problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
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[ [Sphere touching the edges or sides of the pyramid]
The base of the pyramid $S A B C$ is an equilateral triangle $A B C$ with side $2 \sqrt{2}$. Edges $S B$ and $S C$ are equal. A sphere touches the sides of the base, the plane of the face $S B C$, and the edge $S A$. What is the radius of the sphere if $S A=\frac{3... | Let a sphere of radius $R$ touch the side $BC$ of the base at point $D$ (Fig.1). Then $D$ is the only common point of the sphere with the plane of the face $SBC$, i.e., the sphere touches this plane at point $D$. If the sphere touches the edges $AB$ and $AC$ at points $M$ and $N$ respectively, then the center $O$ of th... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
In triangle $A B C$, it is known that $A B=10, B C=24$, and the median $B D$ is 13. The circles inscribed in triangles $A B D$ and $B D C$ touch the median $B D$ at points $M$ and $N$ respectively. Find $M N$. | Let's first prove the following statement. If a circle is inscribed in a triangle $X Y Z$, and $x$ is the distance from vertex $X$ to the point of tangency of the circle with side $X Y$, and $Y Z=a$, then $x=p-a$, where $p$ is the semiperimeter of the triangle.
Let the points of tangency of the inscribed circle with s... | 7 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[
$[$ Inscribed and Circumscribed Circles $]$
In an isosceles triangle, the height is 20, and the base is to the lateral side as $4: 3$. Find the radius of the inscribed circle.
# | Find the ratio in which the bisector of the angle at the base divides the height.
## Solution
Let $C M$ be the height of the given triangle $A B C$, $C M=20$, $A C=B C$, and $O$ be the center of the inscribed circle. Then $O M$ is the radius of this circle.
Since $A O$ is the bisector of angle $C A B$, then $\frac{O... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Properties and characteristics of tangents ] [ Pythagorean Theorem (direct and inverse) ]
Given two parallel lines at a distance of 15 from each other; between them is a point $M$ at a distance of 3 from one of them. A circle is drawn through point $M$, touching both lines. Find the distance between the projections ... | Drop a perpendicular from point $M$ to the radius of the circle, drawn to one of the points of tangency.
## Solution
Let $A$ and $B$ be the projections of point $M$ and the center $O$ of the circle onto one of the lines, with $M A=3$. Then $O B$ is the radius of the circle and $O B=\frac{15}{2}$.
Let $P$ be the proj... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
10,11
In a regular truncated quadrilateral pyramid, the height is 2, and the sides of the bases are 3 and 5. Find the diagonal of the truncated pyramid. | Let's make a section through the opposite lateral edges $A A_{1}$ and $C C_{1}$ of the given truncated pyramid $A B C D A_{1} B_{1} C_{1} D_{1}$ with bases $A B C D$ and $A_{1} B_{1} C_{1} D_{1}\left(A B=5, A_{1} B_{1}=3\right)$. Let $O$ and $O_{1}$ be the centers of the bases $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$, r... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Zhendarov R.G.
Each side of an equilateral triangle is divided into $n$ equal segments, and lines parallel to the sides are drawn through all the division points. This triangle is divided into $n^{2}$ small triangular cells. Triangles located between two adjacent parallel lines form a strip.
a) What is the maximum nu... | a) Examples are shown in the figures.
Estimate. First method. Cut the triangle into four equal triangles using the midlines (left figure). Suppose we managed to mark a certain number of cells... | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
8,9}
The medians of a triangle are 3, 4, and 5. Find the area of the triangle.
# | Prove that the area of the triangle formed by the medians of a given triangle is $\frac{3}{4}$ of the area of the given triangle.
## Solution
Let $B_{1}$ be the midpoint of side $A C$ of triangle $A B C$, and $M$ be the point of intersection of its medians. On the extension of median $B B_{1}$ beyond point $B_{1}$, l... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Given a circle with center $O$ and radius 1. From point $A$, tangents $A B$ and $A C$ are drawn to the circle. Point $M$, lying on the circle, is such that the quadrilaterals $O B M C$ and $A B M C$ have equal areas. Find $M A$. | Note that point $M$ lies on the smaller arc $BC$ (otherwise, $OBMC$ is either inside $ABMC$ or is not a quadrilateral at all). Then
$S_{OBMC} - S_{ABMC} = S_{OBC} + 2S_{MBC} - S_{ABC}$. Therefore, the geometric locus of points for which $S_{OBMC} = S_{ABMC}$ is the perpendicular bisector of segment $OA$. Hence, $AM = ... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
The distance from a fixed point $P$ on the plane to two vertices $A, B$ of an equilateral triangle $A B C$ are $A P=2 ; B P=3$. Determine the maximum value that the segment $P C$ can have.
# | Let $A, B, C$ and $P$ be points on a plane such that $AB = BC = CA$, $AP = 2$, and $BP = 3$. Draw a ray $BM$ from point $B$ such that $\angle CBM = \angle ABP$, and mark a segment $BP' = PB$ on this ray. From the equality of angles: $\angle CBM = \angle ABP$, it follows that $\angle PBP' = \angle ABC = 60^\circ$, and t... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
What is the smallest number of non-overlapping tetrahedra into which a cube can be divided?
# | Answer: 5. If a tetrahedron $A^{\prime} B C^{\prime} D$ is cut out from the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, the remaining part of the cube splits into 4 tetrahedra, i.e., the cube can be cut into 5 tetrahedra. We will prove that it is impossible to cut the cube into fewer than 5 tetrahedra. ... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
8,9
The volume of the pyramid $A B C D$ is 5. A plane is drawn through the midpoints of the edges $A D$ and $B C$, intersecting the edge $C D$ at point $M$. In this case, $D M: M C=2: 3$. Find the area of the section of the pyramid by the specified plane, if the distance from it to the vertex $A$ is 1. | Since the secant plane passes through the midpoints of opposite edges of the tetrahedron, it divides its volume in half. Let the secant plane intersect edge $AB$ at point $K$, and let $P$ and $Q$ be the midpoints of edges $AD$ and $BC$. Then the volume of the polyhedron $PMQKAC$ is $\frac{5}{2}$. On the other hand, it ... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
$[$ [extension of a tetrahedron to a parallelepiped]
Segment $A B(A B=1)$, being a chord of a sphere with radius 1, is positioned at an angle of $60^{\circ}$ to the diameter $C D$ of this sphere. The distance from the end $C$ of the diameter to the nearest end $A$ of the chord $A B$ is $\sqrt{2}$. Find $B D$.
