A_{1} A_{2} P_{i}, A_{3} A_{4} P_{i}, \ldots, A_{2 n-1} A_{2 n} P_{i}$ is the same number $c$ for $i=1,2,3$, then for any internal point $... | 13.53. Let $\boldsymbol{a}_{j}=\overrightarrow{P_{1} A_{j}}$. Then the doubled sum of the areas of the specified triangles for any internal point $P$ is
$$
\left(\boldsymbol{x}+\boldsymbol{a}_{1}\right) \vee\left(\boldsymbol{x}+\boldsymbol{a}_{2}\right)+\left(\boldsymbol{x}+\boldsymbol{a}_{3}\right) \vee\left(\boldsym... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,668 |
13.54*. Given a triangle $A B C$ and a point $P$. Point $Q$ is such that $C Q \| A P$, and point $R$ is such that $A R \| B Q$ and $C R \| B P$. Prove that $S_{A B C}=S_{P Q R}$. | 13.54. Let $\boldsymbol{a}=\overrightarrow{A P}, \boldsymbol{b}=\overrightarrow{B Q}$ and $\boldsymbol{c}=\overrightarrow{C R}$. Then $\overrightarrow{Q C}=\alpha \boldsymbol{a}, \overrightarrow{R A}=\beta \boldsymbol{b}$ and $\overrightarrow{P B}=$ $=\gamma \boldsymbol{c}$, and $(1+\alpha) \boldsymbol{a}+(1+\beta) \bo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,669 |
13.55*. Let $H_{1}, H_{2}$ and $H_{3}$ be the orthocenters of triangles $A_{2} A_{3} A_{4}$, $A_{1} A_{3} A_{4}$ and $A_{1} A_{2} A_{4}$. Prove that the areas of triangles $A_{1} A_{2} A_{3}$ and $H_{1} H_{2} H_{3}$ are equal. | 13.55. Let $\boldsymbol{a}_{i}=\overrightarrow{A_{4} A_{i}}$ and $\boldsymbol{w}_{i}=\overrightarrow{A_{4} H_{i}}$. According to problem 13.49, b) it is sufficient to check that $\boldsymbol{a}_{1} \vee \boldsymbol{a}_{2}+\boldsymbol{a}_{2} \vee \boldsymbol{a}_{3}+\boldsymbol{a}_{3} \vee \boldsymbol{a}_{1}=\boldsymbol{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,670 |
13.56*. In a convex pentagon $A B C D E$, the area of which is $S$, the areas of triangles $A B C, B C D, C D E, D E A$ and $E A B$ are $a, b$, $c, d$ and $e$. Prove that
$$
S^{2}-S(a+b+c+d+e)+a b+b c+c d+d e+e a=0
$$
## Problems for independent solving | 13.56. Let $\boldsymbol{x}=x_{1} \boldsymbol{e}_{1}+x_{2} \boldsymbol{e}_{2}$. Then $\boldsymbol{e}_{1} \vee \boldsymbol{x}=x_{2}\left(\boldsymbol{e}_{1} \vee \boldsymbol{e}_{2}\right)$ and $\boldsymbol{x} \vee \boldsymbol{e}_{2}=x_{1}\left(\boldsymbol{e}_{1} \vee \boldsymbol{e}_{2}\right)$, i.e.
$$
\boldsymbol{x}=\le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,671 |
14.1. a) Prove that the center of mass exists and is unique for any system of points.
b) Prove that if $X$ is an arbitrary point, and $O$ is the center of mass of points $X_{1}, \ldots, X_{n}$ with masses $m_{1}, \ldots, m_{n}$, then $\overrightarrow{X O}=\frac{1}{m_{1}+\ldots+m_{n}}\left(m_{1} \overrightarrow{X X_{1}... | 14.1. Let $X$ and $O$ be arbitrary points. Then $m_{1} \overrightarrow{O X_{1}}+\ldots+m_{n} \overrightarrow{O X_{n}}=$ $=\left(m_{1}+\ldots+m_{n}\right) \overrightarrow{O X}+m_{1} \overrightarrow{X X_{1}}+\ldots+m_{n} \overrightarrow{X X_{n}}$, therefore the point $O$ is the center of mass of the given system of point... | proof | Algebra | proof | Yes | Yes | olympiads | false | 27,672 |
14.2. Prove that the center of mass of the system of points $X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{m}$ with masses $a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{m}$ coincides with the center of mass of two points, the center of mass $X$ of the first system with mass $a_{1}+\ldots+a_{n}$ and the center of mass $Y$ of the s... | 14.2. Let $Z-$ be an arbitrary point, $a=a_{1}+\ldots+a_{n}, b=b_{1}+\ldots+b_{m}$. Then $\overrightarrow{Z X}=\left(a_{1} \overrightarrow{Z X_{1}}+\ldots+a_{n} \overrightarrow{Z X_{n}}\right) / a$ and $\overrightarrow{Z Y}=\left(b_{1} \overrightarrow{Z Y_{1}}+\ldots+b_{m} \overrightarrow{Z Y_{m}}\right) / b$. If $O-$ ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 27,673 |
14.3. Prove that the center of mass of points $A$ and $B$ with masses $a$ and $b$ lies on the segment $A B$ and divides it in the ratio $b: a$.
