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324. Prove that the perimeter of the triangle, the vertices of which are the bases of the altitudes of a given acute-angled triangle, does not exceed half the perimeter of the given triangle. | 324. If $p_{1}$ is the semiperimeter of the triangle with vertices at the bases of the altitudes of a given triangle, and $p, S, r$, and $R$ are the semiperimeter, area, inradius, and circumradius of the given triangle, respectively, then $S = p r$ and, moreover, $S = p_{1} R$ (the latter follows from the fact that the... | p_{1}\leqslant\frac{1}{2}p | Geometry | proof | Yes | Yes | olympiads | false | 23,710 |
326. Let $A B C D$ be a convex quadrilateral. Prove that at least one of the four angles $B A C, D B C, A C D, B D A$ does not exceed $\pi / 4$. | 326. Let $O$ be the point of intersection of the diagonals of quadrilateral $ABCD$. Suppose all angles specified in the condition are greater than $\pi / 4$. Then on segments $OB$ and $OC$, we can take points $B_{1}$ and $C_{1}$ respectively, such that $\angle B_{1} A O=\angle O B_{1} C_{1}=\pi / 4$. Let $\angle B O A=... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,712 |
327. Prove that the median to the largest side of a triangle forms angles with the sides enclosing it, each of which is not less than half the smallest angle of the triangle. | 327. Let in $\triangle A B C$ the sides are related by $c \leqslant b \leqslant a$. Take a point $M$ on $C B$ such that $\angle C A M=\frac{1}{2} \angle C$. We need to prove that $|C M| \leqslant \frac{a}{2}$. By the Law of Sines for $\triangle C A M$ we have: $|C M|=$ $=\frac{b \sin \frac{C}{2}}{\sin \frac{3 C}{2}}=\f... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,713 |
328. Prove that if in triangle $ABC$ angle $B$ is obtuse and $|AB|=|AC| / 2$, then $\angle C>\angle A / 2$. | 328. Let $D$ be the midpoint of $A C$. Construct a perpendicular to $A C$ at $D$ and denote by $M$ the point of intersection of this perpendicular with $B C$; $\triangle A M C$ is isosceles, so $\angle M A C = \angle B C A$. By the problem's condition, $\triangle A B D$ is also isosceles, $\angle A B D = \angle B D A$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,714 |
329. Prove that the circumcircle of a triangle cannot pass through the center of an excircle. | 329. If a circle touches the extensions of sides $A B$ and $A C$ of triangle $A B C$ and its center is $O$, it is easy to find that $\angle B O C=90^{\circ}-$ $-\frac{1}{2} \angle A$. Therefore, $\angle B O C+\angle A=90^{\circ}+\frac{1}{2} \angle A \neq 180^{\circ}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,715 |
330. In a triangle, from vertex $A$ there are a median, a bisector, and an altitude. Which angle is larger: between the median and the bisector or between the bisector and the altitude, if angle $A$ is given? | 330. Let $A D$ be the altitude, $A L$ the angle bisector, and $A M$ the median. Extend the angle bisector to intersect the circumcircle of the triangle at point $A_{1}$. Since $M A_{1} \| A D$, then $\angle M A_{1} A=\angle L A D$. Answer: if $\angle A=90^{\circ}$, the angles are equal; if $\angle A \neq 90^{\circ}$, t... | if\\angleA=90,\the\angles\\equal;\if\\angleA\neq90,\then\it\is\the\opposite | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,716 |
331. Prove that if the medians drawn from vertices $B$ and $C$ of triangle $ABC$ are perpendicular, then $\operatorname{ctg} B + \operatorname{ctg} C \geqslant \geqslant 2 / 3$. | 331. If $A D$ is the altitude, $A N$ is the median, and $M$ is the point of intersection of the medians, then $\operatorname{ctg} B+\operatorname{ctg} C=\frac{|D B|}{|A D|}+\frac{|C D|}{|A D|}=\frac{|C B|}{|A D|} \geqslant \frac{|C B|}{|A N|}=\frac{|C B|}{3|M N|}=\frac{2}{3}$. | \frac{2}{3} | Geometry | proof | Yes | Yes | olympiads | false | 23,717 |
333. From an external point $A$ to a circle, two tangents $A B$ and $A C$ are drawn, and the midpoints of these tangents, $D$ and $E$, are connected by a line $D E$. Prove that this line does not intersect the circle. | 333. If $|O A|=a, R$ is the radius of the circle, and $K$ is the intersection point of $O A$ and $D E$, it is easy to find that $|O K|=a-\frac{a^{2}-R^{2}}{2 a}=\frac{a^{2}+R^{2}}{2 a}>R$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,719 |
335. In triangle $A B C$, the angles are related by $3 \angle A-\angle C<\pi$. Angle $B$ is divided into four equal parts by lines intersecting side $A C$. Prove that the third segment, counting from vertex $A$, into which side $A C$ is divided, is less than $|A C| / 4$. | 335. Let $K, L$ and $M$ be the points of intersection of the drawn lines with $AC$. Denote: $|AC|=b$, $|BC|=a$, $|AB|=c$, $|BL|=$ $=l$. By the angle bisector theorem for the internal angle,
we find: $|LC|=\frac{ab}{a+c}$; applying this theorem again for $\triangle BCL$, we find: $|LM|=\frac{ba}{a+c} \cdot \frac{l}{l+a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,721 |
336. Let $a, b, c, d$ be the consecutive sides of a quadrilateral. Prove that if $S$ is its area, then $S \leqslant(a c + b d) / 2$, and equality holds only for a cyclic quadrilateral with perpendicular diagonals. | 336. Let $A B C D$ be the given quadrilateral. Consider the quadrilateral $A B_{1} C D$, where $B_{1}$ is symmetric to $B$ with respect to the perpendicular bisector of the diagonal $A C$. Obviously, the areas of $A B C D$ and $A B_{1} C D$ are equal, and the sides of $A B_{1} C D$ in order are $b, a, c, d$. The inequa... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,722 |
337. Prove that if the lengths of the bisectors of a triangle are less than 1, then its area is less than $\sqrt{3} / 3$. | 337. Let's consider two cases.
