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486. For the theorems on measuring angles by arcs [1) central angle, 2) inscribed angle, 3) angle formed by a tangent and a chord at the point of tangency, 4) angle formed by two secants intersecting inside the circle, 5) angle formed by a tangent and a secant not passing through the point of tangency, 6) angle formed ... | 486. An angle whose sides are secants or tangents of a given circle is measured by the half-sum or half-difference of the arcs intercepted between its sides, depending on whether the vertex of the angle lies inside or outside the circle; when the vertex lies on the circle, one of the arcs reduces to zero. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,245 |
489. The more sides a regular polygon has, the more convex its periphery becomes; to what limit does the internal (external) angle tend when the number of sides increases indefinitely? | 489. The angle of a regular $n$-sided polygon $\frac{180^{\circ}(n-2)}{n}=180^{\circ}$ $\left(1-\frac{2}{n}\right)$ approaches $180^{\circ}$ as $n \rightarrow \infty$. The polygon approaches the shape of a circle, and the angle (interior) between two consecutive sides approaches the angle (at $180^{\circ}$) formed by t... | 180 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,246 |
498. How many elements are necessary and sufficient to determine a) a triangle? b) a quadrilateral? c) a general n-gon? | 498. To determine an $n$-gon, it is necessary to specify $2n-3$ of its $2n$ elements ($n$ sides and $n$ angles), including no fewer than $n-2$ sides. Proof by induction. | 2n-3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,252 |
520. Given 5 points, any 4 of which do not lie in the same plane. a) How many tetrahedrons exist, the vertices of which are 4 of these points? b) How many planes pass through triples of these points? c) How many lines are determined by pairs of these points? | 520. a) $C_{5}^{4}=C_{5}^{1}=5$; b) the number of planes passing through 3 of these 5 points is $C_{5}^{3}=C_{5}^{2}=10$; c) the number of lines passing through 2 of these points is $C_{5}^{2}=C_{5}^{3}=10$, because if no 4 points lie in the same plane, then no 3 lie on the same line. | 5,10,10 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 40,262 |
521. Through a point lying outside a plane, how many: a) lines perpendicular to the plane can be drawn? b) lines parallel to it? c) planes perpendicular to this plane? d) planes parallel to it? | 521. a) One, b) an infinite set - generating a plane parallel to the given one, c) an infinite set - passing through the perpendicular dropped from this point, d) one (see b). | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,263 |
522. How many planes perpendicular to a given line can be drawn through a point lying outside this line? How many planes parallel to this line can be drawn? | 522. a) One, b) an infinite set - intersecting along a line parallel to the given line, passing through the given point. | )One,b)an\inftyiniteset | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,264 |
523. Can more than one perpendicular be erected to a line at a point taken on it? Can more than one perpendicular be dropped from a point taken outside the line to it? Can more than one parallel be drawn from a point outside the line to it? | 523. a) An infinite set - arranged in a fan (bundle) and forming a plane perpendicular to the given line, b) only one, c) one. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,265 |
525. In space, two lines are given; how many lines exist that intersect both given lines at right angles? All cases. | 525. 1) Skew lines: 1 common perpendicular; 2) parallel lines: $\infty$ common perpendiculars; 3) intersecting lines: 1 common perpendicular (erected to the plane defined by the given lines at their point of intersection). | 1,\infty,1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,267 |
526. If three lines are given at random in space, there exists an unlimited number of lines that intersect all these three lines (how to construct them?); is there such a special position of three lines for which there is not a single line that intersects all of them? | 526. Let the lines $p_{1}, p_{2}, p_{3}$ be given; through one of them, for example, $p_{1}$, we draw a plane which, generally, intersects the lines $p_{2}$ and $p_{3}$ at points $P_{2}$ and $P_{3}$, respectively; the line $\boldsymbol{P}_{2} P_{3}$ generally intersects the line $p_{1}$ at point $\boldsymbol{P}_{1} ; \... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,268 |
527. In a plane, three lines are given; how many circles exist that touch all three lines? Investigate all possible cases. | 527. In general 4; if all three lines pass through one point, or if all three are parallel, then none; if 2 are parallel and intersected by the third, then 2 circles. | 4;none;2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,269 |
528. If the centers of two circles $O_{1}$ and $O_{2}$ are given, then their radii $\boldsymbol{R}_{1}$ and $\boldsymbol{R}_{2}$ can be chosen so that the circles intersect and have 2 common tangents. In what special (limiting) case is this statement not true? How many common tangents can there be in this case? | 528. If $O_{1}$ and $O_{2}$ coincide, then the circles must be concentric. There would be an infinite number of common tangents (if taking $\boldsymbol{R}_{1}=\boldsymbol{R}_{2}$ ). | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,270 |
529. How many common tangents do two circles have? Investigate, taking into account the case of equal circles, as well as the limiting cases when one of them or both reduce to a point. | 529. a) The circles lie outside each other: 4 common tangents; b) external tangency: 3; c) the circles intersect: 2; d) internal tangency: 1; e) one inside the other: 0. If the circles are equal, then instead of d) we have a complete merging of both and an infinite number of common tangents. If one of the circles reduc... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,271 |
531. Where should the source of light (a luminous point) be located so that exactly half of the surface of the sphere is illuminated? | 531. In "infinity"; this means: there is no such position of the light source that would illuminate exactly half of the sphere, but the farther we move the light source, the less the illuminated part of the sphere will differ from a hemisphere. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,273 |
533. In space, a circle and a point are given. Where should the point lie so that a surface can be passed through it and the circle a) of a circular cylinder? b) of a circular cone? c) of a sphere? | 533. a) On the perpendicular erected to the plane of the circle at any point of its circumference; b) anywhere inside the cylindrical surface perpendicular to the plane of the circle, c) at any point in space (excluding only the plane of the circle). | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,275 |
534. Find the necessary and sufficient conditions that two circles in space must satisfy for a spherical surface to be able to pass through them. | 534. In general: the perpendiculars erected at the centers of both circles must intersect at a point equidistant from two arbitrarily taken points on each of the given circles.