## §2. Theorem of Mass Grouping
## | 14.3. Let $O$ be the center of mass of the given system. Then $a \overrightarrow{O A}+b \overrightarrow{O B}=\overrightarrow{0}$, so point $O$ lies on the segment $A B$ and $a O A=b O B$, i.e., $A O: O B=b: a$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,674 |
14.4. Prove that the medians of triangle $ABC$ intersect at one point and are divided by it in the ratio $2: 1$, counting from the vertex. | 14.4. Place unit masses at points $A, B$, and $C$. Let $O$ be the center of mass of this system of points. Point $O$ is also the center of mass of point $A$ with mass 1 and point $A_{1}$ with mass 2, where $A_{1}$ is the center of mass of points $B$ and $C$ with unit masses, i.e., $A_{1}$ is the midpoint of segment $B ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,675 |
14.5. Let $A B C D$ be a convex quadrilateral, and $K, L, M$ and $N$ be the midpoints of sides $A B, B C, C D$ and $D A$. Prove that the intersection point of segments $K M$ and $L N$ is the midpoint of these segments, as well as the midpoint of the segment connecting the midpoints of the diagonals. | 14.5. Place unit masses at the vertices of the quadrilateral \(ABCD\). Let \(O\) be the center of mass of this system of points. It is sufficient to prove that point \(O\) is the midpoint of segments \(KM\) and \(LN\) and the midpoint of the segment connecting the midpoints of the diagonals. Clearly, \(K\) is the cente... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,676 |
14.6. Let $A_{1}, B_{1}, \ldots, F_{1}$ be the midpoints of the sides $A B, B C, \ldots, F A$ of an arbitrary hexagon. Prove that the points of intersection of the medians of triangles $A_{1} C_{1} E_{1}$ and $B_{1} D_{1} F_{1}$ coincide. | 14.6. Place unit masses at the vertices of the hexagon; let $O$ be the center of mass of the resulting system of points. Since points $A_{1}, C_{1}$ and $E_{1}$ are the centers of mass of the pairs of points $(A, B),(C, D)$ and $(E, F)$, the point $O$ is the center of mass of the system of points $A_{1}, C_{1}$ and $E_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,677 |
14.8*. On the sides $A B, B C, C D$ and $D A$ of a convex quadrilateral $A B C D$, points $K, L, M$ and $N$ are taken respectively, such that $A K: K B=$ $=D M: M C=\alpha$ and $B L: L C=A N: N D=\beta$. Let $P$ be the point of intersection of segments $K M$ and $L N$. Prove that $N P: P L=\alpha$ and $K P: P M=\beta$. | 14.8. Place masses $1, \alpha, \alpha \beta$ and $\beta$ at points $A, B, C$ and $D$ respectively. Then the points $K, L, M$ and $N$ are the centers of mass of the pairs of points $(A, B),(B, C),(C, D)$ and $(D, A)$ respectively. Let $O$ be the center of mass of the points $A, B, C$ and $D$ with the specified masses. T... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,679 |
14.9*. Find a point \( O \) inside triangle \( ABC \) that has the following property: for any line passing through \( O \) and intersecting side \( AB \) at point \( K \) and side \( BC \) at point \( L \), the equality \( p \frac{AK}{KB} + q \frac{CL}{LB} = 1 \) holds, where \( p \) and \( q \) are given positive num... | 14.9. Place masses \( p, 1 \), and \( q \) at vertices \( A, B \), and \( C \) respectively. Let \( O \) be the center of mass of this system of points. We will consider the point with mass 1 as two coinciding points with masses \( x_{a} \) and \( x_{c} \), where \( x_{a} + x_{c} = 1 \). Let \( K \) be the center of ma... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,680 |