1) The given triangle $(A B C)$ is acute-angled. Let $\angle B$ be the largest: $60^{\circ} \leqslant \angle B<90^{\circ}$. Since the angle bisectors of angles $A$ and $C$ are less than 1, the heights of these angles $h_{A}$ and $h_{C}$ are also less than 1. We have $S_{A B C}=$ $=\frac{h... | \frac{\sqrt{3}}{3} | Geometry | proof | Yes | Yes | olympiads | false | 23,723 |
338. Prove that a triangle will be acute, right, or obtuse depending on whether the expression $a^{2}+b^{2}+c^{2}-8 R^{2}$ is positive, zero, or negative ( $a, b, c$ - sides of the triangle, $R$ - radius of the circumscribed circle). | 338. Let $c$ be the largest side, opposite vertex $\cdot C$. If $a^{2}+b^{2}+c^{2}-8 R^{2}>0$, then $a^{2}+b^{2}>8 R^{2}-c^{2} \geqslant c^{2}$ (since $c \leqslant 2 R$), i.e., the triangle is acute-angled. Conversely, let the triangle be acute-angled; then $a^{2}+b^{2}+c^{2}=2 m_{c}^{2}+\frac{3}{2} c^{2}$ ($m_{c}$ is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,724 |
339. Prove that a triangle will be acute, right, or obtuse depending on whether its semiperimeter is respectively greater than, equal to, or less than the sum of the diameter of the circumscribed circle and the radius of the inscribed circle. | 339. By substituting $R$ and $r$ using the formulas $R=\frac{a b c}{4 S}, r=\frac{S}{p}$, use Heron's formula for $S$ and the equality
$$
\begin{aligned}
& 4 S^{2}\left(p-\frac{a b c}{2 S}-\frac{S}{p}\right)\left(p+\frac{a b c}{2 S}+\frac{S}{p}\right)= \\
& \quad=\frac{1}{8}\left(a^{2}+b^{2}-c^{2}\right)\left(a^{2}-b^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,725 |
340. Prove that if the lengths of the sides of a triangle are related by the inequality $a^{2}+b^{2}>5 c^{2}$, then $c-$ is the smallest side. | 340. Suppose the opposite, for example, that $c \geqslant a$; then $2 c \geqslant c+a >$ $>b$; squaring the inequalities and adding them, we get: $5 c^{2}>a^{2}+$ $+b^{2}$ - a contradiction. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,726 |
341. In triangle $ABC$, angle $B$ is the middle in size: $\angle A < \angle B < \angle C, I$ is the center of the inscribed circle, $O$ is the center of the circumscribed circle, $H$ is the intersection point of the altitudes. Prove that $I$ lies inside triangle $BOH$. | 341. The bisector of angle $B$ is the bisector of $\angle O B H$, and the bisector of angle $A$ is the bisector of $\angle O A H$. Further, $\angle B A H=90^{\circ}-\angle B|B H|$. If $K$ and $M-$ are the points of intersection of the bisectors of angles $A$ and $B$ with $O H$, then $\frac{|H K|}{|K O|}=$ $=\frac{|A H|... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,727 |
343. In triangle $ABC$, point $M$ lies on side $BC$. Prove that $(|AM|-|AC|)|BC| \leqslant(|AB|-|AC|)|MC|$. | 343. Draw a line through $M$ parallel to $AC$ until it intersects $AB$ at point $K$. It is easy to find: $|AK| = |CM| \cdot \frac{|AB|}{|CB|}, |MK| = |MB| \cdot \frac{|AC|}{|CB|}$. Since $|AM| \leqslant |AK| + |KM|$, by substituting $|AK|$ and $|KM|$, we get:
$|AM| \leqslant \frac{|CM| \cdot |AB|}{|BC|} + \frac{|MB| \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,729 |
345. The sides of an angle equal to $\alpha$ are the edges of a billiard table. What is the maximum number of reflections from the edges that a billiard ball (the size of the ball can be neglected) can make? | 345. "Straightening" the path of the ball: instead of "reflecting" the ball off the side, we will reflect the billiard table itself mirror-like relative to this side. We will obtain a system of rays with a common vertex; any two adjacent rays form an angle $\alpha$. The maximum number of rays in the system that a strai... | [\frac{\pi}{\alpha}]+1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,730 |
346. Four villages are located at the vertices of a square with a side length of 2 km. The villages are connected by roads in such a way that it is possible to travel from any village to any other. Can the total length of the roads be less than 5.5 km? | 346. If the roads are built as shown in Fig. 61 (A, B, C, and D are villages, roads are solid lines), then their total length will be $2+2 \sqrt{3}<5.5$. It can be shown that the specified arrangement of roads realizes the minimum of their total length. | 2+2\sqrt{3}<5.5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,731 |
347. Point $A$ is located between two parallel lines at distances $a$ and $b$ from them. This point serves as the vertex of an angle equal to $\alpha$ for all possible triangles, the other two vertices of which lie on the given lines. Find the minimum value of the area of such triangles. | 347. If one of the sides of the triangle passing through $A$ forms an angle $\varphi$ with a line perpendicular to the given parallel lines, then the other side forms an angle $180^{\circ}-\varphi-\alpha$; finding these sides, we get that the area of the triangle is $-\frac{a b \sin \alpha}{2 \cos \varphi \cos (\varphi... | \operatorname{ctg}\frac{\alpha}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,732 |
348. Given a circle of radius $R$ with center at point $O$, $AB$ is its diameter, point $M$ is on the radius $OA$, and $|AM|:|MO|=k$. A chord $CD$ is drawn through point $M$. What is the maximum value of the area of quadrilateral $ACBD$? | 348. We have: $S_{A C B D}=\frac{|A B|}{|M O|} S_{O C D}=2(k+1) S_{O C D}$. Therefore, $S_{A C B D}$ will be the largest when the area of triangle $O C D$ is the largest. But triangle $O C D$ is isosceles with the lateral side equal to $R$; hence, its area is maximized when the sine of the angle at vertex $O$ reaches i... | \frac{S_{\max}=(k+1)R^{2}ifk\leqslant\sqrt{2}-1,S_{\max}=2R^{2}\sqrt{k(k+2)}}{(k+1)ifk>\sqrt{2}-1} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,733 |
349. Given an angle with vertex $A$ and two points $M$ and $N$ inside it. A line through $M$ intersects the sides of the angle at points $B$ and $C$. Prove that for the area of quadrilateral $A B N C$ to be the smallest, it is necessary and sufficient that the line $B C$ intersects $A N$ at a point $P$ such that $|B P|... | 349. Let the line $BC$ satisfy the condition of the problem: $|BP| = |MC|$ (the order of the points is $B, P, M, C$). We will prove that the area of the quadrilateral $ABNC$ is the smallest. Draw another line intersecting the sides of the angle at points $B_1$ and $C_1$. Suppose point $B$ lies between points $A$ and $B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,734 |
351. Given the result of the previous problem, solve the following. Inside an angle with vertex $O$, a point $A$ is taken. The line $O A$ forms angles $\varphi$ and $\psi$ with the sides of the angle. Find points $M$ and $N$ on the sides of the angle such that $\angle M A N=\beta(\varphi+\psi+\beta<\pi)$ and the area o... | 351. Given the result of the previous problem, we need to determine under what conditions points \( M \) and \( N \) can be found on the sides of the angle such that \(\angle M A N = \beta\) and \( |M A| = |A N| \). Let's describe a circle around triangle \( M O N \) (Fig. 63). Since \(\varphi + \psi + \beta = \angle L... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,736 |
353. Let $A B C D$ be a cyclic quadrilateral. The diagonal $A C$ is equal to $a$ and forms angles $\alpha$ and $\beta$ respectively with the sides $A B$ and $A D$. Prove that the area of the quadrilateral is between $\frac{a^{2} \sin (\alpha+\beta) \sin \beta}{2 \sin \alpha}$ and $\frac{a^{2} \sin (\alpha+\beta) \sin \... | 353. Let $\sin \alpha \geqslant \sin \beta$ for definiteness; take a point $K$ on the extension of $A B$ such that $\angle B K C=\beta$; since $\angle C B K=\angle A D C$ (because quadrilateral $A B C D$ is cyclic), $\triangle K B C$ is similar to $\triangle A C D$. But $|B C| \geqslant|C D|$, therefore, $S_{B C K} \ge... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,738 |
354. Given an angle $\alpha$ with vertex at point $O$ and a point $A$ inside it. Consider all possible quadrilaterals OMAN, where vertices $M$ and $N$ are located on the sides of the angle and such that $\angle M A N=\beta(\alpha+\beta>\pi)$. Prove that if among these quadrilaterals there is a convex quadrilateral such... | 354. Consider another position of points $M_{1}$ and $N_{1}\left(\angle M_{1} A N_{1}=\beta\right)$ and show, given the condition $\alpha+\beta>180^{\circ}$, that the "added" triangle has a greater area than the triangle whose area is reduced (analogous to the solution of problem II.350). | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,739 |
355. Inside an angle with vertex $O$, there is a point $A$ such that $O A$ forms angles $\varphi$ and $\psi$ with the sides of the given angle. Find points $M$ and $N$ on the sides of the angle such that $\angle M A N=\beta(\varphi+\psi+\beta>\pi)$ and the area of the quadrilateral $O M A N$ is minimized. | 355. Given the result of the previous problem and reasoning in the same way as in problem II. 351, we get that if $\varphi>90^{\circ}-\frac{\beta}{2}$ and $\psi>90^{\circ}-$
$-\frac{\beta}{2}$, then the quadrilateral of the smallest area exists and for it