If the given circles lie in parallel planes and the line connecting their centers is $\perp$ to these planes, then a spherical surface can alw... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,276 |
536. Given a conical funnel, the angle of the axial section of which is equal to $2 \alpha$, and two spheres of radii $\boldsymbol{R}$ and $r$. Find the necessary and sufficient conditions for both spheres to be placed in the funnel so that both are in contact with its lateral surface. | 536. By making an axial section, we reduce the problem to planimetry; the required condition is:
$$
\sin \alpha \leqslant \frac{R-r}{R}
$$ | \sin\alpha\leqslant\frac{R-r}{R} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,278 |
537. Can a sphere be described around every circular cone and can a sphere be inscribed in each one? The same questions for a circular cylinder. | 537. The answer to both questions about the cone is positive, since the axial section is a triangle. A sphere can be described around any cylinder of revolution, but a sphere can only be inscribed in one where the axial section is a square. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,279 |
539. a) Around which prisms can a sphere be circumscribed? b) Into which prisms can a sphere be inscribed? | 539. a) Around right prisms, the bases of which are describable polygons; b) in right prisms, into which circles can be inscribed in the bases, if the height of the prism is equal to the diameter of these circles. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,281 |
543. a) How can a right triangle be divided into two similar triangles? b) In how many ways can this be done? | 543. The perpendicular dropped from the vertex of the right angle to the hypotenuse. This is the only way to perform the required decomposition. | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,285 |
548. Which special angles can be divided into 3 equal parts using a compass and a straightedge? | 548. All angles of the form $3 \alpha$, where $\alpha$ denotes an angle that can be constructed with a compass and straightedge (see question 481). Since we can construct angles of $60^{\circ}, 45^{\circ}, 30^{\circ}, 2 \frac{1}{2}^{\circ}, 15^{\circ}, 71^{\circ}, \ldots 37 \frac{1}{2}^{\circ}$, $52 \frac{1}{2}^{\circ}... | \ | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,290 |
549. Given a polygon. How to construct two polygons similar to it, the sum of the areas of which would be equal to the area of the given polygon? Number of solutions? | 549. On one of the sides of the given polygon, as on the hypotenuse, we construct a right triangle and on the legs we construct polygons similar to the given one. The number of solutions, obviously, is unlimited. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,291 |
551. On one of the sides of triangle $A B C$ there is a point $P$; draw a line through $\boldsymbol{P}$ that cuts off from triangle $\boldsymbol{A} \boldsymbol{B} \boldsymbol{C}$ a triangle similar to it. How many solutions? | 551. In general, there are 4 solutions. If the triangle is isosceles or equilateral, the number of solutions decreases accordingly. | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,292 |
568. Given the problem: "Through a point $C$, taken on the extension of the diameter $\boldsymbol{A} \boldsymbol{B}$ of a circle with radius $\boldsymbol{R}$, draw a secant such that the sum $A E^{2}+B D^{2}$ equals a given number $m^{2}$." By calling $C O=a$ and taking as the unknown $x$ the distance of the chord $D E... | 568. In the limiting case, if $\boldsymbol{C D E}$ is a tangent, then $m^{2}=4 \boldsymbol{R}^{2}$, so $x=\boldsymbol{R}$. | \boldsymbol{R} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,303 |
573. How many right angles can a convex pentagon have at most? A hexagon? An octagon? Find a general theorem and prove it. | 573. A convex $n$-gon ( $n>4$ ) can have no more than three right angles. Indeed, let there be $x$ right angles. Obviously, for $n>4$ we have $x \leq n$. The polygon is convex by position; therefore, each of the $n-x$ non-right angles is less than $180^{\circ}$. Hence:
the sum of the angles or $180^{\circ}(n-2)<180^{\... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,306 |
574. What is the maximum number of acute, right, and obtuse angles that a triangle, a quadrilateral... an $n$-sided polygon can have? (The question refers only to convex polygons). | 574. The answer (which needs to be justified) is contained in the following table:
| Angles. | Triangle. | Quadrilateral. | Pentagon. | $n$-gon. |
| :---: | :---: | :---: | :---: | :---: |
