id
stringlengths 4
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stringlengths 77
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| query
stringlengths 7
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| answer
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| choice
stringlengths 4
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| question_type
stringclasses 2
values |
|---|---|---|---|---|---|
func326
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/326.png
|
As shown in the figure, in rhombus ABCO, where point C has the coordinates (3, 4), what are the coordinates of vertex B ______.
|
(8, 4)
|
NULL
|
free_form
|
func327
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/327.png
|
As shown in the figure, place the chessboard in a Cartesian coordinate system. The position of White Chess 2 is (-5, -2), and the position of White Chess 1 is (-4, -5). What are the coordinates of Black Chess 3? __________.
|
(-2, -3)
|
NULL
|
free_form
|
func328
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/328.png
|
As shown in the figure, in the rectangular coordinate system, the graph of the straight line y₁ = 1/2 x - 2 passes through A(4,0) and B(0,-2) and intersects with the graph of the direct proportion function y₂ = 5/6 x at the point (m, -5/4). When y₁ < y₂, the range of values for the real number x is __________.
|
x < 3/2
|
NULL
|
free_form
|
func329
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/329.png
|
5. As shown in the figure, in the rectangular coordinate system, the vertices of rectangle ABCD are given: A(-4, -4), B(12, 6), D(-8, 2). Segment CD intersects the y-axis at point P. What are the coordinates of point P? ________.
|
(0, 7)
|
NULL
|
free_form
|
func33
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/33.png
|
As shown in the figure, given the functions y = -x + 2 and y = kx + 3 intersect at point P, and the y-coordinate of point P is 2.5, then the solution of the system of equations {x + y = 2, kx - y = -3} is ______.
|
x = -0.5, y = 2.5
|
NULL
|
free_form
|
func330
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/330.png
|
As shown in the figure, the line y=2x intersects with the line y=kx+b at point A(m,4). Find the solution set of the inequality 2x > kx + b ________.
|
x>2
|
NULL
|
free_form
|
func331
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/331.png
|
As shown in the figure, in the plane Cartesian coordinate system xOy, it is known that point A(2, n+2), B(k, n+2), C(4, n+4), D(2, n+k). Express the area of the quadrilateral ABCD as a formula involving k.
|
2k−4
|
NULL
|
free_form
|
func332
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/332.png
|
Using an 8m long aluminum alloy profile to make a rectangular window frame as shown in the figure, the maximum light-transmitting area of the window frame is ______ m². (The light-transmitting area refers to the total rectangular area)
|
2
|
NULL
|
free_form
|
func333
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/333.png
|
In a plane rectangular coordinate system, the graphs of the linear functions y₁ = kx (k is a constant, k ≠ 0) and y₂ = mx + n (m is a constant, m ≠ 0) are shown as in the figure. Then the solution set of the inequality (k − m)x ≥ n with respect to x is ______.
|
x ≤ −3
|
NULL
|
free_form
|
func334
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/334.png
|
As shown in the figure, in a plane Cartesian coordinate system, point A(1, 3) is given. Taking point O as the center of the circle and the length of OA as the radius, draw a circle that intersects the grid line at point B(a, 2). Then ① The length of OA is ______. ② The value of a is ______.
|
√10 √6
|
NULL
|
free_form
|
func336
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/336.png
|
In a plane rectangular coordinate system, point A is on the positive half of the x-axis, line AC is parallel to the x-axis, the x-coordinates of points B and C are both 4, and BC = 3. Point D is on AC, and its x-coordinate is 1. If the graph of the inverse proportional function y = k / x (x > 0) passes through points B and D, what is the value of k?
|
4
|
NULL
|
free_form
|
func337
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/337.png
|
As shown in the figure, the straight line AB passes through the origin O, point C lies on the y-axis, D is a moving point on line segment AB. If A(2,m), B(-3,n), C(0,-2), and AB=10, then the minimum length of CD is ________.
|
1
|
NULL
|
free_form
|
func338
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/338.png
|
Definition: The Manhattan distance between two distinct points P(a, b) and Q(c, d) on the coordinate plane is defined as s(P, Q) = |a - c| + |b - d|. In the figure, the Manhattan distance between point M(-5, 2) and point N(2, -1) is s(M, N) = |−5 − 2| + |2 − (−1)| = 10. If point P(2, 5), point Q(x, −3), and E(x, P, Q) = 9, then x is ________.