$|M A|=|A N|$. If this condition is not met, then the desired q... | |MA|=|AN| | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,740 |
357. Find the radius of the largest circle that can be covered by three circles of radius $R$. Solve the problem in the general case when the radii are equal to $R_{1}, R_{2}, R_{3}$. | 357. The radius of the largest circle is equal to the radius of the circumscribed circle around an equilateral triangle with side $2 R$, i.e., $2 R / \sqrt{3}$. (Let's take such a triangle and construct circles on its sides as diameters.) For any circle with a larger radius, if it were covered by the given circles, the... | \frac{2R}{\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,742 |
359. What is the maximum area of a regular triangle that can be covered by three regular triangles with a side length of 1? | 359. First, note that the side of the smallest equilateral triangle covering a rhombus with side $a$ and acute angle $60^{\circ}$ is equal to $2a$. Indeed, if the vertices of the acute angles $M$ and $N$ of the rhombus are on the sides $A B$ and $B C$ of the equilateral triangle $A B C$ and $\angle B N M=\alpha, 30^{\c... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,744 |
360. In triangle $ABC$, points $M$ and $N$ are taken on sides $AC$ and $BC$, and point $L$ is taken on segment $MN$. Let the areas of triangles $ABC$, $AML$, and $BNL$ be $S$, $P$, and $Q$ respectively. Prove that $\sqrt[3]{S} \geqslant \sqrt[3]{P} + \sqrt[3]{Q}$. | 360. Let the ratios $\frac{|A M|}{|M C|}, \frac{|C N|}{|N B|}$ and $\frac{|M L|}{|L M|}$ be denoted by $\alpha, \beta$ and $\gamma$. Then (see the solution of problem I.221) $P=Q \alpha \beta \gamma, S=Q(\alpha+1)(\beta+1)(\gamma+1)$. Next, we will use the inequality $(\alpha+1)(\beta+1)(\gamma+1) \geqslant(\sqrt[3]{\a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,745 |
361. Let $a, b, c, S$ be the sides and area of a certain triangle, and $\alpha, \beta, \gamma$ be the angles of another triangle. Prove that $a^{2} \operatorname{ctg} \alpha+b^{2} \operatorname{ctg} \beta+c^{2} \operatorname{ctg} \gamma \geqslant 4 S$, with equality holding only when both triangles are similar. | 361. Let $\operatorname{ctg} \alpha=x, \operatorname{ctg} \beta=y$, then $\operatorname{ctg} \gamma=\frac{-x y+1}{x+y}=\frac{x^{2}+1}{x+y}-x$, $a^{2} \operatorname{ctg} \alpha+b^{2} \operatorname{ctg} \beta+c^{2} \operatorname{ctg} \gamma=\left(a^{2}-b^{2}-c^{2}\right) x+b^{2}(x+y)+c^{2} \frac{x^{2}+1}{x+y}$. The minim... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,746 |
362. Prove the inequality $a^{2}+b^{2}+c^{2} \geqslant 4 S \sqrt{3}+(a-b)^{2}+$ $+(b-c)^{2}+(c-a)^{2}$, where $a, b, c, S-$ are respectively the sides and area of a triangle (Finsler, Hadwiger). | 362. Let's denote: $p-a=x, p-b=y, p-c=z$ ( $p-$ is the semiperimeter). Leaving $4 S \sqrt{3}$ on the right side of the inequality, after transforming the left side (for example, $a^{2}-(b-c)^{2}=4(p-b)(p-c)=4 y z$) and substituting $S$ using Heron's formula, we get the inequality $x y+y z+z x \geqslant \geqslant \sqrt{... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,747 |
363. Given a triangle with sides $a, b$, and $c$. Determine the area of the largest regular triangle circumscribed around the given triangle, and the area of the smallest regular triangle inscribed in it. | 363. There exist two families of equilateral triangles circumscribed around a given triangle (see problem II.305). Construct equilateral triangles $A B C_{1}, B C A_{1}, C A B_{1}$ on the sides of triangle $A B C$ outwardly and describe circles around them. The vertices of the triangles of the first family are located ... | S_{0}=\frac{\sqrt{3}(^{2}+b^{2}+^{2})}{6}+2S,\quadS_{1}=\frac{S^{2}}{S_{0}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,748 |
364. Let $M$ be an arbitrary point inside triangle $ABC$. The line $AM$ intersects the circumcircle of $ABC$ at point $A_1$. Prove that $\frac{|BM| \cdot |CM|}{|A_1 M|} \geqslant 2r$, where $r$ is the radius of the inscribed circle, and equality is achieved when $M$ coincides with the center of the inscribed circle. | 364. Describe a circle around triangle $A M C$. All triangles $A_{1} M C$, obtained by moving $M$ along the arc $A C$, are similar to each other, so the ratio $\frac{|C M|}{\left|A_{1} M\right|}$ will be the same for them. Therefore, if $M$ is the point of minimum of the expression $f(M)=$ $=\frac{|B M| \cdot|C M|}{\le... | 2r | Geometry | proof | Yes | Yes | olympiads | false | 23,749 |
365. Let $M$ be an arbitrary point inside triangle $ABC$. Prove that $|AM| \sin \angle BMC + |BM| \sin \angle AMC + |CM| \sin \angle AMB \leqslant p$ (where $p$ is the semiperimeter of triangle $ABC$) and equality is achieved when $M$ coincides with the incenter of the inscribed circle. | 365. Let's take points \(C_1\) and \(B_1\) on rays \(MB\) and \(MC\) respectively, such that \(|MC_1| = |MC|\) and \(|MB_1| = |MB|\) (\(\triangle MC_1B_1\) is symmetric to \(\triangle MBC\) with respect to the bisector of angle \(BMC\)), \(C_2\) and \(B_2\) are the projections of \(C_1\) and \(B_1\) onto the line \(AM\... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,750 |
366. Let $h_{1}, h_{2}, h_{3}$ be the altitudes of triangle $ABC$, and $u, v, w$ be the distances from a point $M$ inside triangle $ABC$ to the corresponding sides. Prove the inequalities:
a) $\frac{h_{1}}{u}+\frac{h_{2}}{v}+\frac{h_{3}}{w} \geqslant 9$;
b) $h_{1} h_{2} h_{3} \geqslant 27 u v w$
c) $\left(h_{1}-u\ri... | 366. a) Let's first solve the following problem. Let $M$ be a point on side $AB$ of triangle $ABC$, the distances from $M$ to sides $BC$ and $AC$ are $u$ and $v$ respectively, and $h_{1}$ and $h_{2}$ are the altitudes dropped to $BC$ and $AC$ respectively. Prove that the expression $\frac{h_{1}}{u}+\frac{h_{2}}{v}$ rea... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,751 |
367. Let $h$ be the length of the greatest altitude of an acute-angled triangle, $R$ and $r$ be the radii of the circumcircle and the incircle, respectively. Prove that $R + r \leqslant h$ (Erdős). | 367. Let for an acute-angled triangle $A B C$ the inequality $|A C| \leqslant|A B| \leqslant|B C|$ holds; $B D$ is the altitude, $O$ is the circumcenter, $I$ is the incenter of $\triangle A B C, E$ is the projection of $I$ onto $B D$. Since $|E D|=r$, it is necessary to prove that $|B E| \geqslant R=|B O|$. But $B I$ i... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,752 |
368. Prove that the radius of the circle circumscribed around the triangle formed by the medians of an acute-angled triangle is greater than $5 / 6$ of the radius of the circle circumscribed around the original triangle. | 368. Since the area of a triangle formed by the medians of another triangle is $3 / 4$ of the area of the original triangle, and for any triangle $a b c=4 R S$, it is necessary to prove that for an acute-angled triangle, the inequality
$$
m_{a} m_{b} m_{c}>\frac{5}{8} a b c
$$
holds. For convenience of calculations, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,753 |
369. Prove that the sum of the squares of the distances from an arbitrary point in the plane to the sides of a triangle takes its minimum value for a point inside the triangle for which the distances to the corresponding sides are proportional to these sides. Also prove that this point is the intersection of the symmed... | 369. Let $M$ lie inside $ABC$ at distances $x, y$, and $z$ from the sides $BC, CA$, and $AB$, respectively. The problem is to find the minimum of $x^{2}+y^{2}+z^{2}$ under the condition $ax + by + cz = 2S_{ABC}$. It is clear that this minimum is achieved at the same values of $x, y, z$ as the minimum of $x^{2}+y^{2}+z^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,754 |
370. Given a triangle, all angles of which are less than $120^{\circ}$. Prove that the sum of the distances from an arbitrary point to the vertices of this triangle takes the minimum value for a point inside it from which each side of the triangle is seen at an angle of $120^{\circ}$. | 370. Let $M$ be a point inside triangle $ABC$, all angles of which are less than $120^{\circ}$. Rotate $\triangle AMC$ around point $A$ by an angle of $60^{\circ}$ outward relative to $\triangle ABC$. In this case, point $C$ will move to point $C_{1}$, and point $M$ will move to point $M_{1}$. The sum $|AM| + |BM| + |C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,755 |
371. Prove that among all triangles inscribed in a given acute triangle, the one with the smallest perimeter has its vertices at the feet of the altitudes of the given triangle. | 371. Let $ABC$ be a given acute-angled triangle, $A_1$ a point on side $BC$, $B_1$ on $CA$, $C_1$ on $AB$, $A_2$ and $A_3$ the points symmetric to $A_1$ with respect to sides $AB$ and $AC$ (respectively). The broken line $A_2 C_1 B_1 A_3$ equals the perimeter of triangle $A_1 B_1 C_1$; therefore, this perimeter, for a ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,756 |
373. For an arbitrary triangle, prove the inequality (using standard notation) $\frac{b c \cos A}{b+c}+a<p<\frac{b c+a^{2}}{a}$. | 373. Let's prove the right part of the inequality. Let for definiteness $b \geqslant c$.