| Acute $\left\{\begin{array}{l}\text { max. } \\ \text { min. }\end{array}\right.$ | 3 2 | 3 - | 3 - | 3 - |
| Right $\left\{... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,307 |
580. Given the proportion:
$$
\frac{\text { area } \boldsymbol{S}}{\text { area } \boldsymbol{S}^{\prime \prime}}=\frac{\text { segment } \boldsymbol{a}}{\text { segment } \boldsymbol{a}^{\prime}}
$$
do the following formulas then make sense, and if they do, what sense do they make:
$$
\frac{\text { area } S}{\text ... | 580. The meaning of formula (1) is as follows: the side of a rectangle, the area of which $=S$, and the other side $=a$, is equal to the side of another rectangle, the area of which $=S^{\prime}$, and the other side $=a^{\prime}$; the meaning of formula (2): the volume of a prism, the base of which is equal to $S$, and... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,310 |
589. Can parallel lines be drawn in two non-parallel planes? | 589. Yes, as many as you like, parallel to the straight line of intersection of the planes. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,315 |
593. A point moves along an arc of a circle and then transitions to a straight line; what should be the position of the line relative to the arc so that this transition occurs smoothly, without a jolt? | 593. The line must be tangent to the curve. | The\line\must\be\tangent\to\the\curve | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,316 |
596. In a plane, two points $\boldsymbol{A}$ and $\boldsymbol{B}$ are given, and an axis that does not contain either of them. A curve connecting these points intersects the axis at an even number of points. What can be concluded about the positions of points $\boldsymbol{A}$ and $\boldsymbol{B}$ relative to the axis? ... | 596. If the number of intersection points is even (0 is also considered even), then points $\boldsymbol{A}$ and $\boldsymbol{B}$ lie on the same side of the axis; if odd, then on opposite sides. A point of tangency is equivalent to two (coalesced) intersection points. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,317 |
599. Which of the four notable points of a triangle (the centers of the inscribed and circumscribed circles, the intersection point of the altitudes, the centroid) can lie outside the triangle or on its periphery, and which must lie inside the triangle? | 599. The center of the inscribed circle and the point of intersection of the medians (center of gravity) always lie inside the triangle; the point of intersection of the altitudes can lie outside (obtuse triangle) or at one of the vertices (vertex of the right angle); the center of the circumscribed circle can lie outs... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,318 |
600. Can some of the four notable points of a triangle coincide with others? What can be said about a triangle in which the centers of the circumscribed and inscribed circles coincide? | 600. If any two of the 4 notable points coincide, then all 4 coincide, and the triangle is equilateral. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,319 |
604. Which of the regular polygons can completely and densely cover a plane without gaps? | 604. Let there be a regular $n$-gon. As is known, the angle of this $n$-gon is $180^{\circ}-\frac{360^{\circ}}{n}$ or $180^{\circ}\left(1-\frac{2}{n}\right)$. It is required that at each vertex, the polygon, together with adjacent polygons, should tightly cover the plane. If $k$ regular polygons meet at one vertex, the... | 3,4,6 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,320 |
612. Give examples of lines dividing the plane a) into two and only two regions; b) into exactly three regions; c) into four regions. | 612. a) Line, circle, ellipse, parabola; b) hyperbola; c) system of two intersecting circles, system of three intersecting circles, system of three parallel lines. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,324 |
613. Both an unlimited straight line and a circle divide the plane into two regions; what similarities and differences can be noted between the two cases of dividing the plane into two regions? | 613. Similarities: a line connecting point $A$, located in one region, with point $\boldsymbol{B}$, located in another region, must intersect the boundary at least at one point (generally at an odd number of points); a line connecting points $\boldsymbol{A}$ and $\boldsymbol{A}^{\prime}$, both lying in the same region,... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,325 |
623. From a point, a certain number of rays originate; at least how many colors are needed to color each angle so that two adjacent angles differ in color?