|
1 or 3
|
NULL
|
free_form
|
func339
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/339.png
|
As shown in the figure, it is known that the parabola passes through the points (-2, -3) and (3, -3). If the points (1, y1) and (2, y2) lie on this parabola, then y1 ______ y2. (Fill in '>', '<', or '=')
|
>
|
NULL
|
free_form
|
func34
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/34.png
|
It is known that the graph of the quadratic function y = ax^2 + bx + c is as shown. Then, the point P(ab, c) is located in the first quadrant.
|
Two
|
NULL
|
free_form
|
func340
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/340.png
|
If it is a set of regular patterns composed of equilateral triangles with equal side lengths, the first pattern has 4 triangles, the second pattern has 7 triangles, the third pattern has 10 triangles, and this pattern continues. If the nth pattern contains y triangles, what is the relationship between y and n?
|
y = 3n + 1
|
NULL
|
free_form
|
func342
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/342.png
|
As shown in the figure, in the plane rectangular coordinate system, A(-2,1) and B(3,4), if there is a point P on the x-axis such that the value of PA + PB is minimized, what are the coordinates of point P?
|
(-1,0)
|
NULL
|
free_form
|
func343
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/343.png
|
As shown in the figure, in a rectangular coordinate plane, it is known that point A(1,0) and point B(0,1).
(1) Write directly the coordinates of point C in the figure: ____;
(2) If the y-coordinate of point D is 1 and BD=2, what are the coordinates of point D: ____.
|
(-2,3), (-2,1) or (-2,1), (2,1)
|
NULL
|
free_form
|
func344
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/344.png
|
The activity center needs to print a batch of flyers, as shown in Figure ①, to promote the summer camp activities. The cost y (in yuan) of printing these flyers and the number of flyers x (in sheets) for the two nearby printing centers are represented by the function graphs shown in Figure ②. What is the range of x when the cost of Printing Center B is less than that of Printing Center A?
|
0 < x < 500
|
NULL
|
free_form
|
func345
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/345.png
|
8. Straight line l₁: y = k₁x + b₁ and straight line l₂: y = k₂x + b₂ are shown in the graph in the same plane rectangular coordinate system. Then the solution set of the inequality k₁x + b₁ > k₂x + b₂ with respect to x is ______.
|
x < -1
|
NULL
|
free_form
|
func347
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/347.png
|
As shown in the figure, in a 2D Cartesian coordinate system, a moving point starts from the origin O and translates in the following directions sequentially: upward, rightward, downward, and rightward again. Each move is one unit, resulting in the points A1(0,1), A2(1,1), A3(1,0), A4(2,0), ..., and so on. What are the coordinates of point A4104?
|
(7, 1) (1012, 0)
|
NULL
|
free_form
|
func348
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/348.png
|
As shown in the figure, the straight line y = kx + b (where k and b are constants, and k < 0) passes through point A(2, 1). Then the solution set for the inequality kx + b < 1/2 with respect to x is ________.
|
x > 2
|
NULL
|
free_form
|
func35
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/35.png
|
As shown in the figure, in the Cartesian coordinate plane, the graphs of the functions y = mx + n and y = kx + b intersect at the point P(-2, 1). Therefore, the solution to the system of equations {y = mx + n, y = kx + b = 0} is ________.
|
x = -2, y = 1
|
NULL
|
free_form
|
func350
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/350.png
|
As shown in the figure, point P moves in the Cartesian coordinate plane following the direction of the arrows shown in the diagram. It first moves from the origin to the point (2,2), then moves to the point (4,0) for the second time, and then moves to the point (6,1) for the third time... Following this movement pattern, what will be the coordinates of point P after the 2023rd move?
|
(4046, 1)
|
NULL
|
free_form
|
func351
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/351.png
|
As shown in the figure, point A1(0,1), point A2(2,0), point A3(3,2), point A4(5,1), …, following this pattern, the coordinates of point A13 are _______, and the coordinates of point A2024 are _______.
|
(20,6) (3036,1013)
|
NULL
|
free_form
|
func353
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/353.png
|
5. Gomoku is a two-player board game with the following rules: one player uses black pieces, and the other uses white pieces. The black pieces always go first, and the players take turns placing their pieces on the board, one piece per turn. The winner is determined by forming an unbroken line (horizontal, vertical, or diagonal) of five consecutive pieces of the same color. As shown in the diagram, this is a Gomoku problem to analyze the current board state. Observe the board: (1) Using point O as the origin, establish a two-dimensional Cartesian coordinate system such that each piece corresponds to a point. Given that the coordinates of black piece A are (7,5), what are the coordinates of the white piece? At this point, it is the black player's turn. To ensure that the white player cannot win within two moves (inclusive), where should the black piece, denoted as C, be placed? Please provide the coordinates.
