id
stringlengths 36
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|---|---|---|---|---|
23025af5-475f-4069-9627-9619cb875bb5
|
<image>A random sample of 100 cars passing through a radar speed detection area was taken to record their speeds (unit: $$km/h$$), and the data was plotted as a frequency distribution histogram as shown in the figure. The speed range of these 100 cars is $$[35,85]$$, and the data is grouped into $$[35,45), [45,55), [55,65), [65,75), [75,85)$$ intervals. Based on this, estimate the average speed of vehicles passing through this area as ______.
|
$$61.5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/23025af5-475f-4069-9627-9619cb875bb5.png"
}
] |
57005835-6ab8-454c-8af2-1610013a79cd
|
<image>As shown in the figure, in $\vartriangle ABC$, the two medians BE and CD intersect at point O. The ratio of the perimeters of $\vartriangle EOD$ and $\vartriangle BOC$ is.
|
1:2
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/57005835-6ab8-454c-8af2-1610013a79cd.png"
}
] |
d22c6047-c23a-4a6f-af64-06764aa54b30
|
<image>As shown in the figure, if ∠BOC:∠AOC=1:2, ∠AOB=63°, then ∠AOC=
|
42°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d22c6047-c23a-4a6f-af64-06764aa54b30.png"
}
] |
ccc35b58-1c10-46e0-8751-dc721d22bf00
|
<image>In the Cartesian coordinate system shown in the figure, point P is a moving point on the line $y=x$, and A(1,0) and B(2,0) are two points on the x-axis. The minimum value of $PA+PB$ is.
|
$\sqrt{5}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ccc35b58-1c10-46e0-8751-dc721d22bf00.png"
}
] |
57119e38-64f7-489e-9693-387787505c51
|
<image>As shown in the figure, in $\vartriangle ABC$, $\text{CO}$ is the median of side $AB$, and ${CG}^\rightharpoonup = 2 {GO}^\rightharpoonup$. If ${BD}^\rightharpoonup // {AG}^\rightharpoonup$ and ${AD}^\rightharpoonup = \lambda {AB}^\rightharpoonup + \frac{2}{7} {AC}^\rightharpoonup (\lambda \in R)$, then the value of $\lambda$ is.
|
$\frac{9}{7}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/57119e38-64f7-489e-9693-387787505c51.png"
}
] |
f4c5cf86-d32a-4b04-8be7-9125526b886e
|
<image>The image shows the price tag of 'Piaoyang' shampoo at 'Le购 Supermarket'. Please fill in the blank with its original price.
|
30
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f4c5cf86-d32a-4b04-8be7-9125526b886e.png"
}
] |
ec15c31e-b384-4eda-8cc6-b3b61be2271c
|
<image>As shown in the figure, the right triangle $Rt\Delta OAB$ is placed in a Cartesian coordinate system, where point $A$ has coordinates $\left( 0, 4 \right)$, and point $B$ has coordinates $\left( 3, 0 \right)$. Point $P$ is the center of the incircle of $Rt\Delta OAB$. The triangle $Rt\Delta OAB$ is rolled without slipping along the positive direction of the x-axis, such that its three sides sequentially coincide with the x-axis. After the first roll, the center of the circle is ${{P}_{1}}$, after the second roll, it is ${{P}_{2}}$, and so on. Following this pattern, what are the coordinates of the center ${{P}_{2020}}$ of the incircle of $Rt\Delta OAB$ after the 2020th roll?
|
(8081, 1)
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ec15c31e-b384-4eda-8cc6-b3b61be2271c.png"
}
] |
269fb3e8-fdd7-47c5-9f55-8da76a4028dd
|
<image>As shown in the figure, the perimeter of ABCD is 20 cm, AC intersects BD at point O, OE is perpendicular to AC and intersects AD at E. The perimeter of triangle CDE is cm.
|
10
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/269fb3e8-fdd7-47c5-9f55-8da76a4028dd.png"
}
] |
352da0ac-8678-42e9-9797-e541227c0e68
|
<image>In triangle ABC, ∠ABC = $\frac{\pi}{3}$, side BC lies in plane α, and vertex A is outside plane α. The angle between line AB and plane α is θ. If the dihedral angle between plane ABC and plane α is $\frac{\pi}{3}$, then sinθ =.
|
$\frac{3}{4}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/352da0ac-8678-42e9-9797-e541227c0e68.png"
}
] |
3673b05d-9f7e-4781-818a-bdecb9349006
|
<image>As shown in the figure, the cross-section of a part is a hexagon. The sum of the interior angles of this hexagon is ___.
|
$$720^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3673b05d-9f7e-4781-818a-bdecb9349006.png"
}
] |
63081215-f6e6-4479-b3d4-06265199d54e
|
<image>Given y = , then the arithmetic square root of xy is ______.
|
6
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/63081215-f6e6-4479-b3d4-06265199d54e.png"
}
] |
218f8e17-8e4b-414b-8dda-280182b35999
|
<image>For vectors $$\overrightarrow{PA_{i}}(i=1,2,\cdots ,n)$$, the point $$P$$ that minimizes $$\left \lvert\overrightarrow{PA_{1}} \right \rvert+\left \lvert \overrightarrow{PA_{2}}\right \rvert+\cdot \cdot \cdot +\left \lvert \overrightarrow{PA_{n-1}}\right \rvert+\left \lvert\overrightarrow{PA_{n}} \right \rvert$$ is called the 'balance point' of $$A_{i}(i=1,2,3,\cdots,n)$$. As shown in the figure, the diagonals of rectangle $$ABCD$$ intersect at point $$O$$, and $$BC$$ is extended to $$E$$ such that $$BC=CE$$. Connecting $$AE$$, it intersects $$BD$$ and $$CD$$ at points $$F$$ and $$G$$, respectively. Among the following conclusions, the correct one is ___. 1. The 'balance point' of $$A$$ and $$C$$ must be $$O$$; 2. The 'balance point' of $$D$$, $$C$$, and $$E$$ is the midpoint of segment $$DE$$; 3. The 'balance point' of $$A$$, $$F$$, $$G$$, and $$E$$ exists and is unique; 4. The 'balance point' of $$A$$, $$B$$, $$E$$, and $$D$$ must be $$F$$.