# | Complete the tetrahedron $ABCD$ to a parallelepiped $AKBLNDMC (AN\|KD\|BM\|LC)$ by drawing pairs of parallel planes through its opposite edges. Let $O$ and $Q$ be the centers of the faces $NDMC$ and $AKBL$ respectively. Then $O$ is the center of a sphere, $KL=CD=2, OQ \perp AB, OA=OB=1$ as radii of the sphere, $KL \| C... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ $\underline{\text { Cone }}$
Through the edge $B C$ of the triangular pyramid $P A B C$ and the point $M$, the midpoint of the edge $P A$, a section $B C M$ is drawn. The vertex of the cone coincides with the vertex $P$ of the pyramid, and the circle of the base is inscribed in the triangle $B C M$, touching the sid... | Let $E$ and $D$ be the points of intersection of the medians of triangles $A P B$ and $A P C$, $K$ be the midpoint of $B C$, $O$ be the center of the circle inscribed in triangle $B M C$, and $r$ be the radius of the base of the cone, with $2 r$ being its height. Then
$$
P D=P E=P K=\sqrt{r^{2}+4 r^{2}}=r \sqrt{5}
$$
... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Distance between skew lines]
On the line $l$ in space, points $A, B$ and $C$ are sequentially located, with $A B=18$ and $B C=14$. Find the distance between the lines $l$ and $m$, if the distances from points $A, B$ and $C$ to the line $m$ are 12, 15 and 20, respectively.
# | Let $A 1, B 1$ and $C 1$ be the feet of the perpendiculars dropped from points $A, B$ and $C$ to the line $m$ respectively (Fig.1). According to the problem,
$$
A A 1=12, B B 1=15, C C 1=20 .
$$
Assume that lines $l$ and $m$ lie in the same plane. It is clear that they cannot be parallel. If the intersection point of... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
S. Rukhin
A closed eight-segment broken line is drawn on the surface of a cube, with its vertices coinciding with the vertices of the cube.
What is the minimum number of segments of this broken line that can coincide with the edges of the cube?
# | The segments of the broken line are either edges of the cube or diagonals of its faces. To do this, let's color the vertices of the cube in two colors in a checkerboard pattern (see fig.). Note that an edge of the cube connects vertices of different colors, while a diagonal connects vertices of the same color. For the ... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
$3+$ [ The transfer helps solve the task_ ]
On the side AB of the square ABCD, an equilateral triangle AKB was constructed (outside). Find the radius of the circle circumscribed around triangle CKD, if $\mathrm{AB}=1$.
# | Construct an equilateral triangle on side $\mathrm{CD}$ inside the square $\mathrm{ABCD}$.
## Solution
Construct an equilateral triangle CDM on side CD inside the square ABCD. Then CM=1 and DM=1. Moreover, triangle ABK is obtained from triangle CDM by a parallel translation by vector CB, the length of which is 1. The... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[
Diameter, main properties
On the leg $B C$ of the right triangle $A B C$ as a diameter, a circle is constructed, intersecting the hypotenuse at point $D$ such that $A D: B D=1: 3$. The height dropped from the vertex $C$ of the right angle to the hypotenuse is 3. Find the leg $B C$.
# | $C D$ - the height of triangle $A C D$.
## Solution
Since angle $B D C$ is inscribed in the given circle and rests on its diameter $B C$, then $\angle B D C=90^{\circ}$. Therefore, $C D$ is the height of triangle $A B C$.
Let $A D=x, B D=3 x$. Since $C D^{2}=A D \cdot D B$, then $3 x^{2}=9$. From this, we find that ... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
$\left[\begin{array}{l}{[\text { Inscribed quadrilaterals }]} \\ \text { Area of a trapezoid }]\end{array}\right.$
The midline of an isosceles trapezoid is 5. It is known that a circle can be inscribed in the trapezoid.
The midline of the trapezoid divides it into two parts, the ratio of the areas of which is $\frac{... | Solve the system of equations with two unknowns - the lengths of the bases of the trapezoid.
## Solution
Let $x$ and $y$ be the bases of the trapezoid. Then
$$
\left\{\begin{array}{l}
x+y=10 \\
\frac{x+5}{y+5}=\frac{7}{13}
\end{array}\right.
$$
From this, we find that $x=2$ and $y=8$. Therefore, the lateral side of... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
10,11
In the pyramid $ABCD$, the edges are given: $AB=7, BC=8, CD=4$. Find the edge $DA$, if it is known that the lines $AC$ and $BD$ are perpendicular. | Let's draw the height $B K$ of triangle $A B C$. The line $A C$ is perpendicular to two intersecting lines $B K$ and $B D$ in the plane $B D K$. Therefore, the line $A C$ is perpendicular to each line in this plane, particularly to the line $D K$. Hence, $D K$ is the height of triangle $A D C$. We can apply the Pythago... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[Sum of interior and exterior angles of a polygon]
Find the number $n$ of sides of a convex $n$-gon if each of its interior angles is not less than $143^{\circ}$ and not more than $146^{\circ}$.
# | The sum of the interior angles of a convex $n$-gon is $180^{\circ}(n-2)$, therefore
from which we find that $9 \frac{27}{37} \leqslant n \leqslant 10 \frac{20}{27}$. This inequality is satisfied by the unique natural number 10.
## Answer
10.00 | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Given points $A, B, C$, and $D$ such that segments $A C$ and $B D$ intersect at point $E$. Segment $A E$ is 1 cm shorter than segment $A B$, $A E = D C$, $A D = B E$,
$\angle A D C = \angle D E C$. Find the length of $E C$. | $\angle B E A=\angle D E C=\angle A D C$, therefore triangles $A D C$ and $B E A$ are equal by two sides and the included angle. Thus, $A B=A C$. Therefore,
$E C=A C-A E=A B-A E=1$.
## Answer
1 cm. | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[
Find the height of a right triangle dropped to the hypotenuse, given that the base of this height divides the hypotenuse into segments of 1 and 4.
# | Let $A B C$ be a right triangle with hypotenuse $A B$, and $C H$ be its altitude, $B H=1, A H=4$.
According to the altitude theorem of a right triangle drawn from the vertex of the right angle,
$$
C H^2=B H \cdot A H=1 \cdot 4=4
$$
Therefore, $C H=2$.
## Answer
2. | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[Ratio of areas of triangles with a common angle]
On the sides $AB$ and $AC$ of triangle $ABC$, whose area is 50, points $M$ and $K$ are taken such that $AM: MB=1: 5$, and $AK: KC=3: 2$. Find the area of triangle $AMK$.