Translating the text into English, while preserving the original text's line breaks and format, results in the above translation. | 623. If the number of rays (and thus angles) is even, then 2 colors are necessary and sufficient; if odd, then 3. | 2or3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,327 |
628. In a plane, there are 4 points $\boldsymbol{A}, \boldsymbol{B}, C, D$, arranged such that the line connecting any two of them is perpendicular to the line connecting the other two points. What is $D$ in relation to the triangle $\boldsymbol{A} \boldsymbol{B} C$? What would each of the four points be in relation to... | 628. Each point is the intersection of the altitudes of the triangle formed by the other three points. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,329 |
633. Prove that a bounded geometric figure cannot have two distinct centers of symmetry. | 633. Suppose a bounded figure has two centers of symmetry $O_{1}$ and $O_{2}$, and let $M$ denote any point of the figure. Let the point $M_{1}$ be the point symmetric to point $M$ with respect to $O_{1}$, and $\boldsymbol{M}_{2}$ the point symmetric to point $\boldsymbol{M}_{1}$ with respect to $O_{2}$. It is easy to ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,331 |
634. Prove that if a bounded figure has an axis of symmetry and a center of symmetry, then the center of symmetry lies on the axis of symmetry. | 634. We could apply a method similar to the one applied to the previous question, taking points successively symmetric with respect to the center of symmetry and the axis of symmetry. By leaving this to the reader, we will provide here a purely analytical proof. Let's take the axis of symmetry as the x-axis, and the y-... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,332 |
635. Prove that if a figure has an axis of symmetry and a center of symmetry lying on it, then it has another axis of symmetry, perpendicular to the first and passing through the center of symmetry. | 635. Let $M$ be an arbitrary point of the figure, $\boldsymbol{M}_{1}$ the point symmetric to $\boldsymbol{M}$ with respect to the axis, $\boldsymbol{M}_{2}$ the point symmetric to $\boldsymbol{M}$ with respect to the center of symmetry. Then the point $\boldsymbol{M}_{2}$ will be symmetric to $M$ with respect to the a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,333 |
640. Do there exist limited figures with three axes of symmetry? With a larger number of axes of symmetry? | 640. An equilateral triangle; a curvilinear triangle formed by three equal arcs described with the same radius on the sides of an equilateral triangle from centers lying on perpendiculars erected at their midpoints; a figure formed by three equal sectors of a given circle, the midpoints of the arcs of which are $120^{\... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,337 |
641. Study the symmetric properties of an equilateral triangle. | 641. 3 axes of symmetry; no center of symmetry, for if such existed, it could only be (see question 634) the point of intersection of the altitudes. But it divides the altitudes in the ratio $1: 2$. | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,338 |
645. What symmetrical properties do regular polygons have? | 645. In a regular $n$-sided polygon: a) if $n$ is an even number, there are $n$ axes of symmetry (specifically $\frac{n}{2}$ lines connecting opposite vertices, and the same number of lines connecting the midpoints of opposite sides) and a center of symmetry; b) if $n$ is an odd number, the $n$-sided polygon has $n$ ax... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,340 |
651. What symmetrical properties does a system of two circles lying in the same plane possess? What special cases need to be investigated? | 651. If the radii of the circles are not equal, then the line of centers is the only axis of symmetry, and there is no center of symmetry; in the special case where the circles are concentric, there is a center of symmetry and infinitely many axes of symmetry. If the radii of the circles are equal, then there are two p... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,345 |
652. Study the symmetry properties of the figure formed by three equal circles touching each other in pairs. | 652. There are three axes of symmetry (lines connecting the points of tangency of each of the two circles with the center of the third circle). It is easy to show that the common point of intersection of these three axes is not the center of symmetry; therefore, the figure has no center of symmetry. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,346 |
659. Study the symmetry properties of a cube. Then assume that a) one, b) two, c) three faces of a cube made of white wood are painted black, and study the symmetry properties of these bodies. In cases b) and c), investigate all possible scenarios. (Borel). | 659. In the general case, there are 9 planes of symmetry, 9 axes of symmetry, and a center of symmetry. In special cases:
a) 4 planes of symmetry, one axis, no center of symmetry;
$b_{1}$ ) painted faces are parallel: 5 planes of symmetry, 3 axes, center of symmetry;
$b_{2}$ ) painted faces are adjacent: 2 planes of ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,352 |
660. If in the previous question we also considered the symmetry properties of the cube, four faces of which are painted black, would we obtain a type of symmetry different from types a), b) and c)? | 660. A case identical to $b_{1}$ or $b_{2}$. | b_{1} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 40,353 |
661. Study the symmetry of a cube, two faces of which are painted white, two black, and two red. Investigate all possible cases. | 661. a) All three pairs of like-colored faces are parallel: 3 planes of symmetry, 3 axes, center.
b) One pair of like-colored faces is parallel, two are adjacent: 1 plane of symmetry, no axes or center.
c) No planes, axes, or center of symmetry. | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 40,354 |
669. The angle at which two circles intersect is called the angle formed by the tangents to these circles at the point of intersection. When do two circles intersect at an angle of $0^{\circ}$? $90^{\circ}$? $180^{\circ}$? | 669. $0^{\circ}$ at internal tangency of circles, $180^{\circ}$ at external tangency of circles (gears rotate in the same direction when in internal engagement, and in opposite directions when in external engagement). When intersecting at a right angle, the tangents at the point of intersection, and therefore the radii... | \sqrt{\boldsymbol{R}^{2}+\boldsymbol{r}^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,358 |
673. What properties of a geometric figure are independent of its size? | 673. Magnitudes of angles and ratios of linear measurements (segments). | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,360 |
680. The sides of three regular octagons are respectively equal to 3 l, $4 \boldsymbol{\mu}, 12 \boldsymbol{}$. What is the length of the side of a regular octagon whose area is equal to the sum of the areas of the three given octagons? | 680. Let the ratio of linear measurements of three similar figures $F_{1}, F_{2}, F_{3}$ (in this case, regular octagons) be $k: l: m$ (in this case $3: 4: 12$). Denote the areas of the figures as $S_{1}, S_{2}, S_{3}$, and let $F_{4}$ be a figure similar to $F_{1}, F_{2}, F_{3}$ with area $S_{4}=S_{1}+S_{2}+S_{3}$.