1) If $a \leqslant b$, then $2 p=a+b+c=(b-a)+c+2 a<2 c+2 a \leqslant 2 \frac{b}{a} c+$ $+2 a=2 \frac{b c+a^{2}}{a}$.
2) If $a \geqslant b \geqslant c$, then $a<2 b$ and $2 p=a+b+c=(b+c-a)+2 a \leqslant c+$ $+2 a<\frac{2 b c}{a}+2 a... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,758 |
374. Let $K$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$, $L$ a point on side $AD$, $N$ on side $BC$, and $M$ on diagonal $AC$, such that $KL$ and $MN$ are parallel to $AB$, and $LM$ is parallel to $DC$. Prove that $KLMN$ is a parallelogram and its area is less than $8/27$ of the area... | 374. Given: $\frac{|B N|}{|N C|}=\frac{|A M|}{|M C|}=\frac{|A L|}{|L D|}=\frac{|B K|}{|K D|}$, i.e., $K N$ is parallel to $C D$, and quadrilateral $K L M N$ is a parallelogram. Let $|A K|=a$, $|K C|=b$, $|B K|=x$, $|K D|=y$, and $\frac{x}{y} \geqslant \frac{a}{b}$; then
$$
\begin{aligned}
& S_{K L M}=S_{A L M}-S_{A K ... | \frac{8}{27}S_{ABCD} | Geometry | proof | Yes | Yes | olympiads | false | 23,759 |
376. Given a triangle $A B C$, the angles of which are $\alpha, \beta$ and $\gamma$. Triangle $D E F$ is circumscribed around triangle $A B C$ such that vertices $A, B$ and $C$ are located on sides $E F$, $F D$ and $D E$ respectively, and $\angle E C A = \angle D B C = \angle F A B = \varphi$. Determine the value of th... | 376. Let's describe circles around triangles $A B F, B C D$ and $C A E$. They have a common point $M$. Since the angles of triangle $DEF$ are constant, $\angle D=\gamma, \angle E=\alpha, \angle F=\beta$, the constructed circles and point $M$ do not depend on $\varphi$. The side $D F$ (and therefore $E F$ and $E D$) wil... | \operatorname{tg}\varphi_{0}=\operatorname{ctg}\alpha+\operatorname{ctg}\beta+\operatorname{ctg}\gamma | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,761 |
377. On the sides $B C, C A$ and $A B$ of triangle $A B C$, points $A_{1}, B_{1}, C_{1}$ are taken respectively. Prove that the area of triangle $A_{1} B_{1} C_{1}$ is not less than the area of at least one of the three triangles: $A B_{1} C_{1}, A_{1} B C_{1}, A_{1} B_{1} C$. | 377. Let: $\frac{\left|A C_{1}\right|}{|A B|}=x, \frac{\left|B A_{1}\right|}{|B C|}=y, \frac{\left|C B_{1}\right|}{|C A|}=z$. We will assume that $x \leqslant 1 / 2$. If we assume that the areas of triangles $A B_{1} C_{1}$, $B C_{1} A_{1}$, and $C A_{1} B_{1}$ are greater than the area of triangle $A_{1} B_{1} C_{1}$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,762 |
378. Let $O, I, H$ be the centers of the circumcircle, incircle, and the intersection point of the altitudes of a certain triangle, respectively. Prove that $|O H| \geqslant|I H| \sqrt{2}$. | 378. Let $Q$ be the midpoint of $O H$, and $Q$ is the center of the nine-point circle (see problem II.160). We have: $|O H|^{2}+4|Q I|^{2}=2|O I|^{2}+2|H I|^{2}$. Since $|Q I|=R / 2-r$ (based on Feuerbach's theorem, problem II.287), $|O I|^{2}=R^{2}-2 R r$ (Euler's formula, problem II.193), taking into account that $R ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,763 |
379. Let $M$ be an arbitrary point inside triangle $ABC$; $x, y$, and $z$ be the distances from $M$ to $A, B$, and $C$ respectively; $u, v$, and $w$ be the distances from $M$ to the sides $BC, CA$, and $AB$ respectively; $a, b, c$ be the corresponding sides of triangle $ABC$; $S$ be its area; $R$ and $r$ be the radii o... | 379. An elegant idea for proving inequalities of this type was proposed by Kazarinoff (Michigan Mathematical Journal, 1957, No. 2, pp. $97-98$). The essence of it is as follows. Take points $B_{1}$ and $C_{1}$ on rays $A B$ and $A C$ respectively. It is obvious that the sum of the areas of the parallelograms constructe... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,764 |
4. Prove that if three angles of a quadrilateral are obtuse, then the diagonal passing through the vertex of the acute angle is greater than the second diagonal. | 4. Let angle $A$ of quadrilateral $A B C D$ be acute, and the other three angles be obtuse. We construct a circle with segment $A C$ as its diameter (Fig. 33). Since $\angle B$ and $\angle D$ are obtuse, points $B$ and $D$ will be inside this circle, from which it follows that the distance $B D$ between them will be le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,770 |
5. Sides $A B$ and $A D$ of parallelogram $A B C D$ are equal to 1 and $a$, respectively, $\angle B A D=\alpha$, and triangle $A B D$ is acute. For which $a$ and $\alpha$ will four circles of radius 1 with centers at the vertices of the parallelogram completely cover the parallelogram? | 5. Let's describe .. around the acute-angled triangle $ABD \cdot$ circle $K$; let $O$ be the center of this circle (Fig. 34). We will show that if the radius $R$ of circle $K$ is greater than 1, then point $O$ will not be covered by any of the four circles with centers at $A, B, C, D$ and radius 1. Indeed, $AO = BO = D... | \leq\cos\alpha+\sqrt{3}\sin\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,771 |
7. Inside triangle $A B C$, a point $O$ is chosen; on the rays $O A, O B$, and $O C$, vectors of length 1 are laid out. Prove that the length of the sum of these vectors is less than 1. | 7. Since rays $A O$ and $B O$ intersect sides $B C$ and $A C$ of triangle $A B C$, ray $O C=O C_{0}$ lies inside angle $A^{\prime} O B^{\prime}$, which is the vertical angle of $A O B$ (Fig. 37). We construct two rhombi $O A_{0} D_{0} B_{0}$ and $O A^{\prime} D B^{\prime}$, symmetric with respect to point $O$, with sid... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,773 |
8. a) How to choose some number of vectors from $\overline{O A}_{1}, \overline{O A}_{2}, \overline{O A}_{3}, \ldots, \overline{O A}_{25}$, where $O$ is the center of a regular 25-gon $A_{1} A_{2} A_{3} \ldots A_{25}$, so that the length of the sum of these vectors is as large as possible?