|
(5,1), (3,7) or (7,3)
|
NULL
|
free_form
|
func354
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/354.png
|
For the inverse proportional function y = k/x (k ≠ 0), its graph is as shown in the figure. Then the possible value of k is ________. (Only provide one value that meets the condition.)
|
3 (Answer is not unique)
|
NULL
|
free_form
|
func356
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/356.png
|
In the rectangular coordinate system Oxy, for any two points P1(x1, y1) and P2(x2, y2), the following definitions are given: √((x1−x2)^2+(y1−y2)^2) is called the 'linear distance' between point P1 and point P2; |x1−x2|+|y1−y2| is called the 'zigzag distance' between point P1 and point P2. As shown in the figure, the 'zigzag distance' between P1 and P2 is the sum of the lengths of segments P1R and RP2. For example: given P1(1,1) and P2(3,4), the 'linear distance' between point P1 and point P2 is √((1−3)^2+(1−4)^2)=√13, and the 'zigzag distance' between point P1 and point P2 is |1−3|+|1−4|=5. Given point M(1,1) and N(2,5), the 'linear distance' between M and N is _______; the 'zigzag distance' between M and N is _______.
|
√17 5
|
NULL
|
free_form
|
func357
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/357.png
|
As shown in the figure, in a plane rectangular coordinate system, a square ABCD with a side length of 2 has its side AB lying on the x-axis, and the midpoint of AB is the origin O. Fixed points A and B are given. When the square is pushed along the direction of the arrow such that point D falls on point D' of the positive half of the y-axis, the corresponding coordinates of point C's new position, C', are _______________.
|
(2, √3)
|
NULL
|
free_form
|
func358
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/358.png
|
9. As shown in the figure, for the inverse proportional functions y = 6/x and y = 2/x, their graphs in the first quadrant are C1 and C2 respectively. P is a point on C1, PA ⊥ x-axis at point A and intersects C2 at point B. Connect OB and OP. The area of △POB is ______.
|
2
|
NULL
|
free_form
|
func359
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/359.png
|
In recent years, more and more tourists have been visiting Chongqing to experience the spicy and flavorful hot pot, and they often purchase hot pot base as a souvenir. A shop in Hongyadong launched a promotion: if you buy more than 5 packs (not including 5 packs) of hot pot base at a time, the portion exceeding 5 packs will be discounted. Based on this, the payment amount y (in yuan) and the number of hot pot base packs purchased x are related as shown in the graph. How much would it cost to buy 18 packs of hot pot base?
|
190
|
NULL
|
free_form
|
func36
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/36.png
|
As shown in the figure, the parabola y = -x² + bx + c intersects the y-axis at the point (0,5), and the axis of symmetry is the line x = -2. If y ≥ 5, then the range of x is ______.
|
-4 ≤ x ≤ 0
|
NULL
|
free_form
|
func361
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/361.png
|
As shown in the figure, the line l1: y − x + 1 and the line l2: y = mx + 5 intersect at P(2, a). Then the solution to the system of linear equations {x − y + 1 = 0, mx − y + 5 = 0} is ________.
|
x = 2, y = 3
|
NULL
|
free_form
|
func362
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/362.png
|
As shown in the figure, in the plane Cartesian coordinate system (xOy), the graphs of the functions y1=kx and y2=ax+3 intersect at point A(-1, 2). Then the solution set of the inequality kx ≤ ax+3 with respect to x is ______.
|
x > -1
|
NULL
|
free_form
|
func363
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/363.png
|
As shown in the figure, in the quadrilateral ABCD, point A is on the x-axis, point B is on the y-axis, BC lies on the x-axis, and point D is on the graph of an inverse proportion function. Given that the area of quadrilateral ABCD is 5, the expression of the inverse proportion function is ______.
|
y = 5/x
|
NULL
|
free_form
|
func364
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/364.png
|
There is an English word with its letters corresponding to the ordered pairs in the figure as (1,2), (5,1), (5,2), and (1,3). Please write down this English word as ______.