|
4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/218f8e17-8e4b-414b-8dda-280182b35999.png"
}
] |
ee01809e-a50b-4482-8de8-1295fc0b4432
|
<image>As shown in the figure, the radius of circle $$\odot O$$ is $$2$$, and points $$A$$ and $$B$$ are on $$\odot O$$ with $$\angle AOB=90^{\circ}$$. The area of the shaded region is ___.
|
$$\pi -2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ee01809e-a50b-4482-8de8-1295fc0b4432.png"
}
] |
58ec36aa-ecc2-4bf8-b17d-a72c20f895f3
|
<image>Given that the diagonals $$AC$$ and $$BD$$ of parallelogram $$ABCD$$ intersect at point $$O$$, and $$P$$ is a point outside the plane of $$ABCD$$ such that $$PA=PC$$ and $$PB=PD$$, then the angle formed by $$PO$$ and the plane $$ABCD$$ is ___.
|
$$90^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/58ec36aa-ecc2-4bf8-b17d-a72c20f895f3.png"
}
] |
5e0b143c-1ab6-410e-a6e5-2a33f354930d
|
<image>As shown in the figure, the vertices of $$\triangle ABC$$ are all on the grid points of the grid paper. Then $$cosA=$$ ___.
|
$$\dfrac{2\sqrt{5}}{5}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5e0b143c-1ab6-410e-a6e5-2a33f354930d.png"
}
] |
1dd4b8ec-ad20-4906-bae4-993f55c69a6f
|
<image>In the figure, the diameter of the semicircle $$AB=6$$, $$O$$ is the center, and $$C$$ is any point on the semicircle different from $$A$$ and $$B$$. If $$P$$ is a moving point on the radius $$OC$$, then the minimum value of $$\left(\overrightarrow{PA}+\overrightarrow{PB}\right)\cdot \overrightarrow{PC}$$ is ___.
|
$$-\dfrac{9}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1dd4b8ec-ad20-4906-bae4-993f55c69a6f.png"
}
] |
89f875e0-f94c-4c16-b535-3acda3316244
|
<image>As shown in the figure, $$OB$$ is the bisector of $$\angle AOC$$, and $$OD$$ is the bisector of $$\angle COE$$. If $$\angle AOB=40^{ \circ }$$ and $$\angle COE=60^{ \circ }$$, then the measure of $$\angle BOD$$ is ___.
|
$$70^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/89f875e0-f94c-4c16-b535-3acda3316244.png"
}
] |
1f0421d6-81d8-477b-bb1c-90270045807a
|
<image>In the flowchart shown, the number of times the loop body is executed is ___.
|
$$499$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1f0421d6-81d8-477b-bb1c-90270045807a.png"
}
] |
6fdea15e-7327-4b38-8e3a-5263b1376552
|
<image>"Proofs without words" involve presenting mathematical propositions using simple, creative, and easily understandable geometric figures. Please use the area relationships in Figure A, Figure B, and Figure C to write the trigonometric identity verified by the figures: ___.
|
$$\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta $$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6fdea15e-7327-4b38-8e3a-5263b1376552.png"
}
] |
7706d597-cb23-4bed-9516-59cfe264f781
|
<image>As shown in the figure, the perpendicular bisector of $$BC$$, $$DE$$, intersects $$BC$$ at point $$D$$ and $$AB$$ at point $$E$$. If the perimeter of $$\triangle EDC$$ is $$24$$, and the difference in the perimeters of $$\triangle ABC$$ and quadrilateral $$AEDC$$ is $$12$$, then the length of segment $$DE$$ is ___.
|
$$6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7706d597-cb23-4bed-9516-59cfe264f781.png"
}
] |
43613b20-f74f-4806-8c49-b4657039b4ca
|
<image>Arrange positive integers $$1$$, $$2$$, $$3$$, $$4$$, ... in a triangular array as shown in the figure. The 10th number from the left in the 10th row is ___.
|
$$91$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/43613b20-f74f-4806-8c49-b4657039b4ca.png"
}
] |
a8354c57-e377-4b7d-90dc-3e03203343b1
|
<image>As shown in the figure, with vertex $$A$$ of $$\triangle ABC$$ as the center and the length of $$BC$$ as the radius, an arc is drawn. Then, with vertex $$C$$ as the center and the length of $$AB$$ as the radius, another arc is drawn, intersecting at point $$D$$. Connect $$AD$$ and $$CD$$. If $$\angle B=65^{\circ}$$, then the measure of $$\angle ADC$$ is ___.
|
$$65^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a8354c57-e377-4b7d-90dc-3e03203343b1.png"
}
] |
ee5c0e5f-9b47-4e1b-a5a8-57ff8fb852fb
|
<image>The figure shown is the 'Zhao Shuang String Diagram'. $$\triangle ABH$$, $$\triangle BCG$$, $$\triangle CDF$$, and $$\triangle DAE$$ are four congruent right triangles, and quadrilaterals $$ABCD$$ and $$EFGH$$ are both squares. If $$AB=10$$ and $$EF=2$$, then $$AH$$ equals ___.
|
$$6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ee5c0e5f-9b47-4e1b-a5a8-57ff8fb852fb.png"
}
] |
6f085c43-e71c-4406-bb47-b5229c51c166
|
<image>As shown in the figure, the side length of the regular pentagon $$ABCDE$$ is $$1$$, and circle $$\odot B$$ passes through the vertices $$A$$ and $$C$$ of the pentagon. The length of the minor arc $$AC$$ is ___.
|
$$\dfrac{3 \pi }{5}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6f085c43-e71c-4406-bb47-b5229c51c166.png"
}
] |
6d4f9221-5c12-4614-b382-5e4ea3f0b21f
|
<image>As shown in the figure: Given the line $$l$$: $$4x-3y+6=0$$, and the parabola $$C$$: $$y^{2}=4x$$, the minimum value of the sum of the distances from a moving point $$P$$ on the parabola to line $$l$$ and to the $$y$$-axis is ___.