# | $$
S_{\triangle A M K}=\frac{A M}{A B} \cdot \frac{A K}{A C} S_{\triangle A B C}=\frac{1}{6} \cdot \frac{3}{5} \cdot 50=5
$$
## Answer
5. | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
$[$ Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part
A point $M$ is connected to a circle by two lines. One of them touches the circle at point $A$, while the other intersects the circle at points $B$ and $C$, with $B C=7$ and $B M=9$. Find $A M$.
# | Use the Tangent-Secant Theorem. Consider two cases.
## Solution
Let point $B$ lie between points $M$ and $C$. According to the Tangent-Secant Theorem,
$$
A M^{2}=M C \cdot M B=(9+7) 9=16 \cdot 9=12^{2}
$$
Therefore, $A M=12$.
If point $C$ lies between points $B$ and $M$, then similarly we get $A M=3 \sqrt{2}$.
##... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Rectangles and Squares. Properties and Characteristics ]
In a plane, there is a square with vertices $A, B, C, D$ in sequence and a point $O$ outside the square. It is known that $A O=O B=5$ and $O D=\sqrt{13}$. Find the area of the square. | Compose an equation with respect to the side of the square. One of the solutions of this equation contradicts the condition of the problem.
## Solution
Let $x$ be the side of the square. Let $P$ be the projection of point $O$ onto the line $A D$. From the condition of the problem, it follows that points $O$ and $A$ l... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
$[$ Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part
From a point $M$, located outside the circle at a distance of $\sqrt{7}$ from the center, a secant is drawn, the internal part of which is half the external part and equal to the radius of the circle.
Find the... | Apply the tangent-secant theorem.
## Solution
Let $r$ be the radius of the circle. According to the tangent-secant theorem, $2 r \cdot 3 r = (\sqrt{7} - r)(\sqrt{7} + r) = 7 - r^2$, from which we get $r = 1$.
## Answer
1. Send a comment
Point $B$ lies on segment $AC$, which is equal to 5. Find the distance between... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Measurement of segment lengths and angle measures. Adjacent angles.]
On a straight line, points $A, B, C$ and $D$ are marked sequentially, and $A B=B C=C D=6$.
Find the distance between the midpoints of segments $A B$ and $C D$.
# | Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$ respectively. Then $MN = MB + BC + CN = 3 + 6 + 3 = 12$.
## Answer
12. | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[Measuring lengths of segments and measures of angles. Adjacent angles.] [Arithmetic operations. Numerical identities]
On a ruler 9 cm long there are no divisions.
Mark three intermediate divisions on it so that it can measure distances from 1 to 9 cm with an accuracy of 1 cm
# | It is sufficient to make intermediate divisions at the points 1 cm, 4 cm, 7 cm. Then we have 4 segments: 1 cm, 3 cm, 3 cm, and 2 cm. It is not difficult to verify that with such a ruler, one can measure any whole distance from 1 to 9 cm.
Send a comment | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Inscribed and Circumscribed Circles ]
The hypotenuse of a right triangle is 4 m. Find the radius of the circumscribed circle.
# | The center of the circle circumscribed around a right triangle coincides with the midpoint of the hypotenuse.
## Solution
The center of the circle circumscribed around a right triangle coincides with the midpoint of the hypotenuse. Therefore, the radius of the circle is half the hypotenuse, i.e., 2 m.
.]
Two people had two square cakes. Each made 2 straight cuts from edge to edge on their cake. As a result, one ended up with three pieces, and the other with four. How could this happen? | Note: Cuts can intersect.
## Solution
This could happen if in the first case the cuts did not intersect each other, while in the second case they did. For example, if in the first case the cuts were parallel to each other, and in the second case they were perpendicular.
## Answer
In the first case, the cuts were pa... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
[ Angles between angle bisectors $\quad]$
In triangle $ABC$, the angle bisectors of the angles at vertices $A$ and $C$ intersect at point $D$. Find the radius of the circumcircle of triangle $ABC$, if the radius of the circumcircle of triangle $ADC$ with center at point $O$ is $R=6$, and $\angle ACO=30^{\circ}$. | Prove that $\angle A D C=90^{\circ}+\frac{1}{2} \angle B$.
## Solution
Since $\angle B A C+\angle A C B=180^{\circ}-\angle B$, then $\angle D A C+\angle A C D=90^{\circ}-\frac{1}{2} \angle B$. Therefore,
$$
\angle A D C=180^{\circ}-\left(90^{\circ}-\frac{1}{2} \angle B=90^{\circ}+\frac{1}{2} \angle B>90^{\circ}-\ang... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Triangle Inequality
Point $C$ divides the chord $A B$ of a circle with radius 6 into segments $A C=4$ and $C B=5$. Find the minimum distance from point $C$ to the points on the circle.
# | Let $O$ be the center of the circle. Extend the segment $OC$ beyond point $C$ to intersect the circle at point $M$, and using the triangle inequality, prove that the length of the segment $CM$ is the smallest of the distances from point $C$ to the points on the circle.
## Solution
Let $O$ be the center of the circle.... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[Mutual Position of Two Circles]
What is the mutual position of two circles if:
a) the distance between the centers is 10, and the radii are 8 and 2;
b) the distance between the centers is 4, and the radii are 11 and 17;
c) the distance between the centers is 12, and the radii are 5 and 3? | Let $O_{1}$ and $O_{2}$ be the centers of the circles, $R$ and $r$ be their radii, and $R \geqslant r$.
If $R+r=O_{1} O_{2}$, then the circles touch.
If $R+r<O_{1} O_{2}$, then one circle is located outside the other.
If $O_{1} O_{2}<R-r$, then one circle is located inside the other.
## Answer
a) touch; b) one ins... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Bogov and I.I.
The distance between two cells on an infinite chessboard is defined as the minimum number of moves in the path of a king between these cells. On the board, three cells are marked, the pairwise distances between which are 100. How many cells exist such that the distances from them to all three marked cel... | Consider two arbitrary cells $A$ and $B$. Let the difference in the abscissas of their centers be $x \geq 0$, and the difference in the ordinates be $y \geq 0$. Then the distance $\rho(A, B)$ between these cells is $\max \{x, y\}$.
Let cells $A, B, C$ be marked. Then for each pair of cells, there exists a coordinate i... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5.
Segments connecting an inner point of a convex non-equilateral $n$-gon with its vertices divide the $n$-gon into $n$ equal triangles.
For what smallest $n$ is this possible? | Let's prove that the specified situation is impossible for $n=3,4$.