|
HELLO
|
NULL
|
free_form
|
func365
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/365.png
|
As shown in the figure, in a plane Cartesian coordinate system, given point A(-3, 0) and point B(0, 4), line R (AB) rolls along the x-axis without slipping. After rolling for the first time to position ①, the point corresponding to point A is marked as point A1. After rolling a second time to position ②, the point corresponding to point A1 is marked as point A2. After rolling a third time to position ③, the point corresponding to point A2 is marked as point A3, and so on in this sequence. It is observed that the coordinates of the points are as follows: A(-3, 0), A1(0, 3), A2(9, 0), ... What are the coordinates of point A2024?
|
(8097, 0)
|
NULL
|
free_form
|
func366
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/366.png
|
As shown in the figure, points A and B lie on the graph of the inverse proportion function y=6/x. Perpendicular lines are drawn from points A and B to the two coordinate axes. Given that the shaded area S = 2, then the value of the unshaded area S1 + S2 is ______.
|
8
|
NULL
|
free_form
|
func367
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/367.png
|
As shown in the figure, the graphs of the functions y₁ = -2x and y₂ = ax + 3 intersect at point A(-1, 2). What is the solution set for the inequality -2x > ax + 3?
|
-3 < x < -1
|
NULL
|
free_form
|
func368
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/368.png
|
As shown in the figure, point O(0, 0) and point B0(1, 0) are two vertices of the square OB0B1C1. Using its diagonal OB as a side, construct square OB1B2C2. Using the diagonal OB of square OB1B2C2 as a side, construct square OB2B3C3. Similarly, using the diagonal OB of square OB2B3C3 as a side, construct square OB3B4C4, and so on. What are the coordinates of point B2024?
|
(0, 2012)
|
NULL
|
free_form
|
func369
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/369.png
|
On the sides of a square ABCD with a side length of 4, there is a moving point P. Starting from point A, P moves along the broken line AB → BC → CD → DA for one cycle, and then continues periodically. Let the distance traveled by point P be x, and the area of △PAC be y. Based on the function graph in the figure, calculate: when x = 2024, the value of y is ______.
|
0
|
NULL
|
free_form
|
func37
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/37.png
|
As shown in the figure, a moving point P moves in the direction indicated by the arrows in a Cartesian coordinate system. For the 1st time, it moves from the origin to the point (1,1); for the 2nd time, it moves to the point (2,0); for the 3rd time, it moves to the point (3,2); ... Following this pattern, after the 2024th movement, the coordinates of the moving point P are ________.
|
(2024,0)
|
NULL
|
free_form
|
func370
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/370.png
|
Square A1B1C1O1, square A2B2C2O2, square A3B3C3O3, ..., are placed in the plane Cartesian coordinate system as shown in the diagram. If the points A1, A2, A3, ... and C1, C2, C3, ... lie respectively on the line y = x + 1 and the x-axis, then the coordinates of point B2024 are ________.
|
(2^2024 - 1, 2^2023)
|
NULL
|
free_form
|
func372
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/372.png
|
As shown in the figure, point A is on the graph of the inverse proportional function y = 8 / x (x > 0). A line through point A intersects the graph of the parabola y = k / x (k ≠ 0, x > 0) at segment AC at point B. If the area of ΔAOB is 3, then the value of k is ________.
|
2
|
NULL
|
free_form
|
func373
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/373.png
|
As shown in the figure, the inverse proportional function y = k / x (k ≠ 0) passes through point A and point B, then m = ______.
|
2
|
NULL
|
free_form
|
func374
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/374.png
|
As shown in the figure, in △ABC, point P moves from point A towards point C. During this movement, let x represent the length of line segment AP, y represent the area of △ABP, and the distance from point B to line AC is 6. What is the relationship between y and x? __________
|
y = 3x
|
NULL
|
free_form
|
func375
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/375.png
|
As shown in the figure, the graph of the linear function y = ax + b intersects with the graph of y = cx + d at point P. The following conclusions: ① b = b; ② ac < 0; ③ When x < 1, ax + b > cx + d; ④ a − c = d − b. How many of these conclusions are correct?
|
3
|
NULL
|
free_form
|
func376
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/376.png
|
As shown in the figure, a small bug crawls horizontally 4 units to the right from point A to reach point B(m, n). Then m ________ n. (Fill in '>' or '=')
|
=
|
NULL
|
free_form
|
func377
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/377.png
|
As shown in the figure, in the plane Cartesian coordinate system, the line segment OA connecting point A(4, 3) and the origin O forms an angle α with the positive x-axis. What is the value of sinα?