|
$$1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6d4f9221-5c12-4614-b382-5e4ea3f0b21f.png"
}
] |
7951820b-1b5f-4068-9ad6-828976bf3558
|
<image>If the program flowchart shown in the figure is executed, with input $$x=-1$$, $$n=3$$, what is the output number $$S=$$___?
|
$$-4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7951820b-1b5f-4068-9ad6-828976bf3558.png"
}
] |
9883a767-fe98-4668-a21e-bb0f773eb2f1
|
<image>As shown in the figure, $$AB$$ is the diameter of circle $$⊙O$$, and $$BC$$ is a chord of $$⊙O$$. If $$∠AOC=80^{\circ}$$, then $$∠B=$$ ___.
|
$$40^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9883a767-fe98-4668-a21e-bb0f773eb2f1.png"
}
] |
8f811eec-ccf8-4a80-b4a2-e9fefc0055e9
|
<image>As shown in the figure, $$AB\parallel CD$$, $$BC\parallel DE$$. If $$∠B=50^{\circ}$$, then the measure of $$∠D$$ is ___.
|
$$130^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8f811eec-ccf8-4a80-b4a2-e9fefc0055e9.png"
}
] |
cb8516bf-0a61-410d-a265-4eacc4ef710b
|
<image>If positive integers are arranged according to the pattern shown in the figure, and the ordered pair (a, b) represents the b-th number from left to right in the a-th row. For example, (4, 3) represents the number 9, then (7, 2) represents the number ______ .
|
23
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cb8516bf-0a61-410d-a265-4eacc4ef710b.png"
}
] |
dfb563d3-7a2d-4223-a15a-c399797daf44
|
<image>Zhuangzi said: 'A foot-long rod, if half of it is taken away each day, will never be exhausted.' This statement (verbal language) expresses the ancient idea of dividing things infinitely, which is represented graphically as Figure 1. Following this method of division, an equation (symbolic language) can be derived: $$1=\dfrac{1}{2}+\dfrac{1}{2^{2}}+\dfrac{1}{2^{3}}+\cdot \cdot \cdot +\dfrac{1}{2^{n}}+\cdots $$. Figure 2 is another form of infinite division: In $$\triangle ABC$$, $$\angle C=90^{ \circ }$$, $$\angle B=30^{ \circ }$$, draw $$CC_{1} \perp AB$$ at point $$C_{1}$$, then draw $$C_{1}C_{2}\perp BC$$ at point $$C_{2}$$, and again draw $$C_{2}C_{3}\perp AB$$ at point $$C_{3}$$, and so on, continuing infinitely. This divides $$\triangle ABC$$ into $$\triangle ACC_{1}$$, $$\triangle CC_{1}C_{2}$$, $$\triangle C_{1}C_{2}C_{3}$$, $$\triangle C_{2}C_{3}C_{4}$$, $$\cdots$$, $$\triangle C_{n-2}C_{n-1}C_{n}$$, $$\cdots$$. Assuming $$AC=2$$, the equation representing the sum of the areas of these triangles is ___.
|
$$2\sqrt{3}=\dfrac{\sqrt{3}}{2}\left \lbrack 1+\dfrac{3}{4}+\left (\dfrac{3}{4} \right )^{2}+\left ( \dfrac{3}{4}\right )^{3}+\cdots + \left (\dfrac{3}{4} \right )^{n-1} + \cdots \right \rbrack$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/dfb563d3-7a2d-4223-a15a-c399797daf44.png"
}
] |
4bf9b661-79f4-476e-b613-6ed20743ea75
|
<image>A rectangular piece of paper is folded in the manner shown in the figure, with BC and BD being the creases. The degree measure of ∠CBD is ( ______ ).
|
90°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4bf9b661-79f4-476e-b613-6ed20743ea75.png"
}
] |
6bdaf4a0-e725-49e8-afd2-28838f7fa823
|
<image>The correct simplified graph of the function $$y= \sin \left(2x-\dfrac{ \pi }{3}\right)$$ in the interval $$\left \lbrack-\dfrac{ \pi }{2}, \pi \right \rbrack$$ is ___ (fill in the number).
|
1
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6bdaf4a0-e725-49e8-afd2-28838f7fa823.png"
}
] |
65e392a6-8601-49ee-a838-0f7ed421451c
|
<image>As shown in the figure, circle $$⊙O$$ passes through points $$B$$ and $$C$$. The center $$O$$ is inside the isosceles right triangle $$\triangle ABC$$, with $$∠BAC=90^{\circ}$$, $$OA=1$$, and $$BC=6$$. The radius of $$⊙O$$ is ___.
|
$$\sqrt{13}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/65e392a6-8601-49ee-a838-0f7ed421451c.png"
}
] |
0edce498-8b6a-4082-ade4-99391e627769
|
<image>Execute the program flowchart (algorithm flowchart) as shown in the figure, the output of $$n$$ is ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0edce498-8b6a-4082-ade4-99391e627769.png"
}
] |
579b5edb-17f5-48ff-a658-58fb255f64c5
|
<image>The following is a schematic diagram of several major scenic spots in Yongzhou City. According to the information in the diagram, the coordinates of the center point $$C$$ of Jiuyi Mountain can be determined as ___.
|
$$(3,1)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/579b5edb-17f5-48ff-a658-58fb255f64c5.png"
}
] |
dafef36e-470e-42ab-b5c2-72aa165b5056
|
<image>The function of the pseudocode below is to input two numbers and output the larger one. Therefore, the code at position 1 should be ___.
|
$$b$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/dafef36e-470e-42ab-b5c2-72aa165b5056.png"
}
] |
29a59743-f698-4a2f-84ee-9b4419e0fe41
|
<image>As shown in the figure, the length of the rectangle is $$6$$, and the width is $$3$$. If $$300$$ yellow beans are randomly scattered within the rectangle, and $$125$$ of them fall into the shaded area, then we can estimate the area of the shaded part to be ___.