b) * What is this length if a... | 8. a) Let $\overline{O S}$ be the desired sum, $X Y$ be a line passing through point $O$ perpendicular to line $O S$. If the system of selected vectors contains a vector $\overline{O A}_{i}$ (where $i=1,2,3, \ldots$, or 25), located on the line $X Y$ or on the other side of the line $X Y$ compared to vector $\overline{... | 7.97 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,774 |
14. What is the smallest number of triangular pyramids (tetrahedrons) into which a cube can be divided? | 14. It is easy to see that the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ can be divided into five tetrahedra: if we cut off the tetrahedra $B A C B_{1}$ and $D A C D_{1}$, as well as the tetrahedra $A_{1} B_{1} D_{1} A$ and $C_{1} B_{1} D_{1} C$, we will have one more (fifth) tetrahedron $A C B_{1} D_{1}$ left (Fig. 51; t... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,779 |
15. On the plane, there are a (generally non-convex) quadrilateral and a pentagon, and no vertex of one lies on the side of the other. What is the maximum possible number of intersection points of their sides? | 15. Note that no line can intersect the sides of a polygon in an odd number of points, since otherwise, moving along this line in a certain direction - after entering the polygon for the last time, we would not be able to leave its boundaries. Therefore, each side of the quadrilateral can have no more than four interse... | 16 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,780 |
16. Every two adjacent sides of a flat (self-intersecting!) 14-sided polygon are mutually perpendicular; no two sides of it lie on the same line. What is the maximum possible number of self-intersection points of the sides of such a polygon? | 16. Let's agree to consider that all sides of the 14-sided polygon are either "horizontal" or "vertical." Clearly, in this case, exactly 7 sides are horizontal (and 7 are vertical): after all, from each vertex, one horizontal (and one vertical) side extends. Summing over all 14 vertices, we count 14 horizontal sides; b... | 17 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,781 |
17. What is the maximum number of parts into which the plane can be divided by:
a) two triangles;
b) two rectangles;
c) two convex $n$-gons? | 17. Let's immediately provide the solution to problem (v), namely, we will prove that the maximum number of parts into which two convex $n$-gons can divide the plane is $2n+2$; from this it will follow that two triangles can divide the plane into no more than 8 parts (problem (a)); see Fig. 54, a), and two rectangles c... | 2n+2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,782 |
18. a) Into how many parts can a plane be divided by two closed curves, one of which is a circle and the other is the boundary of a square?
b) $* *$ Into how many parts can space be divided by two closed surfaces, one of which is a sphere and the other is the surface of a cube? | 18. a) The boundary of a square and a circle can divide the plane into parts of the following types:
1) the part located outside the square and outside the circle; such a part is always one;
2) the part located inside the square and inside the circle; if the square does not intersect the circle, then there are no such ... | 16 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,783 |
22. a) Inside a square $A B C D$ with side length 1, there is a convex polygon $M$ with an area greater than $1 / 2$. Prove that there exists a line $l$, parallel to the side $A B$ of the square, that intersects the polygon $M$ along a segment of length greater than $1 / 2$.
b)* Inside a square $A B C D$ with side len... | 22. a) Draw lines through all vertices of the polygon $M$ parallel to the side $AB$. Then the polygon will be divided into a series of triangles and trapezoids (Fig. 67). The area of each of these triangles and trapezoids is equal to the length of the midline multiplied by the height. If all midlines were no more than ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,787 |
23. Inside a square with side 1, a broken line is drawn such that each straight line parallel to the side of the square intersects it in no more than one point. Prove that the length of the broken line is less than 2, and for each number $l<2$ there exists a broken line of length $l$ that satisfies the required conditi... | 23. Let the directions of the sides of the square coincide with the horizontal and vertical directions; denote the consecutive segments of the broken line by $l_{1}, l_{2}, \ldots, l_{n}$, and their projections on the horizontal and vertical sides of the square by $l_{1}^{\prime}$, $l_{2}^{\prime}, \ldots, l_{n}^{\prim... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,788 |
26. On the plane, a triangle $A B C$ is given. A point $D$ in space is chosen such that the height $D P$ of the tetrahedron (triangular pyramid) $A B C D$ is the smallest of its four heights. Where can the point $P$ be located? | 26. Let point $D$ be chosen according to the condition of the problem, $P$ be the foot of the altitude $DP$ of the tetrahedron $ABCD$; $Q$ be the foot of the altitude $CQ$; $M$ and $N$ be the feet of the altitudes $CM$ and $DN$ of triangles $ABC$ and $ABD$ (Fig. 73). Since $V_{ABCD} = \frac{1}{3} DP \cdot S_{\triangle ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,791 |
27. All sides of a convex polygon $m$ with perimeter $p=12$ and area $s$ are moved outward by a distance of 1; the perimeter of the larger polygon $M$ formed by the moved lines (Fig. 5) is denoted by $P$, and its area by $S$. Prove that
a) $P-p>6$
b) $S-s>15$
c) * if the polygon $m$ is a quadrilateral, then $P-p \ge... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,792 | |
29. Can a row of non-overlapping circles be placed in a rectangle of area 1 so that the sum of their radii equals 1962? | 29. It is possible. Let's divide the smaller side of the rectangle into a large number $n$ of equal segments of small length $\alpha$. The larger side is also divided into $m$ segments of length $\alpha$; in this case, we may get a remainder, which, in any case, is less than . Then the entire rectangle is divided into ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,794 |
30. On a plane, there are 25 points; it is known that from any three points, two can be chosen such that the distance between them is less than 1. Prove that among these points, there are 13 that can be covered by a circle of radius 1. | 30. Let $A$ be one of the given points (it does not matter which one!). If all points are at a distance $<1$ from $A$, then all of them can be covered by a circle of radius 1 (with center $A$); therefore, assume that there is a point $B$ among our points such that $A B \geqslant 1$. For each third point $M$ in our syst... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,795 |
31*. On a plane, there are 100 points. Prove that they can be covered by one or several non-overlapping circles, the sum of whose diameters is less than 100, and such that the distance between any two points in different circles exceeds 1. | 31. Consider 100 circles of radius $1 / 2$ with centers at the given points; thus, we cover all points with 100 circles (some of which may intersect or touch each other!), the sum of whose diameters is 100. If any two $K_{1}$ and $K_{2}$ of these circles intersect or touch, we replace them with a larger circle $K$, who... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,796 |
32. Let $m$ be the smallest number of circles of radius 1 that can cover a convex polygon $M$, and $n$ be the largest number of circles of diameter 1 that can be placed such that no two circles intersect and all their centers belong to the polygon $M$. Which number is greater: $-m$ or $n$? | 32. Consider $n$ non-intersecting circles of diameter 1, the centers of which belong to the polygon $M$. Replace each circle with a concentric circle of radius 1. If the circles obtained in this way do not cover some point $A$ of the polygon $M$, then this point will be at a distance of at least 1 from the centers of a... | \leqslantn | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,797 |
33. On a round table with a radius of 25, there are $n$ coins with a radius of 1. Prove that if $n>625$, the coins probably overlap, and if $n<144$, another coin can be placed on the table without overlapping with the previously placed ones.