14.10*. Three flies of equal mass crawl along the sides of a triangle in such a way that their center of mass remains stationary. Prove that it coincides with the point of intersection of the medians of triangle $ABC$, if it is known that one fly has crawled along the entire perimeter of the triangle. | 14.10. Let's denote the center of mass of the flies by $O$. Suppose one fly is at vertex $A$, and $A_{1}$ is the center of mass of the other two flies. It is clear that point $A_{1}$ lies inside triangle $A B C$, and point $O$ lies on segment $A A_{1}$ and divides it in the ratio $A O: O A_{1}=2: 1$. Therefore, point $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,681 |
14.11*. On the sides $AB$, $BC$, and $CA$ of triangle $ABC$, points $C_{1}$, $A_{1}$, and $B_{1}$ are taken such that the lines $CC_{1}$, $AA_{1}$, and $BB_{1}$ intersect at some point $O$. Prove that:
a) $\frac{CO}{OC_{1}}=\frac{CA_{1}}{A_{1}B}+\frac{CB_{1}}{B_{1}A}$
b) $\frac{AO}{OA_{1}} \cdot \frac{BO}{OB_{1}} \cd... | 14.11. a) Let $A B_{1}: B_{1} C=1: p$ and $B A_{1}: A_{1} C=1: q$. Place masses $p, q, 1$ at points $A$, $B, C$ respectively. Then the points $A_{1}$ and $B_{1}$ are the centers of mass of the pairs of points $(B, C)$ and $(A, C)$. Therefore, the center of mass of the system of points $A, B$ and $C$ lies on both segmen... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,682 |
14.12*. On the sides $BC$, $CA$, and $AB$ of triangle $ABC$, points $A_{1}$, $B_{1}$, and $C_{1}$ are taken such that $BA_{1} / A_{1}C = CB_{1} / B_{1}A = AC_{1} / C_{1}B$. Prove that the centroids of triangles $ABC$ and $A_{1}B_{1}C_{1}$ coincide. | 14.12. Let $M$ be the center of mass of triangle $ABC$. Then $\overrightarrow{M A}+\overrightarrow{M B}+\overrightarrow{M C}=\overrightarrow{0}$. Moreover, $\overrightarrow{A B_{1}}+\overrightarrow{B C_{1}}+\overrightarrow{C A_{1}}=k(\overrightarrow{A C}+\overrightarrow{B A}+\overrightarrow{C B})=\overrightarrow{0}$. A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,683 |
14.13*. In the midpoints of the sides of triangle $ABC$, points are placed, the masses of which are equal to the lengths of the sides. Prove that the center of mass of this system of points is located at the center of the inscribed circle of the triangle with vertices at the midpoints of the sides of triangle $ABC$.
N... | 14.13. Let $A_{1}, B_{1}$ and $C_{1}$ be the midpoints of sides $B C, C A$ and $A B$. The center of mass of points $B_{1}$ and $C_{1}$ is at point $K$, for which $B_{1} K: K C_{1}=c: b=B_{1} A_{1}: A_{1} C_{1}$. Therefore, $A_{1} K$ is the bisector of angle $B_{1} A_{1} C_{1}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,684 |
14.14*. On a circle, $n$ points are given. Through the center of mass of $n-2$ points, a line is drawn perpendicular to the chord connecting the two remaining points. Prove that all such lines intersect at one point. | 14.14. Let $M_{1}$ be the center of mass of $n-2$ points, $K$ be the midpoint of the chord connecting the two remaining points, $O$ be the center of the circle, and $M$ be the center of mass of all given points. If the line $O M$ intersects the line passing through point $M_{1}$ at point $P$, then $\overline{O M} / \ov... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,685 |
14.15*. On the lines $B C, C A, A B$, points $A_{1}$ and $A_{2}, B_{1}$ and $B_{2}, C_{1}$ and $C_{2}$ are taken such that $A_{1} B_{2} \| A B, B_{1} C_{2} \| B C$ and $C_{1} A_{2} \| C A$. Let $l_{a}$ be the line connecting the points of intersection of the lines $B B_{1}$ and $C C_{2}, B B_{2}$ and $C C_{1}$; lines $... | 14.15. The parallelism of lines $A_{1} B_{2}$ and $A B$ means that if $B_{2}$ is the center of mass of points $A$ and $C$ with masses 1 and $\gamma$, then $A_{1}$ is the center of mass of points $B$ and $C$ with masses 1 and $\gamma$. We define the numbers $\alpha$ and $\beta$ similarly.