|
3/5
|
NULL
|
free_form
|
func378
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/378.png
|
Given the quadratic function y = x² + bx + c, as shown in the graph, the point P(b, c) is in the ______ quadrant.
|
Third
|
NULL
|
free_form
|
func379
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/379.png
|
Rectangular paper pieces of the same size and shape are arranged in a pattern in the Cartesian coordinate system as shown in the figure. Given A(−2,10), the coordinates of point B are ________.
|
(−28/3, 22/3)
|
NULL
|
free_form
|
func38
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/38.png
|
As shown in the figure, point P lies on the graph of the inverse proportional function \(y = \frac{k}{x} (x < 0)\). Through point P, draw PH perpendicular to the x-axis at point H. Point A lies on the positive half of the y-axis. If the area of △PAH is 4, then the value of k is ______.
|
-8
|
NULL
|
free_form
|
func380
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/380.png
|
As shown in the figure, in the Cartesian coordinate plane, given points A(2,3), B(-2,0), and C(0,-1), a parallelogram is formed with A, B, C, and D as vertices. The coordinates of the fourth vertex D are ______.
|
(4,2) or (0,4) or (-4,-4)
|
NULL
|
free_form
|
func381
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/381.png
|
As shown in the figure, the graphs of the functions y = 2x and y = ax + 4 intersect at point A(m, 4). Then the solution set for the inequality 2x ≥ ax + 4 with respect to x is _______.
|
x ≥ 2
|
NULL
|
free_form
|
func382
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/382.png
|
As shown in the figure, in a Cartesian coordinate plane, point P moves successively in the direction indicated by the arrow. For the first time, it moves from point (-1,0) to point (0,1), for the second time, it moves to point (1,0), for the third time, it moves to point (2,-2), ... Following this pattern of movement, the point P will be at ________ after 2024 moves.
|
(2023,0)
|
NULL
|
free_form
|
func383
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/383.png
|
As shown in the figure, in the Cartesian coordinate system, the coordinates of point A are (0, 4). After △OAB is translated to the right along the x-axis, it becomes △O'A'B'. The corresponding point A' of point A lies on the line y = (4/3)x. What is the distance between point B and its corresponding point B'?
|
3
|
NULL
|
free_form
|
func384
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/384.png
|
As shown in the figure, in the plane rectangular coordinate system, △OAB is a right triangle where ∠AOB = 90°, ∠ABO = 60°, and OB = 8. If the hyperbolic function y = k/x (k ≠ 0) passes through the midpoint C of O, and intersects AB at point D, then k = _______.
|
3√3
|
NULL
|
free_form
|
func385
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/385.png
|
FuGe is an important component of traditional Chinese architectural decoration and an important part of traditional Chinese architectural culture. Figure 1 shows a window grid pattern composed of arc-shaped 'blue tiles' of equal size. Figure 2 is a partial outline illustration of the window grid, placed in a rectangular coordinate system. Points A, B, and C are the endpoints of the arcs. If the coordinates of point A are (-2, √3) and the coordinates of point B are (-3, 0), then the coordinates of point C are ______.
|
(1, -2√3)
|
NULL
|
free_form
|
func386
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/386.png
|
5. Drones have demonstrated excellent performance in both civilian and military applications. DJI, a Chinese drone manufacturer, has been making remarkable strides in the commercial market, holding a 70% market share internationally and achieving the top global sales for 8 consecutive years. The following diagram shows the flight path of a DJI drone during a performance. As shown, the flight path of the drone is placed on a square grid. The drone starts from point O, flies upward 1 unit, and then moves sequentially in the order of right → upward → left → downward, moving 1 unit at a time. The first movement ends at point A_1, the second at point A_2, the third at point A_3, ..., and the n-th movement ends at point A_n. If the coordinates of point O are (0, 0) and A_1 has coordinates (1, 1), what are the coordinates of A_{2025}?
|
(1012, 1)
|
NULL
|
free_form
|
func387
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/387.png
|
In the rectangular coordinate plane, point A(1, 1), B(-1, 1), C(-1, -2), and D(1, -2). Point P starts from point A and moves along the edges of quadrilateral ABCD in the order A→B→C→D→A... . When the distance traveled by point P is 2024 units, the coordinates of the position of point P are ________.
|
(-1, -1)
|
NULL
|
free_form
|
func388
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/388.png
|
As shown in the figure, point A is a point on the graph of the inverse proportion function y = k/x (x > 0). A perpendicular line AB is drawn through point A to intersect the x-axis at point B. Point P is any point on the y-axis, and PA and PB are connected. The area of △ABP is ______.