The first method. For $n=3$, the angles of the triangles in the partition that meet at an internal point are equal, since the sum of any two different angles in them is less than $180^{\circ}$. But then the sides opposite to these angles, which are si... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
The diagonals of the trapezoid are 3 and 5, and the segment connecting the midpoints of the bases is 2. Find the area of the trapezoid.
# | Through the vertex of the smaller base of the trapezoid, draw a line parallel to the diagonal.
## Solution
Let $M$ and $K$ be the midpoints of the bases $BC$ and $AD$ of trapezoid $ABCD$. Through vertex $C$ of the smaller base $BC (AC=3, BD=5)$, draw a line parallel to diagonal $BD$, intersecting line $AD$ at point $... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
A circle with radius $\frac{2}{\sqrt{3}}$ is inscribed in an isosceles trapezoid. The angle between the diagonals of the trapezoid, subtending the larger base, is $2 \operatorname{arctg} \frac{2}{\sqrt{3}}$. Find the segment connecting the points of tangency of the circle with the larger base of the trapezoid and one o... | Denote the halves of the bases of the trapezoid as $x$ and $y$; find $x, y$ and the angle between the lateral side and the larger base.
## Solution
Let $Q$ be the point of intersection of the diagonals $A C$ and $B D$ of trapezoid $A B C D$; $N, M$, and $K$ be the points of tangency of the inscribed circle with the s... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ [tangents drawn from one point] [ Midline of a triangle $\quad$]
A circle of radius 1 is inscribed in triangle $ABC$, where $\cos \angle B=0.8$. This circle touches the midline of triangle $ABC$, parallel to side $AC$. Find the side $AC$. | The distance from the vertex of a triangle to the nearest point of tangency with the inscribed circle is equal to the difference between the semiperimeter and the opposite side.
## Solution
Let $O$ be the center of the inscribed circle, and $K$ be its point of tangency with side $AB$. Then
$$
\operatorname{tg} \frac... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Measuring lengths of segments and measures of angles. Adjacent angles.] Unusual constructions (miscellaneous). $\quad]$
On a wooden ruler, three marks are made: 0, 7, and 11 centimeters. How can you measure with it a segment equal to: a) 8 cm $;$ b) 5 cm
# | a) Using divisions of 7 and 11, you can measure 4 cm. Doing this twice, you will get a segment of 8 cm.
b) First method. Knowing how to measure 8 and 7, you can measure 1 cm. Doing this 5 times, you will get 5 cm.
Second method. $5=3 \cdot 11-4 \cdot 7$, so it is enough to measure 3 times 11 cm and then 4 times 7 in ... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[Measuring lengths of segments and measures of angles. Adjacent angles.] Extreme properties (miscellaneous).
In a village, along a straight road, there are two houses $A$ and $B$ located 50 meters apart from each other.
At what point on the road should a well be dug so that the sum of the distances from the well to t... | Any point $X$ on the road segment between $A$ and $B$ is equally good, since the segments $A X$ and $B X$ together total 50 m. If a point is taken outside this segment, then the sum of the distances from it to points $A$ and $B$ will be greater than $50 \mathrm{M}$.
## Answer
At any point on the segment $A B$. | 0 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
In a square with an area of 18, a rectangle is inscribed such that one vertex of the rectangle lies on each side of the square. The sides of the rectangle are in the ratio $1: 2$.
Find the area of the rectangle. | Let the vertices $K, L, M$ and $N$ of the rectangle $K L M N$ be located on the sides $A B, B C, C D$ and $A D$ of the square $A B C D$, respectively, such that $K N=2 K L$. Let $K L=x, \angle A K N=\alpha$. Then
$M N=x, L M=K N=2 x, \angle C M L=\angle B L K=\angle A K N=\alpha, A K=2 x \cos \alpha, B K=x \sin \alpha... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3 ( A square with an area of 24 has a rectangle inscribed in it such that one vertex of the rectangle lies on each side of the square. The sides of the rectangle are in the ratio $1: 3$.
Find the area of the rectangle.
# | ## Answer
9.
## Problem
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 9 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Shapovalov A.V.
Each face of a cube $6 \times 6 \times 6$ is divided into $1 \times 1$ cells. The cube is covered with $2 \times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by the same number of squares. What is the maximum value that this identical number c... | Evaluation. A cell in the corner of a face can be covered in three ways (entirely within the face, with a fold over one edge of the corner, with a fold over the other edge of the corner). This means that each cell is covered by no more than three squares.
Example. Consider the usual covering of a cube with squares, wh... | 3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
[ Triangle Inequality (other).]
At vertex $A$ of a unit square $A B C D$, there is an ant. It needs to reach point $C$, where the entrance to the ant hill is located. Points $A$ and $C$ are separated by a vertical wall, which has the shape of an isosceles right triangle with hypotenuse $B D$. Find the length of the sh... | Let $K$ be the vertex of the right angle of triangle $B K D$ (a vertical wall). The path of the ant consists of four segments: $A M, M P, P M$, and $M C$, where point $M$ lies on line $B D$, and point $P$ lies on segment $D K$ or $B K$. Clearly, $A M = M C$, so it is sufficient to specify points $M$ and $P$ such that t... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
C
[Median line of a triangle
[ Trigonometric ratios in a right-angled triangle ]
In triangle $ABC$, the median $BM$ and the altitude $AH$ are drawn. It is known that $BM = AH$. Find the angle $MBC$.
# | Drop a perpendicular from point $M$ to line $B C$.
## Solution
Drop the perpendicular $M K$ to line $B C$. Then $M K$ is the midline of triangle $A H C$. Therefore, $M K = 1/2 A H = 1/2 B M$, which means $\angle M B K = 30^{\circ}$. Consequently, $\angle M B C = 30^{\circ}$ or $150^{\circ}$.
 consecutive natural numbers. What is the greatest possible value of the area of tr... | Author: Shestakov S.A.
Let the areas of the triangles be $n, n+1, n+2$, and $n+3$. Then $S_{A B C D}=4 n+6$. $E F$ is the midline of triangle $B C D$, so $S_{B C D}=4 S_{E C F} \geq 4 n$. Therefore, $S_{A B D}=S_{A B C D}-S_{B C D} \leq 6$.
We will show that this area can equal 6. Consider an isosceles trapezoid $A B... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Sergeev I.N.
At the base $A_{1} A_{2} \ldots A_{n}$ of the pyramid $S A_{1} A_{2} \ldots A_{n}$ lies a point $O$, such that $S A_{1}=S A_{2}=\ldots=S A_{n}$ and $\angle S A_{1} O=\angle S A_{2} O=$ $\ldots=\angle S A_{n} O$.