|
$$\dfrac{15}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/29a59743-f698-4a2f-84ee-9b4419e0fe41.png"
}
] |
aed4c153-53ea-4692-9405-1c7d2ef18bc8
|
<image>As shown in the figure, in the right triangle $$\text{Rt}\triangle ABC$$, $$AC \perp BC$$. Point $$C$$ is projected perpendicularly onto $$AB$$ at point $$D$$, and $$AC=3$$, $$AD=2$$. Then, $$AB=$$ ___.
|
$$\dfrac{9}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/aed4c153-53ea-4692-9405-1c7d2ef18bc8.png"
}
] |
fd15d24e-592e-4257-ad0a-6177acea107f
|
<image>Arrange some small circles with the same radius according to the pattern shown in the figure. Carefully observe and determine that the $$n$$th pattern has ___ (express using an algebraic expression containing $$n$$) small circles.
|
$$n^{2}+n+4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/fd15d24e-592e-4257-ad0a-6177acea107f.png"
}
] |
bf299775-19ab-4584-bef3-ed2d2009a068
|
<image>Read the flowchart below and run the corresponding program. The value of $$S$$ that is output is ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bf299775-19ab-4584-bef3-ed2d2009a068.png"
}
] |
21b83e43-7b6d-48af-9efa-f4f7ee8fe854
|
<image>In a Chinese weekly test for a class in the senior year of a certain high school, each student's score falls within the range [100, 128]. The class's test scores are divided into seven groups: [100, 104], [104, 108), [108, 112), [112, 116), [116, 120), [120, 124), [120, 124), [124, 128]. The frequency distribution histogram is shown in the figure below. It is known that the number of students scoring below 112 points is 18. The number of students scoring 120 points or higher is ___.
|
10
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/21b83e43-7b6d-48af-9efa-f4f7ee8fe854.png"
}
] |
3011e40d-3a05-4d5c-a0ae-8f0d47ac6ca6
|
<image>The side length of square $$ABCD$$ is $$a$$. Points $$E$$ and $$F$$ are two points on the diagonal $$BD$$. Lines through points $$E$$ and $$F$$ are drawn parallel to $$AD$$ and $$AB$$, respectively, as shown in the figure. The sum of the areas of the shaded regions is ___.
|
$$\dfrac{1}{2}a^{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3011e40d-3a05-4d5c-a0ae-8f0d47ac6ca6.png"
}
] |
62c4fa7b-a161-40f4-8208-3c93ec3d1a5a
|
<image>As shown in the figure, given that $$CD$$ bisects $$\angle ACB$$, $$DE \parallel AC$$, and $$\angle 1 = 30\degree$$, then $$\angle 2 =$$ ______ degrees.
|
$$60$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/62c4fa7b-a161-40f4-8208-3c93ec3d1a5a.png"
}
] |
a4540e16-2c60-4f66-9261-259aed75036e
|
<image>As shown in the figure, OB⊥OA, line CD passes through point O, and ∠AOC＝25°, then ∠BOC＝ ______ , ∠BOD＝ ______ .
|
65°115°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a4540e16-2c60-4f66-9261-259aed75036e.png"
}
] |
ca114092-e50c-4579-95f8-31a5c75de9b6
|
<image>To understand the weight situation of students at a school who are preparing to apply as pilots this year, the collected data was organized and a frequency distribution histogram was drawn (as shown in the figure). It is known that the frequency ratio of the first 3 groups from left to right is 1:2:3, and the frequency of the second group is 12. The number of students sampled is ___.
|
48
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ca114092-e50c-4579-95f8-31a5c75de9b6.png"
}
] |
089f82f2-7c02-4e59-a116-0bf752544fa0
|
<image>On China Tourism Day, the tourism department of a city conducted a sampling survey on the duration of tourists' visits to Lishui in the first quarter of 2016, with the statistics as follows: If the statistical data is represented in a pie chart, then the central angle of the sector representing a tourism duration of '2～3 days' is ___.
|
144°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/089f82f2-7c02-4e59-a116-0bf752544fa0.png"
}
] |
41252915-9f2d-4914-b740-5a4adc57285e
|
<image>As shown in the figure, there is a circular arch gate in a scenic area. The width of the road $$AB$$ is $$2m$$, and the clear height $$CD$$ is $$5m$$. The radius of the circle to which the circular arch gate belongs is ___ $$m$$.
|
$$2.6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/41252915-9f2d-4914-b740-5a4adc57285e.png"
}
] |
8d5c5e75-1e66-4851-8e78-dc5a770bcb4d
|
<image>As shown in the figure, in the right-angle coordinate system $$S _{\triangle AMF}$$, it is known that point $$A(0,1)$$, point $$P$$ is on line segment $$OA$$, and the circumference of the circle $$\odot P$$ with radius $$AP$$ is $$1$$. Point $$M$$ starts from $$A$$ and moves counterclockwise along $$\odot P$$. The ray $$AM$$ intersects the $$S _{\triangle BEN}$$ axis at point $$N(S_{四边形MNHG},0)$$. Let the distance traveled by point $$M$$ be $$MN(M)$$, and as point $$M$$ moves, when $$N$$ moves from $$AB$$ to $$BN=\sqrt{MN^{2}-AM^{2}}=\sqrt{9-4}=\sqrt{5}$$, the path length traveled by point $$N$$ is ___.
|
$$\dfrac{2}{3}\sqrt{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8d5c5e75-1e66-4851-8e78-dc5a770bcb4d.png"
}
] |
625ae6de-1c44-46b2-a2ea-40eb41876cfc
|
<image>As shown in the figure, a particle moves in the first quadrant. In the first second, it moves from the origin to $$(0,1)$$, and then it continues to move in directions parallel to the $$x$$-axis and $$y$$-axis, moving one unit length per second. Therefore, after $$2000$$ seconds, the coordinates of the position of the particle are ___.