Lines $B B_{1}$ and $C C_{2}$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,686 |
14.17*. On the sides $BC$, $CA$, and $AB$ of triangle $ABC$, points $A_1$, $B_1$, and $C_1$ are taken; lines $B_1 C_1$, $BB_1$, and $CC_1$ intersect line $AA_1$ at points $M$, $P$, and $Q$ respectively. Prove that:
a) $A_1 M / M A = (A_1 P / P A) + (A_1 Q / Q A)$
b) if $P=Q$, then $MC_1 : MB_1 = (BC_1 / AB) : (CB_1 /... | 14.17. a) Place masses $\beta, \gamma$, and $b+c$ at points $B, C$, and $A$ respectively, such that $C A_{1}: B A_{1} = \beta: \gamma$, $B C_{1}: A C_{1} = b: \beta$, and $A B_{1}: C B_{1} = \gamma: c$. Then $M$ is the center of mass of this system, which means $A_{1} M / A M = (b+c) / (\beta+\gamma)$. Point $P$ is the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,688 |
14.18*. On the line $A B$, points $P$ and $P_{1}$ are taken, and on the line $A C$, points $Q$ and $Q_{1}$ are taken. The line connecting point $A$ with the intersection point of lines $P Q$ and $P_{1} Q_{1}$ intersects line $B C$ at point $D$. Prove that
$$
\frac{\overline{B D}}{\overline{C D}}=\frac{(\overline{B P} ... | 14.18. The intersection point of the lines $P Q$ and $P_{1} Q_{1}$ is the center of mass of points $A$, $B$, and $C$ with masses $a$, $b$, and $c$; here, $P$ is the center of mass of points $A$ and $B$ with masses $a-x$ and $b$, and $Q$ is the center of mass of points $A$ and $C$ with masses $x$ and $c$. Let $p=\overli... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,689 |
14.19. Let $O$ be the center of mass of a system of points with total mass $m$. Prove that the moments of inertia of this system relative to the point $O$ and an arbitrary point $X$ are related by the equation $I_{X}=I_{O}+m X O^{2}$. | 14.19. Let's number the points of the given system. Let $\boldsymbol{x}_{i}$ be the vector with its origin at point $O$ and its end at the point with number $i$, and let the mass $m_{i}$ be assigned to this point. Then $\sum m_{i} \boldsymbol{x}_{i}=\mathbf{0}$. Let, further, $\boldsymbol{a}=\overrightarrow{X O}$. Then... | I_{X}=I_{O}+XO^{2} | Algebra | proof | Yes | Yes | olympiads | false | 27,690 |
14.20. a) Prove that the moment of inertia relative to the center of mass of a system of points with unit masses is equal to $\frac{1}{n} \sum_{i<j} a_{i j}^{2}$, where $n$ is the number of points, and $a_{i j}$ is the distance between points with indices $i$ and $j$.
b) Prove that the moment of inertia relative to th... | 14.20. a) Let $\boldsymbol{x}_{i}$ be a vector with its origin at the center of mass $O$ and its end at the point with number $i$. Then $\sum_{i, j}\left(\boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right)^{2}=\sum_{i, j}\left(\boldsymbol{x}_{i}^{2}+\boldsymbol{x}_{j}^{2}\right)-2 \sum_{i, j}\left(\boldsymbol{x}_{i}, \boldsym... | proof | Algebra | proof | Yes | Yes | olympiads | false | 27,691 |
14.21. a) Triangle $A B C$ is equilateral. Find the geometric locus of points $X$ such that $A X^{2}=B X^{2}+C X^{2}$.
b) Prove that for points of the specified locus, the pedal triangle relative to triangle $A B C$ is a right triangle. | 14.21. a) Let $M$ be the point symmetric to point $A$ with respect to the line $BC$. Then $M$ is the center of mass of points $A, B$, and $C$ with masses $-1, 1$, and $1$, respectively, which means $-A X^{2} + B X^{2} + C X^{2} = I_{X} = I_{M} + (-1 + 1 + 1) M X^{2} = (-3 + 1 + 1) a^{2} + M X^{2}$, where $a$ is the sid... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,692 |
14.22. Let $O$ be the center of the circumcircle of triangle $ABC$, and $H$ be the orthocenter. Prove that $a^{2}+b^{2}+c^{2}=9 R^{2}-O H^{2}$. | 14.22. Let $M$ be the center of mass of the vertices of triangle $ABC$ with unit masses. Then $I_{O}=I_{M}+3 M O^{2}=\left(a^{2}+b^{2}+c^{2}\right) / 3+3 M O^{2}$ (see problems 14.19 and $\left.14.20, \mathrm{a}\right)$ ). Since $O A=O B=O C=R$, we have $I_{O}=3 R^{2}$. It remains to note that $O H=3 O M$ (problem 5.11... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,693 |