S... | 13m | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,361 |
683. Can two similar but not congruent triangles be inscribed in the same circle? Can they be circumscribed around the same circle? | 683. No; for if the triangles are similar but not equal, then all their linear elements are not equal, and therefore the radii of the circumscribed (inscribed) circles are not equal either. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,362 |
687. How to cut off from a given triangle, by a straight line parallel to one of its sides, a triangle equal to one fourth of the given one? equal to its $\boldsymbol{n}$-th part? a quadrilateral equal to the $\boldsymbol{n}$-th part of the given triangle? | 687. By a straight line parallel to the base, we cut off a triangle whose height is equal to $\frac{h}{\sqrt{n}}$. To obtain the required quadrilateral, we cut off a triangle, the area of which is equal to $\frac{n-1}{n}$ of the area of the given triangle. | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,363 | |
693. A regular tetrahedron, octahedron, and icosahedron have equal surface areas. How do their edges relate? | 693. $2 \sqrt{10}: \sqrt{10}: 2$. | 2\sqrt{10}:\sqrt{10}:2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,366 |
696. What body will be formed if the midpoints of the faces of a cube are connected by straight lines? What will be the length of the edge of this body if the edge of the cube is $a$? | 696. A regular octahedron, with edge $=\frac{a}{\sqrt{2}}$. | \frac{}{\sqrt{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,369 |
698. From the given regular tetrahedron, the trihedral angles are cut off by planes passing through the midpoints of the edges. What body is obtained? | 698. A regular octahedron, the edge of which is twice smaller than the edge of the given tetrahedron. | A\regular\octahedron,\the\edge\of\which\is\twice\smaller\than\the\edge\of\the\given\tetrahedron | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,371 |
699. What body will result if the midpoints of the faces of a regular octahedron are connected by straight lines? | 699. A cube with an edge $=\frac{\sqrt{\overline{2}}}{3} a$ ( $a$ is the edge of the given octahedron). | withanedge=\frac{\sqrt{2}}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,372 |
700. In this triangle, we connect the midpoints of the sides; in the new triangle obtained, we do the same; and so on. Without performing calculations: a) what sequence do the perimeters of the successive triangles form; b) the areas of these triangles? | 700. An infinite series of similar triangles is obtained, the sides of which are in the ratio $1: \frac{1}{2}: \frac{1}{4}$ and so on. Therefore, infinitely decreasing geometric progressions will be obtained with common ratios a) $\frac{1}{2}$, b) $\frac{1}{4}$. | )\frac{1}{2},b)\frac{1}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,373 |
703. For an equilateral triangle, a square, and a circle with equal areas, which has the largest perimeter and which has the smallest? Try to generalize. | 703. The perimeter of a triangle is greater than that of all others, while the length of a circle is less. This can be established either by direct calculation or by reference to a previous question. Indeed, if a circle and a square have the same perimeter, then the area of the circle is greater than the area of the sq... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,376 |
725. Which regular polygons can be constructed using only a ruler in the form of a right-angled triangle, the shorter leg of which is half the hypotenuse? | 725. Triangle, quadrilateral, 6- and 12-sided polygon. | Triangle,quadrilateral,6-12-sidedpolygon | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,383 |
732. Given the sides of a triangle expressed in numbers. How can one determine with a rather simple calculation whether this triangle is right-angled, acute-angled, or obtuse-angled? Provide numerical examples. | 732. Let a be the largest of the sides, $b$ and $c$ the other sides. If $a^{2}>b^{2}+c^{2}$, then angle $\alpha$ is obtuse; if $a^{2}=b^{2}+c^{2}$, then angle $\alpha$ is right; if $a^{2}<b^{2}+c^{2}$, then angle $\alpha$ is acute. | ^{2}=b^{2}+^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,385 |
739. Can three angles of a convex quadrilateral be equal to the angles of a given triangle, respectively? | 739. No, because in that case the fourth angle would have to equal $180^{\circ}$. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,388 |
740. Show that a rectangle is the only convex polygon in which all angles are right angles. | 740. From the equation $\frac{180^{\circ}(n-2)}{n}=90^{\circ}$ it follows that $n=4$. | 4 | Geometry | proof | Yes | Yes | olympiads | false | 40,389 |
741. Show that a rectangle is the only convex polygon containing four right angles. | 741. Suppose that $n>4$. The sum of the angles of an $n$-gon is $180^{\circ}(n-2)$. If 4 of the angles are right angles, then the sum of the remaining $n-4$ angles must be $180^{\circ}(n-2)-360^{\circ}=180^{\circ}(n-4)$, which would require that each of these $n-4$ angles be $180^{\circ}$, or that some of them be $>180... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,390 |
742. Which parallelograms are inscribed in a circle? Which are circumscribed about a circle? Which are both inscribed and circumscribed? | 742. a) Rectangles, because from $\alpha=\gamma$ and $\alpha+\gamma=180^{\circ}$ it follows that $\alpha=90^{\circ}$; b) rhombuses, because from $a+c=b+d$ and $a=c, b=d$ it follows that $a=b ;$ c) squares. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,391 |
748. The line $\boldsymbol{A}^{\prime} \boldsymbol{B}^{\prime}$, intersecting two sides of triangle $\boldsymbol{A} \boldsymbol{B} \boldsymbol{C}$, divides it into triangle $\boldsymbol{A}^{\prime} \boldsymbol{B}^{\prime} \boldsymbol{C}$ and quadrilateral $\boldsymbol{A} \boldsymbol{B} \boldsymbol{B}^{\prime} \boldsymb... | 748. Let $A^{\prime}$ denote the point lying on $\boldsymbol{A} \boldsymbol{C}, \boldsymbol{B}^{\prime}$ the point on $\boldsymbol{B} \boldsymbol{C}$, and let $\boldsymbol{A}^{\prime} C=b^{\prime}, \boldsymbol{B}^{\prime} C=a^{\prime}$; then $\boldsymbol{A} \boldsymbol{A}^{\prime}+\boldsymbol{B} \boldsymbol{B}^{\prime}... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,396 |
764. How should the projection plane lie so that the rectangular projection of a right angle onto it is also a right angle. | 764. The projection plane must be parallel to at least one side of the angle. | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,400 |
766. The rectangular projection of angle $\alpha$ onto a plane is an obtuse angle $\alpha^{\prime}$. What can be concluded about the measure of angle $\alpha$? What can be concluded about the measure of angle $\alpha$ if angle $\alpha^{\prime}$ is acute? | 766. a) The angle $\alpha$ is obtuse; b) nothing. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,401 |