|
$$(24,44)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/625ae6de-1c44-46b2-a2ea-40eb41876cfc.png"
}
] |
101cd194-df3b-47cf-8a12-0b7e771cd35e
|
<image>As shown in the figure, points A and B are on the number line, corresponding to the numbers -4 and $\frac{2\text{x}+2}{3\text{x-}5}$, respectively. Given that the distances from points A and B to the origin are equal, find x.
|
$\frac{11}{5}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/101cd194-df3b-47cf-8a12-0b7e771cd35e.png"
}
] |
1e458fd7-8849-4936-949b-bfd0607ff701
|
<image>According to the flowchart shown in the figure, after running, the output result is $$63$$. What is the value of the integer $$M$$ in the decision box?
|
$$5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1e458fd7-8849-4936-949b-bfd0607ff701.png"
}
] |
c2dc7aea-9845-41f7-be9e-c6473cea0e06
|
<image>As shown in the figure, in rhombus $$ABCD$$, $$\angle BAD=60^{ \circ }$$, $$AB=4$$, and $$E$$ is any point inside $$\triangle BCD$$. $$AE$$ intersects $$BD$$ at point $$F$$. The minimum value of $$\overrightarrow{AF}\cdot \overrightarrow{BF}$$ is ___.
|
$$-1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c2dc7aea-9845-41f7-be9e-c6473cea0e06.png"
}
] |
38146c4a-77d3-472e-9f18-3530631355d3
|
<image>As shown in the figure, the parabola $$y=x^{2}+bx+c$$ passes through the point $$(0,-3)$$. Please determine a value of $$b$$ such that one of the intersections of the parabola with the $$x$$-axis is between $$(1,0)$$ and $$(3,0)$$. The value of $$b$$ you determine is ___.
|
$$1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/38146c4a-77d3-472e-9f18-3530631355d3.png"
}
] |
53dbbdd4-8f4e-4bf2-8a61-969843016f69
|
<image>Referring to the flowchart shown, run the corresponding program, the output result is ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/53dbbdd4-8f4e-4bf2-8a61-969843016f69.png"
}
] |
9ef96b36-79c2-4c8d-8883-8f2bf181ddf7
|
<image>As shown in the figure, in $$\triangle ABC$$, $$AB=\quantity{5}{cm}$$, $$AC=\quantity{13}{cm}$$, and the median $$AD=\quantity{6}{cm}$$ on side $$BC$$. What is the length of side $$BC$$?
|
$$2\sqrt{61}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9ef96b36-79c2-4c8d-8883-8f2bf181ddf7.png"
}
] |
721280c6-4742-4391-b0f2-7f9e987e3e00
|
<image>As shown in the figure, points $$A$$, $$B$$, $$C$$, and $$D$$ are all on the same vertical plane perpendicular to the horizontal plane. Points $$B$$ and $$D$$ are the tops of two lighthouses on two islands. A measurement ship at point $$A$$ on the water surface measures the angles of elevation to points $$B$$ and $$D$$ as $$75^{\circ}$$ and $$30^{\circ}$$, respectively. At point $$C$$ on the water surface, the angles of elevation to points $$B$$ and $$D$$ are both $$60^{\circ}$$. Given that $$AC=0.1km$$. If $$AB=BD$$, then the distance between $$B$$ and $$D$$ is ___$$km$$.
|
$$\dfrac{3\sqrt{2}+\sqrt{6}}{20}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/721280c6-4742-4391-b0f2-7f9e987e3e00.png"
}
] |
3b128ff5-431e-4b3d-b90a-766999fefe32
|
<image>As shown in the figure, semicircles are drawn with the two legs of the isosceles right triangle $$\text{Rt}\triangle ABC$$ as diameters, and an arc is drawn with $$C$$ as the center and $$AC$$ as the radius. If $$AC=1$$, then the area of the shaded region is ___.
|
$$\dfrac{ \pi }{4}-\dfrac{1}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3b128ff5-431e-4b3d-b90a-766999fefe32.png"
}
] |
f7238428-94ed-4e61-a29f-9d99e568ab25
|
<image>Execute the pseudocode shown in the figure, the output result is ___.
|
$$0$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f7238428-94ed-4e61-a29f-9d99e568ab25.png"
}
] |
903d6863-5e49-49ec-9528-0dc4394102b7
|
<image>As shown in the figure, quadrilateral $$ABCD$$ is a square with a side length of $$3$$. After trisecting each side and connecting the points, 16 intersection points are formed as shown in the figure. If two of these points are selected as the starting and ending points to form a vector, then the number of vectors that are collinear with $$\overrightarrow{AC}$$ and have a length of $$2\sqrt{2}$$ is ___.
|
$$8$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/903d6863-5e49-49ec-9528-0dc4394102b7.png"
}
] |
5799b859-dfb4-41da-adf5-2e949785b3a0
|
<image>Given the graph of a linear equation in two variables as shown, the equation is ___.
|
$$x-3y=3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5799b859-dfb4-41da-adf5-2e949785b3a0.png"
}
] |
ea97f3b3-92c9-416b-9a80-9f895577b2e4
|
<image>As shown in the figure, with vertex $$D$$ of the rectangular prism $$ABCD-A_{1}B_{1}C_{1}D_{1}$$ as the origin, and the three edges passing through $$D$$ as the coordinate axes, a spatial rectangular coordinate system is established. If the coordinates of $$\overrightarrow{DB_{1}}$$ are $$(4,3,2)$$, then the coordinates of $$\overrightarrow{AC_{1}}$$ are ___.
|
$$(-4,3,2)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ea97f3b3-92c9-416b-9a80-9f895577b2e4.png"
}
] |
6f857d7f-4048-4518-9d24-685152302d34
|
<image>If the pseudocode shown in the figure is designed to: determine whether the input number $$x$$ is positive. If it is, output its square; if not, output its opposite. Then the blank line should be filled with ___.