For what least value of $n$ does it follow from this that $S O$ is the height of the pyramid? | By the Law of Sines for triangles $S A_{k} O (k=1,2, \ldots, n)$, we have $\sin \angle S O A_{k} = \frac{S A_{k}}{S O} \cdot \sin \angle S A_{k} O$.
Since the right-hand side of this equality does not depend on the choice of $k=1,2, \ldots, n$, the value of $\sin \angle S O A_{k}$ also does not depend on this choice. ... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Parallelism of lines and planes $]$ [ Systems of segments, lines, and circles ]
In space, there are $n$ segments, no three of which are parallel to the same plane. For any two segments, the line connecting their midpoints is perpendicular to both segments. What is the largest $n$ for which this is possible? | Answer: $n=2$.
Suppose that $n \geq 3$. Take three segments $a, b$, and $c$ and draw three lines connecting the midpoints of each pair. Each of the segments $a, b, c$ is perpendicular to the two lines emanating from its midpoint, and therefore it is perpendicular to the plane passing through all three drawn lines. Thu... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Shapovalov A.V.
For what largest $n$ can one choose $n$ points on the surface of a cube so that not all of them lie in the same face of the cube and are vertices of a regular (planar) $n$-gon. | On one face of a cube, no more than two vertices of a polygon can lie (otherwise, the entire polygon would lie in this face). Therefore, the polygon cannot have more than 12 vertices.
To obtain a 12-sided polygon, recall that there exists a section of the cube by a plane that forms a regular hexagon. This section pass... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Inscribed Quadrilaterals (Miscellaneous) ]
Quadrilateral $ABCD$ is inscribed in a circle. Point $X$ lies on its side $AD$, such that $BX \parallel CD$ and $CX \parallel BA$. Find $BC$, if $AX = 3/2$ and $DX = 6$.
# | Triangles $A B C, B X C$ and $X C D$ are similar.
## Solution
Let the angles at vertices $A, B$, and $X$ of triangle $A B X$ be denoted as $\alpha, \beta$, and $\gamma$ respectively. Since $C X \| B A$ and $B X \| C D$, then
$\angle D C X = \angle B X C = \angle A B X = \beta, \quad \angle C D X = \angle B X A = \ga... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[Sum of angles in a triangle. Theorem about the exterior angle. ] [ Lengths of sides, heights, medians, and angle bisectors ]
In triangle $A B C \angle A=40^{\circ}, \angle B=20^{\circ}$, and $A B-B C=4$. Find the length of the angle bisector of angle $C$. | Let's set aside segment $BD$ on side $AB$, equal to $BC$. Then triangle $BCD$ is isosceles with the angle at the vertex $20^{\circ}$, so the angles at the base are $80^{\circ}$ (see the figure). Let $CE$ be the bisector of angle $C$. Then $\angle BCE = 60^{\circ}$, so $\angle AEC = 20^{\circ} + 60^{\circ} = 80^{\circ}$... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Examples and counterexamples. Constructions ]
Six identical parallelograms of area 1 were used to cover a cube with an edge of 1. Can we assert that all parallelograms are squares? Can we assert that all of them are rectangles? | Examples of such cube nets are shown in Figure 1.
Fig. 1
## [Elementary (basic) constructions with a compass and straightedge] Problem 107725 Topics: [ Construction of triangles by various e... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Two tangents drawn from one point $]$
Given a circle of radius 1. From an external point $M$, two perpendicular tangents $MA$ and $MB$ are drawn to the circle. Between the points of tangency $A$ and $B$ on the smaller arc $AB$, a point $C$ is taken, and a third tangent $KL$ is drawn through it, forming a triangle $K... | Tangents drawn from a single point to a circle are equal to each other.
## Solution
Since $KA = KC$ and $BL = LC$, then
$$
\begin{gathered}
ML + LK + KM = ML + (LC + CK) + KM = \\
= (ML + LC) + (CK + KM) = (ML + LB) + (AK + KM) = \\
= MB + AM = 1 + 1 = 2.
\end{gathered}
$$
, and $P$ is the projection of point $O_{1}$ onto the radius $O_{2}C$ of the smaller circle.
If $AC = x$, then $BC = 3x$. Drop a perpendicular $... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Two circles with radii 5 and 4 touch each other externally. A line tangent to the smaller circle at point $A$ intersects the larger circle at points $B$ and $C$, and
$A B = B C$. Find $A C$. | Let a circle of radius 4 with center $O_{1}$ and a circle of radius 5 with center $O_{2}$ touch each other externally at point $D$ (see the left figure). Then
$O_{1} O_{2}=O_{1} D+O_{2} D=9$
Drop perpendiculars $O_{2} M$ to the chord $B C$ and $O_{1} F$ to the line $O_{2} M$. Then $A O_{1} F M$ is a rectangle, so $M ... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
The height of a right triangle, dropped to the hypotenuse, is 1, and one of the acute angles is $15^{\circ}$.
Find the hypotenuse. | Draw the median from the vertex of the right angle.
## Solution
Let $C H$ be the height of the right triangle $A B C$, drawn from the vertex of the right angle $C, \angle A=15^{\circ}$. Draw the median $C M$. The angle $C M H$ as the external angle of the isosceles triangle $A M C$ is $30^{\circ}$. From the right tri... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Each face of the cube was divided into four equal squares and these squares were painted in three colors so that squares sharing a side were painted in different colors. Prove that 8 squares were painted in each color. | Three squares meeting at a vertex of a cube are painted with different colors. Therefore, each color is used 8 times. | 8 | Combinatorics | proof | Yes | Yes | olympiads | false |
| ethics. Mental arithmetic, etc. |
| :---: |
On a straight line, ten points were placed at equal intervals, and they occupied a segment of length $a$. On another straight line, one hundred points were placed at the same intervals, and they occupied a segment of length $b$. How many times is $b$ greater than $a$? | On a segment of length $a$, 9 intervals fit, and on a segment of length $b-99$. Therefore, $b=11a$.
Author: Folkpor
In a hexagon, five angles are $90^{\circ}$, and one angle is $-270^{\circ}... | 11 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Folklore
What is the smallest value that the perimeter of a scalene triangle with integer side lengths can take?
# | Let $a, b$ and $c$ be the integer lengths of the sides of a triangle and $a>b>c$. According to the triangle inequality, $c>a-b$. Since $a$ and $b$ are different natural numbers, $c \geq 2$, thus $b \geq 3$ and $a \geq 4$. Therefore, $a+b+c \geq 9$. Equality is achieved for a triangle with sides 2, 3, and 4.