768. A line on a plane is defined by two points; under what mutual arrangement of these points will the line be defined as accurately as possible (in drawing)?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 768. When the distance between the points is as large as possible, and at small distances, the movement of the ruler by a slight angle is sensitively reflected in the position of the line. | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,403 |
770. It is required to divide the segment $\boldsymbol{A} \boldsymbol{B}$ in half. How should the radii of the arcs described from points $\boldsymbol{A}$ and $\boldsymbol{B}$, as centers, be chosen so that the construction is as accurate as possible? | 770. The arcs must intersect at a right angle; for this, the radii must be equal to $\frac{A B \sqrt{2}}{2}$ or, by eye, slightly less than $\frac{3}{4} \boldsymbol{A} \boldsymbol{B}$. | \frac{AB\sqrt{2}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,405 |
771. It is enough to glance at a circle to be convinced that it is equally curved everywhere (that it has the same curvature at all points). Are two circles with different radii equally curved? Connect two points with arcs of different curvatures. Can they be connected with arcs of the same curvature? | 771. Of the two circles, the one with the smaller radius has the greater curvature. Two points can be connected by two equally curved symmetrical arcs. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,406 |
772. A glance at the parabola is enough to convince oneself that it is not equally curved everywhere. Where is the curvature the greatest, and where is it the least? The same questions for the ellipse and the hyperbola. | 772. For a parabola, the curvature is greatest at the vertex, and it decreases as you move away from it. The same applies to a hyperbola. For an ellipse, the curvature is greater at the ends of the major axis and less at the ends of the minor axis. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,407 |
773. A segment can slide along a straight line; a circular arc can slide along a circle; are there other plane curves that possess this property? | 773. No, for the circle is the only plane curve of constant curvature (the straight line is the limit of the circle as the radius tends to infinity). | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,408 |
775. Localities $\boldsymbol{A}$ and $\boldsymbol{B}$ on the Earth's surface lie on the same parallel. Will the arc of this parallel be the shortest distance from $\boldsymbol{A}$ to $\boldsymbol{\boldsymbol { B }}$ : The same question for two localities lying on the same meridian. | 775. For a parallel - no; the shortest will be the arc of the great circle of the sphere between $\boldsymbol{A}$ and $\boldsymbol{B}$. - For a meridian - yes. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,409 |
776. What is the shortest distance between two points on the surface: a) of a circular cylinder? b) of a circular cone?
## VI. Motion. | 776. a) In the particular case, if both points $\boldsymbol{A}$ and $\boldsymbol{B}$ lie on the same circular cross-section of the cylindrical surface (perpendicular to the axis of the cylinder), the shortest distance will be the arc of this circle. In the general case, proceed as follows: unfold the lateral surface of... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,410 |
777. Is it possible that during the motion of a body, one and only one point remains stationary? | 777. Yes. In such a motion, all points of the body describe lines lying on concentric spherical surfaces, the common center of which is a fixed point (rotation about a point). | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,411 |
778. Can it happen that during the motion of a body, two and only two points remain stationary? | 778. If two points of a body are stationary, then all other points of the body lying on the line connecting them are also stationary (rotation about an axis). If, in addition to the two given stationary points, no point on the line connecting them belongs to the body, then the answer to the question posed must be affir... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,412 |
779. At least how many points of a body must be fixed in order for the entire body to be immobile? | 779. If three points of a figure, not lying on the same straight line, are made immobile, then the entire figure will be fixed. This condition is both sufficient and necessary. | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,413 |
781. A triangle moves in space so that its center of gravity remains stationary, and one of its vertices describes a circle, with the plane of the triangle remaining constantly perpendicular to the plane of the circle. What lines are described by the vertices of the triangle | 781. Circles. | Circles | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,415 |
783. A plane figure $\boldsymbol{F}$ is subjected to a rotation around point 0 by an angle $\alpha$ and then around another point $0_{1}$ by an angle equal and opposite to it, i.e., by an angle - $\alpha$. By means of which single movement can the figure be translated from its initial position to the final one? | 783. By means of parallel translation. | paralleltranslation | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,417 |
784. Show that two congruent plane figures (congruent in the narrower sense, i.e., non-symmetric) can be brought into coincidence either by a rotation around a known point, or by a parallel translation (where these cases exclude each other). | 784. Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be two arbitrary points of the first figure, $\boldsymbol{A}^{\prime}$ and $\boldsymbol{B}^{\prime}$ the corresponding points of the second figure. There are two cases: a) $\boldsymbol{A} \boldsymbol{B} \| \boldsymbol{A}^{\prime} \boldsymbol{B}^{\prime}$, then a parallel t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,418 |
785. Show that if two congruent (in the narrow sense) plane figures lying in the same plane and not parallel are given, then they can be brought into coincidence by means of two movements: a parallel translation and a rotation around arbitrary points. | 785. It is sufficient to wrap the segment $\boldsymbol{A}^{\prime} \boldsymbol{B}^{\prime}$ around an arbitrarily chosen point in the proper direction by an angle equal to the angle of inclination of the segments $\boldsymbol{A} \boldsymbol{B}$ and $\boldsymbol{A}^{\prime} \boldsymbol{B}^{\prime}$. Then $\boldsymbol{A}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,419 |