The estimates given in problem 33 (in the general case of a table with radius... | 33. It is clear that if \( n > 625 \), then the total area of all the coins is greater than
\[
625 \cdot \left(\pi \cdot 1^{2}\right) = \pi \cdot 625 = \pi \cdot 25^{2},
\]
i.e., greater than the area of the table with a radius of 25; therefore, the coins cannot be placed on the table without overlapping. On the othe... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,798 |
34. An irrigated field has the shape of a square with a side of 12; from a source $J$ located within the field, a system of straight ditches has been laid out such that the distance from any point in the field to the nearest ditch does not exceed 1. Prove that the total length of the ditches is greater than 70 (we negl... | 34. The network of irrigation ditches represents a "branched" broken line (Fig. 87) passing through the field; in this case, the considered broken line is "connected," i.e., consists of one piece, since all ditches are connected to the source J. Let us single out from the system of ditches some "main ditch," i.e., a "s... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,799 |
39. Ship $K_{1}$ notices ship $K_{2}$, which is at a distance $d$ from $K_{1}$ and is moving with a speed $v$ in a direction,
Fig. 8. perpendicular to the line $K_{1} K_{2}$ (Fig. 8). Ship... | 39. a) Let's denote by $\alpha$ the angle between the course of ship $K_{1}$ and (constant) course of ship $K_{2}$ at some moment in time $t$; it is clear that the angle $\alpha$ will change over time. At the moment $t$, ship $K_{1}$ is approaching $K_{2}$ at a speed $v$; as for ship $K_{2}$, it is moving away from $K_... | \frac{\sqrt{v^2-u^2}}{v} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 23,804 |
40. A tourist got lost in the forest; he knows the shape of the forest, but does not know where he is.
a) If the forest is a circle with diameter $d$, then by moving in a straight line in any fixed direction, the tourist will certainly exit the forest after traveling a distance $d$: any segment of length $d$ starting ... | 40. a) It cannot: a straight path $AB$ of length less than $d$ does not guarantee an exit from the forest. Indeed, suppose the tourist's location and the chosen direction of movement at the beginning of the journey are such that the midpoint of the line $AB$ coincides with the center $O$ of the forest. In this case, th... | 6.40d | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,805 |
43. Two sides of triangle $ABC$ are equal to $a$ and $b$. What value can
a) the greatest angle of the triangle;
b) the smallest angle of the triangle;
take? | 43. a) Let, for definiteness, \( AC = b \) and \( a = CB \). It is clear that the largest angle of triangle \( ABC \) is less than \( 180^\circ \) and can be arbitrarily close to \( 180^\circ \) (see triangle \( \bar{A}BC \) in Fig. 112).
 What values can the following quantities take:
1) angle $A$ of triangle $A B C$;
2) angle $B$
3) angle $C$,
if $A \leqslant B \leqslant C$?
b) The area of a triangle with sides $a, b$, and $c$, where $a \leqslant b \leqslant c$, is 1. What values can
1) side $a$ of the triangle;
2) side $b$
3) side $c$
take... | 45. a) It is clear that
1) $0<A \leqslant 60^{\circ}$,
where $A$ is the smallest angle of triangle $ABC$ ($\angle A \leqslant 60^{\circ}$, because $A+B+C=180^{\circ}=3 \cdot 60^{\circ}$ and $A \leqslant B \leqslant C$; $\angle A=60^{\circ}$, if triangle $ABC$ is equilateral);
2) $0<B<90^{\circ}$,
where $B$ is the midd... | 0<A\leqslant60,\quad0<B<90,\quad60\leqslantC<180,\quad0<\infty,\quad\sqrt{2}\leqslant\infty,\quad\frac{2\sqrt[4]{27}}{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,810 |
47. Let the radius $\rho$ of the inscribed sphere of the tetrahedron $ABCD$ (Fig. 10, b) be 1; the radii of its exspheres of the 1st kind (Fig. $11, a$) are denoted by $\rho_{A}, \rho_{B}, \rho_{C}$, and $\rho_{D}$ (where, for example, the sphere with radius $\rho_{A}$ touches the opposite vertex $A$ of the face $BCD$ ... | 47. By connecting the center O of the inscribed sphere of a tetrahedron (see Fig. 10, a in the text) with all the vertices of the tetrahedron, we can easily find that
$$
V=V_{o A B C}+V_{O A B D}+V_{o A C D}+V_{o B C D}
$$
where \( V \) is the volume of the tetrahedron. On the other hand, the triangular pyramids \( o... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,812 |
49. Let $h_{a}=\beta_{b}=m_{c}$, where $h_{a}=A P, \beta_{b}=B L$ and $m_{c}=C F-$ are the altitude, bisector, and median drawn from three different vertices of some triangle $A B C$. What value can the ratio of the greatest side of triangle $A B C$ to its smallest side have? | 49. First of all, note that if for triangle \(ABC\) the inequality \(m_{a}b = CA\) holds. This immediately follows from the known formula \(^{2}\) \(m_{a} = \frac{1}{2} \sqrt{2 b^{2} + 2 c^{2} - a^{2}}\), but it can also be proven without calculations. Translate triangle \(ABC\) parallel to vector \(\overline{ED}\) to ... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,814 |
50. Let $h_{a}=m_{b}$, where $h_{a}=A P$ is the largest altitude of an acute-angled triangle $A B C$, and $m_{b}=B E$ is the median drawn from vertex $B$. Prove that $\angle B \leqslant 60^{\circ}$, | 50. Dropping perpendiculars $E Q$ and $E Q_{1}$ from the midpoint $E$ of side $A C$ of triangle $A B C$ to sides $B C$ and $A B$ (Fig. 129). Since $E Q=\frac{1}{2} A P=\frac{1}{2} h_{a}$ and $B E=m_{b}=h_{a}$, then $\angle E B C=30^{\circ}$. On the other hand, $E Q_{1}=\frac{1}{2} C R=\frac{1}{2} h_{c} \leqslant \frac{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,815 |
51. Let $A P=h_{a}, A K=\boldsymbol{\beta}_{a}$ and $A D=m_{a}$ be the height, bisector, and median of triangle $A B C$ with sides $A C=b$ and $A B=c$. Prove that
a) if $\frac{1}{b}+\frac{1}{c}=\frac{1}{h_{a}}$, then $\angle A \leqslant 120^{\circ}$;
b) if $\frac{1}{b}+\frac{1}{c}=\frac{1}{\beta_{a}}$, then $\angle A... | 51. If $A K=\beta_{a}$ is the angle bisector of triangle $A B C$ (Fig. 130), then obviously,
$$
S_{\triangle B A K}+S_{\triangle C A K}=S_{\triangle A B C},
$$
or
$$
\frac{1}{2} c \beta_{a} \sin \frac{A}{2}+\frac{1}{2} b \beta_{a} \sin \frac{A}{2}=\frac{1}{2} b c \sin A
$$
Therefore,
$$
\beta_{a}=\frac{b c \sin A}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,816 |
52. The base $BC$ of triangle $ABC$ is equal to 1, and angle $A$ is equal to $\alpha$. What values can the length of the median $AD=m_{a}$ of this triangle take? | 52. Let's fix the base $B C$ of the triangle; in this case, vertex $A$ will belong to the arc $B E C$ of the segment constructed on $B C$ that accommodates the angle $\alpha$ (Fig. 131). In the case where $\alpha < 90^{\circ}$, the points $A_{1}$ and $A_{2}$ lie on the same side of the chord $B C$, and for the points $... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,817 | |
53. The medians $A D$ and $B E$ of triangle $A B C$ are perpendicular to each other. What can the angle $C$ of this triangle be? | 53. First solution. This problem can be easily solved algebraically. Let the sides of triangle $ABC$ be denoted by $a, b, c$, and its (perpendicular by the problem's condition) medians $AD$ and $BE$ by $m_a$ and $m_b$ (Fig. $132, a$). Since
$$
m_a^2 = \frac{1}{4} \left(2b^2 + 2c^2 - a^2\right) \quad \text{and} \quad m... | 0<C\leq\arccos\frac{4}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,818 |
58. A convex broken line of length $d$ rotates around a straight line connecting its ends. What can be the area of the surface of revolution thus obtained? | 58. Let $A_{1} A_{2} \ldots A_{n}$ be a given broken line, $A_{1} A_{n}$ - a straight line passing through its ends, $B$ - a point that divides the perimeter of the broken line in half (Fig. 137). The segment to which point $B$ belongs is $A_{i} A_{i+1}$.