## Otвет
... | 9 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
One side of the rectangle was increased by 3 times, and the other side was reduced by 2 times, resulting in a square.
What is the side of the square if the area of the rectangle is 54 m²
# | Let's reduce one side of the given rectangle by half. Then the area of the resulting rectangle will be $27 \mathrm{~m}^{2}$. Next, we will triple the other side. The area of the resulting figure will become $27 \cdot 3=81$ ( $\mathrm{m}^{2}$ ). Since, by the condition, a square is formed, its side is 9 m.
## Answer
9... | 9 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
What is the minimum number of three-cell corners that can be placed in an $8 \times 8$ square so that no more such corners can be placed in this square? | In each $2 \times 2$ square, at least two cells must be covered by corners (otherwise, another corner can fit into such a square).
A $8 \times 8$ square can be divided into 16 squares of size $2 \times 2$ each, meaning that at least 32 cells must be covered by corners, which requires no fewer than 11 corners.
 = a, \quad C N = 2 B C \cos 20^{\circ} = 2 b \cos 20^{\circ... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Touching Circles $\quad$ [
Two circles touch each other externally at point $A$, and a third circle at points $B$ and $C$. The extension of chord $A B$ of the first circle intersects the second circle at point $D$, the extension of chord $A C$ intersects the first circle at point $E$, and the extensions of chords $B... | Let $S 1, S 2$ and $S 3$ be the first, second, and third circles, respectively. Draw the common tangents $l_{a}, l_{b}, l_{c}$ through points $A, B$ and $C$ to the circles $S 1$ and $S 2, S 1$ and $S 3, S 2$ and $S 3$ respectively. Then the tangents $l_{a}$ and $l_{b}$ form equal angles with the chord $A B$. Denote the... | 9 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4 [ Sphere inscribed in a dihedral angle ]
Point $O$ is located in the section $A A^{\prime} C^{\prime} C$ of a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with dimensions $2 \times 6 \times 9$ such that $\angle O A B + \angle O A D + \angle O A A^{\prime} = 180^{\circ}$. A sphere ... | Let $\angle O A B=\alpha, \angle O A D=\beta, \angle O A A^{\prime}=\gamma$. According to the problem, $\alpha+\beta+\gamma=180^{\circ}$.
We mark a segment $A E=1$ on the ray $A O$ (Fig.1). Then the projections of point $E$ onto the lines $A B, A D$, and $A A^{\prime}$ are $\cos \alpha, \cos \beta$, and $\cos \gamma$ ... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Rectangular parallelepipeds ] [ Sphere inscribed in a dihedral angle ]
Point $O$ is located in the section $A C C^{\prime} A^{\prime}$ of a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with dimensions $2 \times 3 \times 6$ such that $\angle O C B + \angle O C D + \angle O C C^{\pr... | Let $\angle O C B=\alpha, \angle O C D=\beta, \angle O C C^{\prime}=\gamma$. According to the problem, $\alpha+\beta+\gamma=180^{\circ}$.
We mark a segment $C E=1$ on the ray $C O$. Then the projections of point $E$ on the lines $C B, C D$, and $C C^{\prime}$ are $\cos \alpha, \cos \beta$, and $\cos \gamma$ respective... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Kalinin A.
Anya and Borya, whose speeds are constant but not necessarily the same, simultaneously set out from villages A and B towards each other. If Anya had set out 30 minutes earlier, they would have met 2 km closer to village B. If Borya had set out 30 minutes earlier, the meeting would have taken place closer to... | Let Kolya (half an hour after Borya) and Tolya (half an hour before Borya) also leave from B at the same speed, and from A - Tanya (half an hour before Anya).
First method. Anya meets Tolya, Borya, and Kolya sequentially at equal time intervals. Therefore, the distances between the adjacent meeting points are also equ... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
|
At a familiar factory, they cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius is 10 mm. Engineer Sidorov believes that a stack of 100 disks wi... | Given $\mathrm{E} R=0.5 \mathrm{~m}, \mathrm{D} R=10^{-4} \mathrm{M}^{2}$. Let's find the expected value of the area of one disk:
$\mathrm{E} S=\mathrm{E}\left(\pi R^{2}\right)=\pi \mathrm{E} R^{2}=\pi\left(\mathrm{D} R+\mathrm{E}^{2} R\right)=\pi\left(10^{-4}+0.25\right)=0.2501 \pi$.
Therefore, the expected value of... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
10,11 | |
A right circular cone with base radius $R$ and height $H=3 R \sqrt{7}$ is laid on its side on a plane and rolled so that its vertex remains stationary. How many revolutions will the base make before the cone returns to its original position? | The base circle rolls along the circumference of a circle of radius $L$, where $L$ is the slant height of the cone. Therefore, the number of revolutions is $\frac{2 \pi L}{2 \pi R}=\frac{\sqrt{H^{2}+R^{2}}}{R}=8$.
## Answer
8 revolutions. | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
The altitudes $A D$ and $B E$ of an acute-angled triangle $A B C$ intersect at point $H$. The circumcircle of triangle $A B H$ intersects sides $A C$ and $B C$ at points $F$ and $G$ respectively. Find $F G$, if $D E=5$ cm. | Let $\angle F A H = \angle H B F = \alpha$. Right triangles $A D C$ and $E C B$ share the common angle $C$, so $\angle E B C = \alpha$.
Thus, $B E$ is both the altitude and the angle bisector of triangle $F B C$, which means this triangle is isosceles and $B E$ is its median, i.e.,
$F E = E C$. Similarly, $C D = D G$... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
On a circle of radius 5, circumscribed around an equilateral triangle, a point $D$ is taken. It is known that the distance from point $D$ to one of the vertices of the triangle is 9. Find the sum of the distances from point $D$ to the other two vertices of the triangle. | Let the distance from vertex $A$ of triangle $A B C$ to point $D$ be 9. Prove that point $D$ lies on the smaller arc $B C$ and $B D + C D = A D$.
## Solution
Let the distance from vertex $A$ of triangle $A B C$ to point $D$ be 9. Note that $B C = 5 \sqrt{3}$. Since
$A D = 9 > B C$, point $D$ lies on the smaller arc ... | 9 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
8,9 |
In an acute triangle $A B C$, altitudes $A L$ and $B M$ are drawn. Then, the line $L M$ is extended to intersect the extension of side $A B$.