|
$$x \leqslant 0$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6f857d7f-4048-4518-9d24-685152302d34.png"
}
] |
72812673-8e82-4b11-99c5-683b2034a073
|
<image>As shown in the figure, in the rectangular paper ABCD, AB = 6 cm, AD = 8 cm. The paper is folded so that side AB falls on the diagonal AC, with point B landing at point F, and the fold line is AE. Then EF = cm.
|
3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/72812673-8e82-4b11-99c5-683b2034a073.png"
}
] |
14666dc5-6d8d-424a-ae92-330206534781
|
<image>As shown in the figure, the three vertices of the equilateral triangle $ABC$ are on the surface of sphere $O$. The distance from the center of the sphere $O$ to the plane $ABC$ is 1, and $AB=3$. What is the surface area of sphere $O$?
|
$16\pi $
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/14666dc5-6d8d-424a-ae92-330206534781.png"
}
] |
7f820538-d48d-4465-992f-c0d6fa374cd6
|
<image>The graph of a linear function is shown below. What is the function equation?
|
$y=-\frac{3}{2}x+3$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7f820538-d48d-4465-992f-c0d6fa374cd6.png"
}
] |
10a2843d-c604-4625-a445-f04245878890
|
<image>As shown in the figure, in the rectangular coordinate system $$xOy$$, it is known that $$A(12,0)$$, $$B(0,9)$$, $$C(3,0)$$, $$D(0,4)$$, and $$Q$$ is a moving point on line segment $$AB$$. $$OQ$$ intersects the circle passing through points $$O$$, $$C$$, and $$D$$ at point $$P$$. The value of $$OP \cdot OQ$$ is ___.
|
$$36$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/10a2843d-c604-4625-a445-f04245878890.png"
}
] |
f47df4ef-fd95-4812-961d-9910736bf561
|
<image>As shown in the figure, $$AB$$ is the diameter of $$\odot O$$, chord $$CD \perp AB$$, with the foot of the perpendicular being $$E$$. $$P$$ is a point on the extension of $$BA$$. Connecting $$PC$$ intersects $$\odot O$$ at point $$F$$. If $$PF=7$$, $$FC=13$$, and $$PA:AE:EB=2:4:1$$, then $$CD=$$ ___.
|
$$4\sqrt{10}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f47df4ef-fd95-4812-961d-9910736bf561.png"
}
] |
b6699669-ca5c-41a6-a501-63ddc6e93111
|
<image>As shown in the figure, it is known that $$\angle AOB$$ and $$\angle BOC$$ are adjacent supplementary angles, $$OD$$ is the bisector of $$\angle AOB$$, $$OE$$ is inside $$\angle BOC$$, $$\angle BOE= \dfrac{1}{2} \angle EOC$$, $$\angle DOE= 72^{ \circ }$$, then $$\angle EOC=$$ ___.
|
$$72^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b6699669-ca5c-41a6-a501-63ddc6e93111.png"
}
] |
e1073579-cb27-4075-b25b-128a49df85ef
|
<image>As shown in the figure, in $$\triangle ABC$$, $$E$$ is a point on $$BC$$ such that $$EC=2BE$$, and point $$D$$ is the midpoint of $$AC$$. Let the areas of $$\triangle ABC$$, $$\triangle ADF$$, and $$\triangle BEF$$ be $$S_{\triangle ABC}$$, $$S_{\triangle ADF}$$, and $$S_{\triangle BEF}$$, respectively, and $$S_{\triangle ABC}=12$$. Then, $$S_{\triangle ADF}-S_{\triangle BEF}=$$ ___.
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e1073579-cb27-4075-b25b-128a49df85ef.png"
}
] |
ea7bfb66-90f3-49fe-aa3d-9e2e22e30891
|
<image>As shown in the figure, a rectangular paper piece $$ABCD$$ is folded so that point $$B$$ falls on point $$D$$ and point $$C$$ falls on point $$C'$$. The crease $$EF$$ intersects $$BD$$ at point $$O$$. Given that $$AB=16$$ and $$AD=12$$, the length of the crease is ___.
|
$$15$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ea7bfb66-90f3-49fe-aa3d-9e2e22e30891.png"
}
] |
f407510b-2410-4bd5-9256-81f02cd0b168
|
<image>As shown in the figure, points $$A_{1}$$, $$A_{2}$$, $$A_{3}$$, ..., $$A_{n}$$, $$A_{n+1}$$ are on the $$x$$-axis, and $$OA_{1} = A_{1}A_{2} = A_{2}A_{3} = ... = A_{n}A_{n+1} = 1$$. Perpendiculars are drawn from points $$A_{1}$$, $$A_{2}$$, $$A_{3}$$, ..., $$A_{n}$$, $$A_{n+1}$$ to the $$x$$-axis, intersecting the line $$y = \dfrac{1}{2}x$$ at points $$B_{1}$$, $$B_{2}$$, $$B_{3}$$, ..., $$B_{n}$$, $$B_{n+1}$$, respectively. Lines $$A_{1}B_{2}$$, $$B_{1}A_{2}$$, $$A_{2}B_{3}$$, $$B_{2}A_{3}$$, ..., $$A_{n}B_{n+1}$$, $$B_{n}A_{n+1}$$ are connected, intersecting at points $$P_{1}$$, $$P_{2}$$, $$P_{3}$$, ..., $$P_{n}$$, respectively. The areas of triangles $$\triangle A_{1}B_{1}P_{1}$$, $$\triangle A_{2}B_{2}P_{2}$$, ..., $$\triangle A_{n}B_{n}P_{n}$$ are denoted as $$S_{1}$$, $$S_{2}$$, $$S_{3}$$, ..., $$S_{n}$$, respectively. Then, $$S_{n} =$$ ___ (please express in terms of $$n$$).
|
$$\dfrac{n^{2}}{8n+4}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f407510b-2410-4bd5-9256-81f02cd0b168.png"
}
] |
943ad2e0-d08f-47a7-a18b-870e6cb48bbe
|
<image>Squares $$A_{1}B_{1}C_{1}O$$ and $$A_{2}B_{2}C_{2}C_{1}$$ are placed as shown in the figure. Points $$A_{1}$$ and $$A_{2}$$ lie on the line $$y=x+1$$, and points $$C_{1}$$ and $$C_{2}$$ lie on the x-axis. Given that the coordinates of point $$A_{1}$$ are $$(0,1)$$, the coordinates of point $$B_{2}$$ are ___.