14.23. Chords $A A_{1}, B B_{1}$, and $C C_{1}$ of a circle with center $O$ intersect at point $X$. Prove that $\left(A X / X A_{1}\right)+\left(B X / X B_{1}\right)+\left(C X / X C_{1}\right)=3$ if and only if point $X$ lies on the circle with diameter $O M$, where $M$ is the centroid of triangle $A B C$. | 14.23. It is clear that $A X / X A_{1}=A X^{2} / A X \cdot X A_{1}=A X^{2} /\left(R^{2}-O X^{2}\right)$. Therefore, we need to check that $A X^{2}+B X^{2}+C X^{2}=3\left(R^{2}-O X^{2}\right)$ if and only if $O M^{2}=O X^{2}+M X^{2}$. For this, it is sufficient to note that $A X^{2}+B X^{2}+$ $+C X^{2}=I_{X}=I_{M}+3 M X... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,694 |
14.24*. On the sides $AB, BC, CA$ of triangle $ABC$, points $A_{1}$ and $B_{2}, B_{1}$ and $C_{2}, C_{1}$ and $A_{2}$ are taken such that segments $A_{1} A_{2}, B_{1} B_{2}$, and $C_{1} C_{2}$ are parallel to the sides of the triangle and intersect at point $P$. Prove that $P A_{1} \cdot P A_{2} + P B_{1} \cdot P B_{2}... | 14.24. Let $P$ be the center of mass of points $A, B$, and $C$ with masses $\alpha, \beta$, and $\gamma$; we can assume that $\alpha+\beta+\gamma=1$. If $K$ is the intersection point of the lines $CP$ and $AB$, then
$$
\frac{BC}{PA_{1}}=\frac{CK}{PK}=\frac{CP+PK}{PK}=1+\frac{CP}{PK}=1+\frac{\alpha+\beta}{\gamma}=\frac... | R^{2}-OP^{2} | Geometry | proof | Yes | Yes | olympiads | false | 27,695 |
14.25*. Inside a circle of radius $R$, there are $n$ points. Prove that the sum of the squares of the pairwise distances between them does not exceed $n^{2} R^{2}$. | 14.25. Place unit masses at these points. According to the result of problem 14.20, a), the sum of the squares of the pairwise distances between these points is equal to $n I$, where $I$ is the moment of inertia of the system of points relative to the center of mass. Now consider the moment of inertia of the system rel... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,696 |
14.26*. Inside triangle $ABC$, a point $P$ is taken. Let $d_{a}, d_{b}$, and $d_{c}$ be the distances from point $P$ to the sides of the triangle, and $R_{a}, R_{b}$, and $R_{c}$ be the distances from it to the vertices. Prove that
$$
3\left(d_{a}^{2}+d_{b}^{2}+d_{c}^{2}\right) \geqslant\left(R_{a} \sin A\right)^{2}+\... | 14.26. Let $A_{1}, B_{1}$ and $C_{1}$ be the projections of point $P$ onto sides $B C, C A$ and $A B$; $M$ be the centroid of triangle $A_{1} B_{1} C_{1}$. Then $3\left(d_{a}^{2}+d_{b}^{2}+d_{c}^{2}\right)=3 I_{P} \geqslant 3 I_{M}=$ $=A_{1} B_{1}^{2}+B_{1} C_{1}^{2}+C_{1} A_{1}^{2}=\left(R_{c} \sin C\right)^{2}+\left(... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,697 |
14.27*. Points $A_{1}, \ldots, A_{n}$ lie on a circle, and $M$ is their center of mass. The lines $M A_{1}, \ldots, M A_{n}$ intersect this circle at points $B_{1}, \ldots, B_{n}$ (distinct from $A_{1}, \ldots, A_{n}$). Prove that $M A_{1}+\ldots+M A_{n} \leqslant M B_{1}+\ldots+M B_{n}$.
## §4. Various Problems
. On the other hand, sin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,698 |
14.28. Prove that if a polygon has several axes of symmetry, then all of them intersect at one point. | 14.28. Place unit masses at the vertices of the polygon. Due to symmetry with respect to the axis of symmetry, this system of points transforms into itself, so its center of mass also transforms into itself. Therefore, all axes of symmetry pass through the center of mass of the vertices with unit masses.
![](https://cd... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,699 |
14.29. A centrally symmetric figure on graph paper consists of $n$ "corners" and $k$ rectangles of size $1 \times 4$, as shown in Fig. 14.1. Prove that $n$ is even. | 14.29. We will place unit masses at the centers of the cells that make up the "corners" and rectangles. We will divide each original cell of the paper into four cells, thus obtaining a new grid paper. It is easy to check that now the center of mass of the corner lies at the center of the new cell, and the center of mas... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 27,700 |
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