What is the maximum number of pairs of similar triangles that can be counted on this diagram, if no pairs of congruent triangles are formed on it? | Points $A, B, M$ and $L$ lie on the same circle.
## Solution
Right triangles $C A L$ and $C B L$ are similar by two angles.
Let $H$ be the orthocenter of triangle $A B C$. Right triangles $A M H$ and $B L H$ are similar by two angles.
Right triangles $A M H$ and $A L C$ are similar by two angles.
Right triangles $... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Dolmatov $C$.
Point $M$ lies on the side $A C$ of an acute-angled triangle $A B C$. Circles are circumscribed around triangles $A B M$ and $C B M$. For what position of point $M$ will the area of the common part of the circles bounded by them be the smallest? | The angles under which the segment $BM$ is seen from the centers of the circles have a constant magnitude.
## Solution
Let $O$ and $O_{1}$ be the centers of the circumcircles of triangles $ABM$ and $CBM$. The common part of the specified circles is the union of two segments with a common chord $BM$.
Since
$$
\angle... | 0 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
In quadrilateral $A B C D$, it is known that $D O=4, B C=5, \angle A B D=45^{\circ}$, where $O$ is the point of intersection of the diagonals. Find $B O$, if the area of quadrilateral $A B C D$ is equal to $\frac{1}{2}(A B \cdot C D+B C \cdot A D)$. | Prove that $ABCD$ is a cyclic quadrilateral with mutually perpendicular diagonals.
## Solution
Let $C_{1}$ be the point symmetric to vertex $C$ with respect to the perpendicular bisector of diagonal $BD$. Then
$$
\begin{gathered}
S_{\triangle \mathrm{ABCD}}=S_{\Delta \mathrm{ABC}_{1} \mathrm{D}}=S_{\Delta \mathrm{AB... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Trigonometric ratios in a right triangle [ Theorem of the length of a tangent and a secant; the product of the entire secant and its external part [ Two tangents drawn from the same point
In parallelogram $K L M N$, side $K L$ is equal to 8. A circle, tangent to sides $N K$ and $N M$, passes through point $L$ and in... | Apply the tangent-secant theorem.
## Solution
Let $A B$ be the hypotenuse of the right triangle $A B C$. Then $A B=2 R$. If $\angle C A B=\boldsymbol{\alpha}$, then
$$
B C=A B \sin \alpha=2 R \sin \alpha .
$$
If $C D$ is the height of the triangle $A B C$, then
$$
C D=B C \cos \angle D C B=B C \cos \alpha=2 R \sin... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Malkin M.i.
On the board, 101 numbers are written: $1^{2}, 2^{2}, \ldots, 101^{2}$. In one operation, it is allowed to erase any two numbers and write down the absolute value of their difference instead.
What is the smallest number that can result from 100 operations? | From four consecutive squares (in three operations), the number 4 can be obtained: $(n+3)^{2}-(n+2)^{2}-((n+$ $\left.1)^{2}-n^{2}\right)=(2 n+5)-(2 n+1)=4$.
We can get 24 such fours from the numbers $6^{2}, 7^{2}, \ldots, 101^{2}$. 20 fours can be turned into zeros by pairwise subtraction. From the numbers $4,9,16,25$... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
8,9 | |
In parallelogram $A B C D$, the diagonals $A C=15, B D=9$ are known. The radius of the circle circumscribed around triangle $A D C$ is 10. Find the radius of the circle circumscribed around triangle $A B D$. | Apply the formula $a=2 R \sin \boldsymbol{\alpha}$.
## Solution
Let $R=10$ be the radius of the circumcircle of triangle $A D C$. Then
$$
\sin \angle A D C=\frac{A C}{2 R}=\frac{15}{20}=\frac{3}{4}
$$
Since $\angle B A D=180^{\circ}-\angle A D C$, then
$$
\sin \angle B A D=\sin \angle A D C=\frac{3}{4}
$$
If $R_{... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Given a triangle with sides $A B=2, B C=3, A C=4$. A circle is inscribed in it, and the point $M$ where the circle touches side $B C$ is connected to point $A$. Circles are inscribed in triangles $A M B$ and $A M C$. Find the distance between the points where these circles touch the line $A M$. | Since $K$ is the point of tangency of the incircle inscribed in triangle $AMB$ with side $AM$, $AK = (AB + AM - BM) / 2$.
Similarly, expressing $AL$ and subtracting the obtained expressions from each other, we get
$KL = (AB - AC - BM + MC) / 2$.
| 3 Topics: | $\left[\begin{array}{lc}\text { [ Cube } & \text { [ } \\... | 0 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Zaslavsky A.A.
Given a circle with center $O$ and a point $P$ not lying on it. Let $X$ be an arbitrary point on the circle, $Y$ the intersection point of the bisector of angle $P O X$ and the perpendicular bisector of segment $P X$. Find the geometric locus of points $Y$. | Let $K, L$ be the projections of point $Y$ onto $O P$ and $O X$. From the definition of point $Y$, it follows that $Y P = Y X$ and $Y K = Y L$. Therefore, triangles $Y K P$ and $Y L X$ are equal, which means
$X L = P K$. Moreover, $O L = O K$. Since the lengths of segments $O P$ and $O X$ are not equal, one of them is... | 0 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Systems of points and segments. Examples and counterexamples ]
$$
\text { [ Combinatorial geometry (miscellaneous). ] }
$$
The grandfather called his grandson to visit him in the village:
- You'll see what an extraordinary garden I've planted! There are four pears growing there, and there are also apple trees, pla... | Various arrangements of apple and pear trees are possible, for example, as shown in the figure on the left. The maximum number of apple trees can be placed if the pears grow densely enough. For instance, if pears are planted in a row every 5 meters, there will be room for 12 apple trees (figure in the center).
.
 | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
How to cut a square into unit squares a) $4 \times 4$ b) $5 \times 5$ with the fewest number of cuts. (Parts can be stacked on top of each other during cutting).
# | a) 4 cuts; b) 6 cuts. | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Geometry (miscellaneous).]
A sphere of radius $\sqrt{5}$ with center at point $O$ touches all sides of triangle $ABC$. The point of tangency $N$ bisects side $AB$. The point of tangency $M$ divides side $AC$ such that $AM=\frac{1}{2} MC$. Find the volume of the pyramid $OABC$, given that $AN=NB=1$. | The radius of the circle inscribed in a triangle is equal to the area of the triangle divided by its semi-perimeter.