|
$$(3,2)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/943ad2e0-d08f-47a7-a18b-870e6cb48bbe.png"
}
] |
d9621766-073f-4072-97a3-9742e69ad011
|
<image>As shown in the figure, the equilateral triangle in the first diagram (Figure 1) is divided by connecting the midpoints of each side, resulting in the second diagram (Figure 2); then, the smallest equilateral triangle in the center of the second diagram is divided in the same manner, resulting in the third diagram (Figure 3); and the smallest equilateral triangle in the center of the third diagram is divided in the same manner, ..., then the nth diagram will have ______ equilateral triangles.
|
4n-3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d9621766-073f-4072-97a3-9742e69ad011.png"
}
] |
3fe4bef6-dd61-4f5c-b715-6570c122bf6f
|
<image>As shown in the figure, $$AD$$ bisects $$\angle BAC$$. To make $$\triangle ABD \cong \triangle ACD$$, which of the following conditions can be added? (Add one condition only)
|
$$AB=AC$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3fe4bef6-dd61-4f5c-b715-6570c122bf6f.png"
}
] |
7cebd5bc-513c-4f94-b140-cc51a770c3cb
|
<image>As shown in the figure, in the Cartesian coordinate system, △ABC and △DEF are homothetic with respect to the origin O, and the similarity ratio k = $\frac{1}{3}$. If B(2, 1), then the coordinates of point E are.
|
(6, 3)
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7cebd5bc-513c-4f94-b140-cc51a770c3cb.png"
}
] |
669d4508-1e3c-45d4-a74d-e8a6b6d4b7e7
|
<image>As shown in the figure, according to the given information, the value of $\frac{BC}{{B}'{C}'}$ is.
|
$\frac{1}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/669d4508-1e3c-45d4-a74d-e8a6b6d4b7e7.png"
}
] |
de645965-6b59-4873-b4a4-22234b7de98f
|
<image>As shown in the figure, in $\vartriangle ABC$, the external angle bisectors of $\angle B$ and $\angle C$ intersect at point $O$. If $\angle A = 74^\circ$, then $\angle O = $ degrees.
|
53
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/de645965-6b59-4873-b4a4-22234b7de98f.png"
}
] |
7563b7da-1956-4169-8e15-03fc476fa9e5
|
<image>M is a fixed point on the circumference of a circle with radius R. A point N is chosen randomly on the circumference. If MN is connected, then the probability that the length of chord MN exceeds √3R is:
|
1/3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7563b7da-1956-4169-8e15-03fc476fa9e5.png"
}
] |
2d443b18-69ea-4df5-b712-0cd510c1c7d7
|
<image>As shown in the figure, each pattern is formed by arranging several chess pieces. According to this pattern, the total number of chess pieces in the $$n$$th pattern can be expressed by an algebraic expression containing $$n$$ as ______.
|
$$n(n+1)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2d443b18-69ea-4df5-b712-0cd510c1c7d7.png"
}
] |
d36ea577-3b12-48f2-a6ff-cdd07594e3be
|
<image>As shown in the figure, the line $$l$$ passing through the origin intersects the graph of the inverse proportion function $$y=-\dfrac{1}{x}$$ at points $$M$$ and $$N$$. Based on the graph, conjecture the minimum length of segment $$MN$$ is ___.
|
$$\sqrt{8}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d36ea577-3b12-48f2-a6ff-cdd07594e3be.png"
}
] |
08907992-7ff9-47c7-9835-f794553c1316
|
<image>To understand the relationship between annual income and annual expenditure of residents in a community, 5 households were randomly surveyed, and the following statistical data table was obtained: According to the table, the linear regression equation is $$\hat{y}=bx+a$$, where $$b=0.76$$ and $$a=\overline{y}-b\overline{x}$$. Based on this, estimate the annual expenditure of a household with an annual income of $$15$$ million yuan to be ___ million yuan.
|
$$\number{11.8}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/08907992-7ff9-47c7-9835-f794553c1316.png"
}
] |
624c7dc7-7306-418b-93d8-391ff4a4cb10
|
<image>As shown in the figure, Hope Middle School has created a pie chart representing the selection of four school courses by students: chess, martial arts, photography, and embroidery. From the chart, the percentage of students choosing embroidery is ______.
|
13%
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/624c7dc7-7306-418b-93d8-391ff4a4cb10.png"
}
] |
68a6b2c0-7ba3-4777-a44a-10f12d55f793
|
<image>As shown in the figure, point $$A$$ is on the hyperbola $$y={k\over x}$$, $$AB\bot x$$ axis at point $$B$$, and the area of $$\triangle AOB$$ is $$2$$. What is the value of $$k$$?
|
$$-4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/68a6b2c0-7ba3-4777-a44a-10f12d55f793.png"
}
] |
0d56592e-c982-427f-a176-78e431f9d5c5
|
<image>As shown in the figure, $$\angle B=\angle D=90\degree$$, $$BC=DC$$, $$\angle 1=40\degree$$, then $$\angle 2=$$______ degrees.
|
$$50$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0d56592e-c982-427f-a176-78e431f9d5c5.png"
}
] |
3a8a2096-beb1-430e-9b44-290aee45cc1d
|
<image>As shown in the figure, in △ABC, ∠ABC = 50°, ∠ACB = 70°. Extend CB to point D such that BD = BA, and extend BC to point E such that CE = CA. Connect AD and AE. What is the measure of ∠DAE in degrees?
|
120
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3a8a2096-beb1-430e-9b44-290aee45cc1d.png"
}
] |
c4998efe-7975-4c7a-a82c-127ba578bf38
|
<image>Execute the program flowchart shown in the figure, then the output result is.