786. By means of what movement can two plane symmetric and symmetrically lying figures be brought into coincidence?
The translation is provided as requested, maintaining the original text's line breaks and format. | 786. A half-turn (in space) around the axis of symmetry (lying in the plane of the given figures). Therefore, to bring two planar symmetrical figures into coincidence, it is necessary to move out of the plane. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,420 |
787. A plane figure $\boldsymbol{F}$, after one parallel translation, occupies the position $\boldsymbol{F}^{\mathbf{l}}$, and after a second parallel translation (which is not a continuation of the first), the position $\boldsymbol{F}^{\prime \prime}$. Show that $\boldsymbol{F}$ can be brought to the position $\boldsy... | 787. The proof is based on the fact that two lines, each parallel to a third, are parallel to each other. In general: $\boldsymbol{n}$ parallel translations that move the figure $\boldsymbol{F}$ to the position $\boldsymbol{F}^{n}$ can be replaced by a single parallel translation. | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,421 |
788. What movements (parallel translations and rotations about an axis) can be used to bring two congruent spatial figures into coincidence? | 788. Let $A B C$ be three derivative points of one figure, not lying on the same straight line, $\boldsymbol{A}^{\prime} \boldsymbol{B}^{\prime} C^{\prime}$ the corresponding points of the second figure. The planes in which these points lie can always be brought to coincide or to be parallel by translation, if these pl... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,422 |
793. All trigonometric functions have a period of $2 \pi$ (or $360^{\circ}$); which of them have a smaller period? | 793. It is enough to cast a glance at the graphical representation of trigonometric functions on a circle with a radius equal to one (or to reproduce this image in one's mind) to see that the tangent and cotangent have a period of $\pi$ (or $180^{\circ}$). | \pi | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,423 |
797. What symmetric properties does a sine wave possess? | 797. Let the sine curve be drawn as the graph of the function $y=\sin x$ with respect to a rectangular system of axes; the origin and all points of the curve lying on the $x$-axis will be centers of symmetry; all lines drawn parallel to the $y$-axis at distances
$$
\frac{\pi}{2}, \frac{3 \pi}{2} \ldots,-\frac{\pi}{2},... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,424 |
798. With respect to which axes are the curves $y=\sin x$ and $y=\cos x$ mutually symmetric? | 798. With respect to the parallels to the $y$-axis, corresponding to the abscissas
$$
x=\frac{\pi}{4}, \frac{5}{4} \pi, \frac{9}{4} \pi \ldots,-\frac{\pi}{4},-\frac{5}{4} \pi,-\frac{9}{4} \pi \ldots
$$ | \frac{\pi}{4},\frac{5}{4}\pi,\frac{9}{4}\pi\ldots,-\frac{\pi}{4},-\frac{5}{4}\pi,-\frac{9}{4}\pi\ldots | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,425 |
799. What does $\sin 5 x$ mean, where 5 is an abstract number (not $5^{\circ}$ or $5^{\prime}$)? How can the value of this function be calculated using ordinary tables of angular functions? | 799. The absólute unit of angle is the radian, i.e., the angle that cuts from a circle an arc equal in length to its radius, or an angle of $57^{\circ} 17^{\prime} 44^{\prime \prime} 8^{\prime \prime \prime}$ or $209284^{\prime \prime}$. Therefore, an angle of 5, expressed in degrees, minutes, and seconds, equals $286... | -0.959 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,426 |
802. Why for all acute angles α
$$
\sin \alpha < \operatorname{tg} \alpha ?