## Solution
Let the given sphere touch the side $BC$ of triangle $ABC$ at point $K$. Then
$$
BK = BN = 1, AM = AN = 1, CM = 2 \cdot AM = 2, CK = CM = 2
$$
The section of the sphere by the plane of tr... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Geometry (other) ]
A sphere with radius $3 / 2$ has its center at point $N$. From point $K$, located at a distance of $3 \sqrt{5} / 2$ from the center of the sphere, two lines $K L$ and $K M$ are drawn, touching the sphere at points $L$ and $M$ respectively. Find the volume of the pyramid $K L M N$, given that $M L=... | From the right triangle $K L N$, we find that
$$
K L=\sqrt{K N^{2}-L N^{2}}=\sqrt{45 / 4-9 / 4}=3
$$
Therefore, $K M=K L=3$. In the isosceles triangles $L N M$ and $L K M$, the medians $N P$ and $K P$ are altitudes, so
$$
\begin{gathered}
N P^{2}=\sqrt{N L^{2}-L P^{2}}=\sqrt{9 / 4-1}=\sqrt{5} / 2 \\
K P^{2}=\sqrt{K ... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Plane, divided by lines ]
How many maximum parts can 5 segments divide a plane into?
# | Each subsequent segment intersects with p already drawn segments in no more than p points.
## Solution
Let $\mathrm{n}$ segments have already been drawn, and they divide the plane into K(n) parts. Draw the ( $\mathrm{n}+1$ )-th segment. It intersects with the n already drawn segments in no more than n points, which c... | 7 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Law of Cosines ]
Find the sum of the squares of the distances from a point $M$, taken on the diameter of a certain circle, to the ends of any chord parallel to this diameter, if the radius of the circle is $R$, and the distance from the point $M$ to the center of the circle is $a$. | Apply the Law of Cosines.
## Solution
Let $O$ be the center of the circle. $AB$ is an arbitrary chord parallel to the given diameter. Denote $\angle BOM = \varphi$. Then $\angle AOM = 180^\circ - \varphi$.
By the Law of Cosines in triangles $BOM$ and $AOM$, we find that
\[
\begin{gathered}
BM^2 = a^2 + R^2 - 2aR \c... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ The product of the lengths of the chord segments and the lengths of the secant segments ]
On a line, points $A, B, C$, and $D$ are arranged in the given order. It is known that $B C = 3, A B = 2 \cdot C D$. A circle is drawn through points $A$ and $C$, and another circle is drawn through points $B$ and $D$. Their co... | Apply the theorem of intersecting chords twice.
## Solution
Let $C D=a, A B=2 a$. Denote $B K=x$. Then
$$
K C=3-x, K D=D C+K C=a+3-x, A K=A B+B K=2 a+x
$$
Let $M N$ be the common chord of the specified circles. Then
$$
A K \cdot K C=M K \cdot N K=B K \cdot K D
$$
from which
From this equation, we find that $x=2$... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
A circle of radius $1+\sqrt{2}$ is circumscribed around an isosceles right triangle. Find the radius of the circle that touches the legs of this triangle and internally touches the circle circumscribed around it. | Let $M$ and $N$ be the points of tangency of the desired circle with the legs $A C$ and $B C$ of triangle $A B C$, and $Q$ be the center of this circle. Then the quadrilateral $Q M C N$ is a square.
## Solution
Let $r$ be the radius of the desired circle, $Q$ its center, $M, N$ the points of tangency with the legs $A... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Common tangent to two circles [ Trigonometric ratios in a right triangle ]
On the plane, there are two circles with radii 12 and 7, centered at points $O_{1}$ and $O_{2}$, touching a certain line at points $M_{1}$ and $M_{2}$ and lying on the same side of this line. The ratio of the length of the segment $M_{1} M_{2... | Consider the right triangle $O_{1} P O_{2}$, where $P$ is the projection of point $O_{2}$ onto $O_{1} M_{1}$.
## Solution
Let $P$ be the projection of point $O_{2}$ onto the line $O_{1} M_{1}$. Denote $\angle O_{1} O_{2} P=\alpha$. Then
$$
\begin{gathered}
\cos \alpha=\frac{O_{2} P}{O_{1} O_{2}}=\frac{M_{1} M_{2}}{O... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
$\left[\begin{array}{l}{[\text { Equilateral (regular) triangle }]} \\ {[\quad \text { Area of a circle, sector, and segment }}\end{array}\right]$
In an equilateral triangle $ABC$, a circle is drawn passing through the center of the triangle and touching side $BC$ at its midpoint $D$. A line is drawn from point $A$, t... | Find $\sin \angle D A E$ from the right triangle $E A Q(Q$ - the center of the given circle).
## Solution
Let $O$ be the center of triangle $A B C, Q$ be the center of the given circle. Denote the side of triangle $A B C$ by $a$. Then
$$
O D=\frac{1}{3} A D=\frac{a \sqrt{3}}{6}, O Q=Q D=\frac{a \sqrt{3}}{12}
$$
$$
... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
## In a convex quadrilateral $A B C D$, the length of the segment connecting the midpoints of sides $A B$ and $C D$ is 1. Lines
$B C$ and $A D$ are perpendicular. Find the length of the segment connecting the midpoints of diagonals $A C$ and $B D$. | Prove that the midpoints of sides $AB$ and $CD$ and the midpoints of diagonals $AC$ and $BD$ are the vertices of a rectangle.
## Solution
Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$ of quadrilateral $ABCD$, and $P$ and $Q$ be the midpoints of its diagonals $AC$ and $BD$, respectively. Then $MP$ is the mid... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
8,9 | |
The height $B K$ of the rhombus $A B C D$, dropped to the side $A D$, intersects the diagonal $A C$ at point $M$. Find $M D$, if it is known that $B K=4, A K: K D=1: 2$. | 
## Solution
Since $AC$ is the perpendicular bisector of diagonal $BD$, then $MD = MB$. Since $AM$ is the angle bisector of triangle $BAK$, then
$$
\frac{BM}{MK} = \frac{AB}{AK} = \frac{AD}{AK... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
[ Auxiliary area. The area helps to solve the problem_] Heron's formula
The sides of the triangle are 13, 14, and 15. Find the radius of the circle that has its center on the middle side and touches the other two sides. | Let the center $O$ of the given circle be located on the side $AB$ of triangle $ABC$. The area of triangle $ABC$ is equal to the sum of the areas of triangles $AOC$ and $BOC$.
## Solution
Let $AC=13$, $AB=14$, $BC=15$, $O$ be the center of the given circle (with $O$ on side $AB$), $R$ be its radius, and $P$ and $Q$ b... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
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