The surface obtained by rotating the broken line around the lin... | \pi^{2}/2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,823 |
59. The area of square $K$ is 1, and the area of the inscribed rectangle $\Pi$ is $s$. What value can $s$ take if
a) $\Pi$ is not a square;
b) $\Pi$ is a square? | 59. If \(A_{1} B_{1} C_{1} D_{1}\) is a rectangle inscribed in the square \(A B C D\) with side 1 (Fig. 141), then
\[
\angle B A_{1} B_{1}=90^{\circ}-\angle B B_{1} A_{1}=180^{\circ}-90^{\circ}-\angle B B_{1} A_{1}=\angle C B_{1} C_{1}
\]
and similarly
\[
\angle B A_{1} B_{1}=\angle C B_{1} C_{1}=\angle D C_{1} D_{1... | )0<<\frac{1}{2},\quadb)\frac{1}{2}\leqslant<1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,824 |
60. The diameter of the circle inscribed in triangle $ABC$ is 1. What can the side $d$ of the square inscribed in the triangle be, with two vertices lying on the base $AB$ of the triangle (not on its extensions!); and the other two on sides $AC$ and $BC?$ | 60. Since angles \( A \) and \( C \) at the base of the triangle must be acute (otherwise one of the vertices of the square would fall on the extension of the base), the points \( D \) and \( E \) of tangency of the sides \( AB \) and \( BC \) with the inscribed circle \( s \) are located on the "upper" semicircle (Fig... | \frac{4}{5}\leq1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,825 |
62. Triangle $A B C$ is divided into two triangles $A B D$ and $C B D$ by the straight line $B D$. Prove that the sum of the radii $r_{1}$ and $r_{2}$ of the circles inscribed in triangles $A B D$ and $C B D$ is greater than the radius $r$ of the circle inscribed in triangle $A B C$. | 62. Let $O, P, Q$ be the centers of the circles inscribed in triangles $ABC, ABD$, and $BCD$; $O', P', Q'$ be the points of tangency of these circles with the line $AC$ (Fig. 145). Draw the line $PK \parallel AC$ ($K$ is a point on the segment $OO'$); we need to prove that $PP' + QQ' > OO'$, or that $QQ' > OK$.
Let's ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,827 |
63. Let $M$ be an arbitrary point inside triangle $ABC$; the points of intersection of $AM$, $BM$, and $CM$ with the opposite sides $BC$, $AC$, and $AB$ of the triangle are denoted by $A_{1}$, $B_{1}$, and $C_{1}$ (Fig. 15). Prove that the sum
$$
M A_{1} + M B_{1} + M C_{1}
$$
does not exceed the length of the larges... | 63. First Solution. Let
$$
a=BC \geqslant b=AC \geqslant c=AB \text{; }
$$
then $BC=a$ is the greatest distance between any two points of triangle $ABC$: if $M$ and $N$ are two points of triangle $ABC$ and $M_{1}, N_{1}$ are the points of intersection of segment $MN$ with the boundary of the triangle, then
$$
MN \le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,828 |
64. Given a triangle $ABC$ and a point $D$ inside it such that
$$
AC - AD \geqslant 1 \quad \text{and} \quad BC - BD \geqslant 1
$$
Prove that for any point $E$ on side $AB$, the inequality
$$
EC - ED \geqslant 1
$$
holds.
## 3. PROBLEMS ON FINDING MAXIMUM AND MINIMUM VALUES OF GEOMETRICAL QUANTITIES
The central ... | 64. It is obvious that segment $E C$ intersects either $B D$ or $A D$; let it, for example, intersect $A D$ at some point $F$ (Fig. 149; point $F$ may coincide with $D$). By a known property of triangles,
$$
E F+D F \geqslant E D, \quad A F+C F \geqslant A C
$$
Adding these two inequalities, we get
$$
E F+D F+A F+C ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,829 |
65. Given an angle and a point inside it. Draw a line through this point so that the following are minimized:
a) the area of the triangle cut off from the angle by the drawn line;
b) the perimeter of this triangle. | 65. a) Let's prove that the desired line has the following property: the segment $L N$ of this line, enclosed between the sides of the angle, is bisected by the given point $A$. Indeed, let $A L = A N$ (Fig. 150$)^{1}$ ). We need to prove that the triangle $L M N$ has the smallest possible area. Draw any other line thr... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,830 |
66*. Inside angle $B A C$, a point $M$ is chosen, and a line $l$ is drawn through this point such that the segment $U V$, cut off by this line on the sides of the angle, has the smallest possible length.
Prove that the foot $N$ of the perpendicular dropped from point $A$ to the line $U V$ is symmetric to point $M$ wit... | 66. Let $U V$ be some line passing through point $M$ and intersecting the sides of the angle
to it (Fig. 152); let us also denote $M U=u, \quad M V=v$, $\angle M U A=\beta, \quad \angle M V A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,831 |
67. The perimeter of triangle $ABC$ is $2p$. What is the greatest value that the length of a segment tangent to the inscribed circle of $ABC$ and parallel to $BC$ can have? For which triangle (triangles) is this value achieved? | 67. Let the inscribed circle $s$ of the triangle touch its
Fig. 154.
sides $B C, A C$, and $A B$ of lengths $a, b$, and $c$ at points $D, E$, and $F$, and the line $K L$ (where $K$ and $L$... | \frac{p}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,832 |
68. a) Given a segment $M N$ and a line $l$. Find a point $X$ on the line $l$ from which the segment $M N$ is seen at the largest angle,
b) Given a segment $M N$ and a circle $S$. Find points $X$ and $Y$ on the circle $S$ from which the segment $M N$ is seen at the largest and the smallest angles. | 68. a) Let's denote the line on which the segment $M N$ lies as $k$. Suppose lines $k$ and $l$ intersect. In this case, there are three possible scenarios.
$1^{\circ}$. The point $O$ of intersection of lines $k$ and $t$ lies outside the segment $M N$. Then, the closer the point $X$ on line $l$ is to point $O$, the lar... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,833 |
69. a) Given a line $l$ and two points $A$ and $B$ on the same side of this line. Find a point $X$ on the line $l$ such that the sum of the distances from $X$ to points $A$ and $B$ is minimized.
b) Given a line $l$ and two points $A$ and $B$ on opposite sides of it. Find a point $X$ on the line $l$ such that the diffe... | 69. a) Let $B^{\prime}$ be the point symmetric to point $B$ with respect to the line $l$ (Fig. 171). If $X^{\prime}$ is an arbitrary point on the line $l$, then $A X^{\prime} + X^{\prime} B = A X^{\prime} + X^{\prime} B^{\prime}$. Therefore, the sum $A X^{\prime} + X^{\prime} B$ will be
 inside the quadrilateral $A B C D$;
b) in the plane of this quadrilateral
the point $X$, the sum of whose distances from the vertices of the quadrilateral is the smallest. | 76. It will be convenient for us to start with the solution of part b).
b) Let \(ABCD\) be an arbitrary quadrilateral, \(M\) the point of intersection of its diagonals, and \(X\) any other point on the plane (Fig. 201, a, b). It is clear that
\[
X A + X C \geqslant A C = M A + M C \quad \text{and} \quad X B + X D \ge... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,839 |
82. Find
a) in the plane of quadrilateral $A B C D$;
b) inside quadrilateral $A B C D$;
c) in three-dimensional space, the point for which the sum of the squares of the distances from the given points $A, B, C$ and $D$ (vertices of the quadrilateral in problems a) and b) and vertices of the given tetrahedron $A B C ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,844 | |
83. Among all triangles inscribed in a given circle, find the one for which the sum of the squares of the sides is maximal. | 83. Let $ABC$ be an arbitrary triangle inscribed in a given circle, and $BK$ be its median. Then
$$
B K^{2}=\frac{1}{2}\left(A B^{2}+B C^{2}-\frac{1}{2} A C^{2}\right)
$$
(see above, p. 158), and therefore,
$$
A B^{2}+B C^{2}=2 B K^{2}+\frac{1}{2} A C^{2}
$$
Thus, for a fixed side $AC$, the sum $A B^{2}+B C^{2}$ wi... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,845 |
85. Two segments $AB$ and $BC$ of lengths $a$ and $b$ are connected by a hinge at point $B$; an equilateral triangle $ACD$ is constructed on segment $AC$ (points $B$ and $D$ are located on opposite sides of line $AC$; see Fig. 18). For what angle $ABC$ will the length of segment $BD$ be
a) the greatest;
b) the smalle... | 85. Let \( AB = a \geqslant b = BC \). Construct an equilateral triangle \( ABE \) on the segment \( AB \), located on the opposite side of \( AB \) from triangle \( ABC \) (Fig. 226). A rotation by \( 60^{\circ} \) around point \( A \) translates the segment \( DB \) into segment \( CE \), and we only need to determin... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,847 | |
86. Two pedestrians, moving with the same speed $w$, start from points $A$ and $B$ along straight roads $O A$ and $O B$ at the same time; the distances $O A=a$ and $O B=b$, and the angle $A O B=\alpha$ are known.