$$
803 In the first quadrant, both sines and tangents increase with the angle, but tangents increase much faster than sines (and the further, the faster); why is this so? | 802. $\operatorname{tg} \alpha=\frac{\sin \alpha}{\cos \alpha}$, but for $0<\alpha<90^{\circ} \cos \alpha<1$, and dividing by a number $<1$ increases $\sin \alpha$. From the graphical representation, it is clearly visible that the line $\operatorname{tg} \alpha$ is longer than the line $\sin \alpha$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 40,427 |
805. In the theory of concave mirrors, the formula is encountered:
$$
x=r \frac{\sin ^{2} \frac{\alpha}{2}}{\cos x}
$$
For small $\alpha$, this formula is approximated with high accuracy by the formula $x=\frac{r \alpha^{2}}{4}$. Justify this. In what units of measure is the angle $\alpha$ expressed? | 805. For $0<\alpha<\frac{\pi}{2}$, there exist inequalities
$$
\begin{aligned}
& 0<\alpha-\sin \alpha<\frac{\alpha^{3}}{2} \\
& 0<1-\cos \alpha<\frac{\alpha^{2}}{2}
\end{aligned}
$$
From the first inequality, it follows that for small $\alpha$, $\sin \alpha$ is almost equal to $\alpha$ and, even more so, $\sin \frac{... | \frac{r\alpha^{2}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,429 |
806. If a sine is given, for the cosine there are two equal and opposite values; explain (without calculation) why this should be the case. | 806. A certain (for example, positive) value of the sine corresponds to two angle values: $\alpha$ in the first quadrant and $180^{\circ}-\alpha$ in the third quadrant; the corresponding cosine values are equal but with opposite signs (mentally reproduce the picture of the graphical representation). | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,430 |
811. Given an arbitrary number $n$; can it be considered as the value of one of the trigonometric functions of a known angle? And for each of the six functions? | 811. Always - tangent and cotangent; if $|n| \leqslant 1$, then (in addition to tangent and cotangent) also sine and cosine; if $|n| \geqslant 1$, then secant and cosecant. | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 40,433 |
813. Show that there is no angle for which the sine is equal to the secant, and that there is no angle for which the cosine is equal to the cosecant. | 813. When $\sin \alpha$ runs through (in absolute value) from 0 to 1, $\cos \alpha$ runs through values from 1 to 0, and $\sec \alpha$ from 1 to $\infty$; therefore, the values of $\sin$ and $\sec$ can never coincide.
Otherwise: suppose $\sin \alpha = \sec \alpha$ or $\sin \alpha = \frac{1}{\cos \alpha}$, from which $... | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,434 |
816. If you double the angle between $0^{\circ}$ and $45^{\circ}$, the sine will be less than twice the sine of the original angle. Why? | 816. $\sin 2 \alpha=2 \sin \alpha \cos \alpha$, but $\cos \alpha<1$; therefore, $\sin 2 \alpha<2 \sin \alpha$. | \sin2\alpha<2\sin\alpha | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 40,435 |
826. How to easily and accurately remember the sine and cosine formulas for angles:
$$
\frac{\pi}{2} \pm \alpha ; \pi \pm \alpha ; \frac{3 \pi}{2} \pm \alpha, 2 \pi-\alpha ?
$$ | 826. Let $\alpha<\frac{\pi}{4}$. Draw in a circle of radius $=1$ the functions of all named angles. There will be 8 equal right triangles. The first one lies on the horizontal axis, its smaller leg is $\sin \alpha$, the larger one is $\cos \alpha$. Each of the angles $\pi \pm \alpha$ is near the horizontal axis, the co... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,439 |
832. Solve the equation $\sin 3 x = \sin 5 x$. | 832. The solutions have the form $x=k \pi$ or $x=\frac{2 k+1}{8} \pi$, where $k$ denotes an integer, positive or negative. | k\pior\frac{2k+1}{8}\pi | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,441 |
833. How many different values does the expression $\sin \frac{k \pi}{3}$ take when $k$ takes all integer values from 0 to $+\infty$? | 833. Let $q$ be the integer quotient of the division of $k$ by $3$, and $r$ the remainder, which lies between -2 and 2, then $k=3 q+r$.
We have $\sin \frac{k \pi}{3}=\sin \left(q \pi+\frac{r \pi}{3}\right)=\sin \left( \pm \frac{r \pi}{3}\right)$. Everything reduces to the number of different values of the seven member... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,442 |
834. How many different values does the expression $\sin \frac{k \pi}{\sqrt{3}}$ take when $k$ is assigned all integer values from 0 to $+\infty$? | 834. All values of $\sin \frac{k \pi}{\sqrt{3}}$ are distinct. Indeed, let $\sin \frac{k \pi}{\sqrt{3}}=\sin \frac{k^{\prime} \pi}{\sqrt{3}}$. Then it would be $\frac{k \pi}{\sqrt{3}}=k^{\prime \prime} \pi \pm \frac{k^{\prime} \pi}{\sqrt{3}} k \mp k^{\prime}=$ $=k^{\prime \prime} \sqrt{3}$, which is impossible, since t... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,443 |
845. When solving a triangle given two sides and the angle opposite the smaller of them, two values can be obtained for the third side. Could this have been foreseen in advance? | 845. Yes, because when constructing, two different triangles are obtained. | Yes | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,448 |
846. When constructing a triangle given two sides and the angle opposite one of them, there can be two solutions, one solution (two coincident solutions), or no solution at all. On the other hand, a quadratic equation can have either two real roots, one (two equal roots), or none. Is this analogy coincidental, or does ... | 846. The mentioned construction problem leads to the determination of the points of intersection of a line and a circle; in the language of analytic geometry, it requires solving a system of two equations with two unknowns, one of which is linear (the equation of the line), the other quadratic (the equation of the circ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,449 |
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