|
8
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c4998efe-7975-4c7a-a82c-127ba578bf38.png"
}
] |
c87ea5d2-9ac4-450b-bc6c-82b0c814fbb6
|
<image>When the pattern shown in the figure is rotated n° around its center, it completely overlaps with the original pattern. What is the minimum value of n?
|
120
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c87ea5d2-9ac4-450b-bc6c-82b0c814fbb6.png"
}
] |
2c1622ac-6997-48d5-a13c-3cc8b459ebd1
|
<image>As shown in the figure, quadrilateral $$ABCD$$ is inscribed in circle $$\odot O$$. Point $$E$$ is on the extension of $$BC$$. If $$\angle BOD=120^{ \circ }$$, then $$\angle DCE=$$ ___.
|
$$60^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2c1622ac-6997-48d5-a13c-3cc8b459ebd1.png"
}
] |
cf878bf4-9829-4b51-94b8-ce9b46a48ac2
|
<image>Given that the area of the figure shown in the diagram is 27, according to the conditions in the diagram, the equation that can be set up is ______ .
|
(x+1)^2=25
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cf878bf4-9829-4b51-94b8-ce9b46a48ac2.png"
}
] |
b614d65b-3b64-49ac-bd0b-712432b63236
|
<image>As shown in the figure, in rectangle $$ABCD$$, $$AD=6$$, $$O$$ is the intersection point of the diagonals of the rectangle, circle $$O$$ intersects side $$AB$$ at points $$M$$ and $$N$$. If $$MN=8$$, then the radius of circle $$O$$ is ___.
|
$$5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b614d65b-3b64-49ac-bd0b-712432b63236.png"
}
] |
28571bd4-b221-4a5a-9ad3-a401c1f10ed5
|
<image>As shown in the figure, points $$B$$ and $$C$$ are on line segment $$AD$$, $$M$$ is the midpoint of $$AB$$, $$N$$ is the midpoint of $$CD$$, $$MN = a$$, $$BC = b$$, then the length of line segment $$AD =$$___.
|
$$2a - b$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/28571bd4-b221-4a5a-9ad3-a401c1f10ed5.png"
}
] |
83dba3a3-2e33-4b5a-b3c2-92fb9e561322
|
<image>As shown in the figure, a circular flower bed is divided into four sections: $$A$$, $$B$$, $$C$$, and $$D$$. There are four different types of flowers to choose from, and the requirement is to plant one type of flower in each section such that no two adjacent sections have the same type of flower. How many different ways can the flowers be planted?
|
$$84$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/83dba3a3-2e33-4b5a-b3c2-92fb9e561322.png"
}
] |
e7b9ef27-b073-41c7-8873-2beefd977c93
|
<image>As shown in the figure, in the right-angled triangle $$ABC$$, $$\angle C=90{{}^\circ}$$, $$AC=5cm$$, $$BC=12cm$$, the angle bisector of $$\angle CAB$$ intersects $$BC$$ at $$D$$, and a perpendicular line from $$D$$ to $$AB$$ meets $$AB$$ at $$E$$. What is the perimeter of $$\triangle BDE$$ in $$cm$$?
|
20
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e7b9ef27-b073-41c7-8873-2beefd977c93.png"
}
] |
9d372750-a9a2-46ac-bc39-e81a9c2747af
|
<image>After a math exam, the scores of 10 randomly selected students from a class were used as a sample, and the stem-and-leaf plot of the score data is shown in the figure. What is the variance of the sample?
|
175
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9d372750-a9a2-46ac-bc39-e81a9c2747af.png"
}
] |
21f8e560-9e92-41df-b884-f503820bb3d1
|
<image>Write the result of running the following program. If the input is 2 when running, then the output result is_____.
|
48,99
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/21f8e560-9e92-41df-b884-f503820bb3d1.png"
}
] |
2dc659b8-c4a9-44c4-a134-bd3f6d7af26a
|
<image>In engineering, steel balls are often used to measure the diameter of small holes in parts. Suppose the diameter of the steel ball is $$\quantity{10}{mm}$$, and the distance from the top of the steel ball to the part's surface is $$\quantity{8}{mm}$$, as shown in the figure. Then the diameter of this small hole is ___ $$\unit{mm}$$.
|
$$8$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2dc659b8-c4a9-44c4-a134-bd3f6d7af26a.png"
}
] |
ed2b429a-d318-4b40-a53b-a31bfd585bc8
|
<image>As shown in the figure, to measure the height of the Eiffel Tower in France, a baseline $$AB$$ is selected on the ground, where $$AB = 324\sqrt{4-\sqrt{3}}\ \unit{m}$$. At point $$A$$, the angle of elevation to the top of the tower $$P$$ is $$\angle OAP = 30^{\circ}$$, and at point $$B$$, the angle of elevation to the top of the tower $$P$$ is $$\angle OBP = 45^{\circ}$$. It is also measured that $$\angle AOB = 60^{\circ}$$. What is the height of the tower in meters?
|
$$324$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ed2b429a-d318-4b40-a53b-a31bfd585bc8.png"
}
] |
e159c092-27a0-4ccd-a475-d62214a139fa
|
<image>Arrange consecutive positive integers according to the following pattern: If the positive integer 565 is located in the a-th row and the b-th column, then a+b=______.
|
147
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e159c092-27a0-4ccd-a475-d62214a139fa.png"
}
] |
b8a581b0-1720-4578-8fbe-2644d8ad7465
|
<image>Execute the program flowchart shown in the figure. If $$p=0.8$$, then the output $$n=$$ ___.
|
$$4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b8a581b0-1720-4578-8fbe-2644d8ad7465.png"
}
] |
5d696c8d-8181-4f58-bfb9-51aa91b7c943
|
<image>As shown in the figure, the side length of square $$ABCD$$ is $$4$$, point $$A$$ has coordinates $$\left (-1,1\right )$$, $$AB \parallel x$$-axis, and $$BC \parallel y$$-axis. The coordinates of point $$C$$ are ___.
|
$$\left (3,5\right )$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5d696c8d-8181-4f58-bfb9-51aa91b7c943.png"
}
] |
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