. After time $t$ from the start of the movement, the pedestrians will have traveled along the roads $A O$ and $B O$ the distances $A M$ and $B N$, where $A M = B N = v t$. Draw through poi... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,848 | |
89. Prove that the sum of the medians of an arbitrary triangle $ABC$ is always less than the perimeter of the triangle and greater than $3/4$ of its perimeter:
$$
2 p > m_{a} + m_{b} + m_{c} > \frac{3}{2} p, \text{ or } 2 > \frac{m_{a} + m_{b} + m_{c}}{p} > 1^{1/2}
$$
Can the (double) inequality
$$
2 > \frac{m_{a} +... | 89. Let $M$ be the point of intersection of the medians $A D=m_{a}, B E=m_{b}$, and $C F=m_{c}$ of triangle $A B C$ (Fig. 230); then $A M=\frac{2}{3} m_{a}, B M=\frac{2}{3} m_{b}$, $C M=\frac{2}{3} m_{c} ; \quad M D=\frac{1}{3} m_{a}, \quad M E=\frac{1}{3} m_{b}, \quad M F=\frac{1}{3} m_{c}$. From the consideration of ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,850 |
90. Prove that for any triangle
$$
A a+B b+C c \geqslant \frac{1}{2}(A b+B a+A c+C a+B c+C b)
$$
In which case does this inequality become an equality? | 90. Let $a \geqslant b \geqslant c$; then, since in a triangle the larger side is opposite the larger angle, $A \geqslant B \geqslant C$. Therefore,
$$
\begin{aligned}
(a-b)(A-B)+(b-c) & (B-C)+ \\
& +(c-a)(C-A) \geqslant 0
\end{aligned}
$$
since each term on the left side is non-negative.
Expanding the brackets and ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,851 |
92. a) Prove that if the bases $BC$ and $B_{1}C_{1}$, as well as the angles $A$ and $A_{1}$, of two triangles $ABC$ and $A_{1}B_{1}C_{1}$ are equal, then the one with the smaller (in absolute value) difference of the angles at the base has the larger perimeter.
b) What perimeter can triangle $ABC$ with a known side $B... | 92. a) Let's align the bases $BC$ and $B_1C_1$ of triangles $ABC$ and $A_1B_1C_1$; then extend their sides $BA$ and $BA_1$ to segments $AD = AC$ and $A_1D_1 = A_1C$ (here we assume that $AB \leq AC$ and $A_1B \leq A_1C$; see Fig. 234). From the isosceles triangles $ACD$ and $A_1CD_1$, we have: $\angle ADC = \frac{\angl... | 2a<2p\leq(1+\operatorname{cosec}\frac{\alpha}{2}) | Geometry | proof | Yes | Yes | olympiads | false | 23,853 |
95. Prove that for any triangle $ABC$
a) $9 r \leqslant h_{a}+h_{b}+h_{c} \leqslant \beta_{a}+\beta_{b}+\beta_{c} \leqslant m_{a}+m_{b}+m_{c} \leqslant \frac{9}{2} R$;
b) $\beta_{a}+\beta_{b}+\beta_{c} \leqslant \sqrt{r_{a} r_{b}}+\sqrt{r_{b} r_{c}}+\sqrt{r_{c} r_{a}} \leqslant p \sqrt{3} \leqslant$
$$
\leqslant r_{... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,855 | |
96*. Prove that for any triangle $ABC$
a) $r_{a}^{2}+r_{b}^{2}+r_{c}^{2} \geqslant 27 r^{2}$
b) $4 R < r_{a}+r_{b}+r_{c} \leqslant 4 \frac{1}{2} R$
Can these inequalities be improved?
Problem 96 b) indicates the range within which the sum $r_{a}+r_{b}+r_{c}$ of the radii of the excircles of a triangle can vary if t... | 96. a) Above (p. 250) we have already seen that
$$
r_{a} r_{b} + r_{b} r_{c} + r_{a} r_{c} = p^{2}
$$
Therefore, using inequality (B) (p. 250) and the inequality derived from the results of problem 95 a), b):
$$
9 r \leqslant p \sqrt{3}, \text{ or } p^{2} \geqslant 27 r^{2}
$$
(see also problem 97 below), we obtain... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,856 |
97. Prove that for any triangle $ABC$
$$
r \leqslant \frac{\sqrt{\sqrt{3} S}}{3} \leqslant \frac{\sqrt{3}}{9} p \leqslant \frac{1}{2} R
$$
In what case do the inequalities in this problem become equalities?
The results of problem 97 deserve a more detailed discussion. By discarding the intermediate terms of the ineq... | 97. In problem 95 b), it was already shown that
$$
27 r^{2} \leqslant p^{2} \leqslant \frac{27}{4} R^{2}
$$
from which
$$
r \leqslant \frac{\sqrt{3} p}{9} \leqslant \frac{1}{2} R
$$
and equality holds only for an equilateral triangle. Thus, we need to prove that
$$
r \leqslant \frac{\sqrt{\sqrt{3}} S}{3} \leq \fra... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,857 |
98. Prove that for any tetrahedron
$$
V \leqslant \frac{8 \sqrt{3}}{27} \mathrm{P}^{3}
$$
In what case does the equality $V=\frac{8 \sqrt{3}}{27} \mathrm{P}^{3}$ hold?
99 * %. Prove that for any tetrahedron
a) $\Delta \geqslant 6 \sqrt[3]{\sqrt{3} V^{2}}$
b) $\Pi \geqslant 3 \sqrt{2} \cdot \sqrt[3]{3 V}$
In what ... | 98. First solution. Let the tetrahedron \(ABCD\) be inscribed in a sphere \(\Sigma\) of radius \(P\). If the base \(BCD\) of this tetrahedron lies in a fixed plane \(\pi\), which is at a distance \(d\) from the center \(O\) of the sphere \(\Sigma\), then the volume \(V\) of the tetrahedron does not exceed \(\frac{1}{3}... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,858 |
100. a) Prove that for any triangle $ABC$
$$
a^{2}+b^{2}+c^{2} \geqslant 4 \sqrt{3} S
$$
where equality holds only for an equilateral triangle $ABC$.
b)* Prove that, moreover,
$$
a^{2}+b^{2}+c^{2}-(a-b)^{2}-(b-c)^{2}-(c-a)^{2} \geqslant 4 \sqrt{3} S
$$
where equality also holds only when $a=b=c$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,859 | |
101*. a) Prove that for each tetrahedron \(ABCD\)
\[
\Pi^{2} \geqslant 3 \sqrt{3} \Delta
\]
where equality holds only if the tetrahedron \(ABCD\) is regular.
b) Prove that, moreover,
\[
\begin{aligned}
& \Pi^{2} \geqslant 3 \sqrt{3} \Delta + 1 / 2\left[\left(a + a_{1} - b - b_{1}\right)^{2} + \left(a + a_{1} - c - ... | 101. Since the inequality of problem a) obviously follows from the stronger inequality of problem b), we will limit ourselves to proving the latter. We apply the inequality of problem 100 b) to each face of the tetrahedron $A B C D$ (Fig. 252):
$$
a_{1}^{2}+b_{1}^{2}+a_{1}^{2}-\left(a_{1}-b_{1}\right)^{2}-\left(b_{1}-... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,860 |
102. Prove that for each triangle $ABC$
a)
$$
3 / 2 r \leqslant \rho_{a}+\rho_{b}+\rho_{c} \leqslant 3 / 4 R
$$
where $\rho_{a}, \rho_{b}, \rho_{c}$ are the radii of the three circles inscribed in the segments $BC, CA, AB$ of the circumcircle $K$ of triangle $ABC$, cut off from $K$ by the sides of the triangle (circl... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,861 |
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