|
3
|
NULL
|
free_form
|
func389
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/389.png
|
As shown in the figure, in parallelogram ABCD, diagonals AC and BD intersect at point O. A perpendicular from D to AB intersects AB at point O and intersects AC at point F. Connecting BF, it is known that ∠BCD = 60° and BD = 6. A coordinate system is established with point O as the origin. The coordinates of point C are ______.
|
(6, 3√3)
|
NULL
|
free_form
|
func39
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/39.png
|
As shown in the figure, in the Cartesian coordinate system, the coordinates of point A are (0, 3). △OAB is translated to the right along the x-axis to obtain △O'A'B'. The corresponding point A' of point A lies on the line y = 3/4x. The distance between point A and its corresponding point A' is _______.
|
4
|
NULL
|
free_form
|
func390
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/390.png
|
As shown in the figure, in △ABC, it is known that BC=8, the height AD on side BC is 5, and the movable point C' moves from point C towards point B along CB (not overlapping with point D). Let the length of C'C' be x, and the area of △ABC' be S. Then the relationship between S and x is ______
|
S = 20 - 5/2x
|
NULL
|
free_form
|
func392
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/392.png
|
As shown in the figure, it is known that a radar detector identified two targets A and B during detection, where the position of A can be represented as (60°, 6) and the position of B as (150°, 4). The distance between A and B is ________.
|
2√13
|
NULL
|
free_form
|
func393
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/393.png
|
As shown in the figure, the line y=2x+2 intersects the line y=kx+6 at point P(3, n). Then, the solution of the system of equations {x+y=3, 2x-y=-6} is ______.
|
x=3, y=8
|
NULL
|
free_form
|
func394
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/394.png
|
A and B are 14 kilometers apart. Both person A (甲) and person B (乙) travel from location A to location B. In the given graph, l1 and l2 represent the distance (in kilometers) traveled by person A and person B, respectively, in relation to time (in hours). How many hours after person B's departure will the two be 2 kilometers apart?
|
1
|
NULL
|
free_form
|
func395
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/395.png
|
As shown in the figure, point A(-2,6) and point B(-4,2), when the line y=kx (k≠0) intersects with the line segment AB, write a value of k that satisfies this condition: ______.
|
-2 (The answer is not unique, as long as -3 ≤ k ≤ -1/2).
|
NULL
|
free_form
|
func396
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/396.png
|
As shown in the figure, the straight line y = kx + b (k and b are constants, and k≠0) passes through points A(-4, 0) and B(3, 2). Then the solution set of the inequality group 0 < kx + b < 2 with respect to x is ______.
|
-4 < x < -3
|
NULL
|
free_form
|
func397
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/397.png
|
In the Cartesian coordinate system, an ant starts from the origin O and moves continuously in the direction of 'up → right → down → right,' with each move being one unit in length. Its movement path is shown in the diagram. The ant reaches point A_1 on its first move, A_2 on its second move, and so on. Determine the coordinates of point A_2024.
|
(1012, 0)
|
NULL
|
free_form
|
func398
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/398.png
|
As shown in the figure, the line y=2x intersects the line y=kx+b (k≠±1) at point M(1,2). Then the solution set of the inequality 2x > kx + b is ______.
|
x > 1
|
NULL
|
free_form
|
func399
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/399.png
|
As shown in Figure 1, in △ABC, a moving point P starts at point A and moves uniformly along the broken line AB → BC → CA until it stops at point A. Let the distance traveled by point P be x, and the length of line segment AP be y. Figure 2 shows the approximate graph of the function relationship between y and x. Point E is the lowest point of the curve DF. The length of CD is ________.
|
√57
|
NULL
|
free_form
|
func4
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/4.png
|
As shown in the figure, point A lies on the graph of the inverse proportional function y = k/x. A vertical line segment is drawn from point A to the x-axis, and another to the y-axis. If the shaded area is 6, what is the value of k?
|
6
|
NULL
|
free_form
|
func40
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/40.png
|
Given that in the Cartesian coordinate plane, the vertices of triangle △ABC are A(0, 3), B(-2, -1), and C(2, 0). The line segment AB intersects the x-axis at point D. What are the coordinates of point D?
|
(-3/2, 0)
|
NULL
|
free_form
|
func400
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/400.png
|
As shown in the figure, P is a point on the hyperbolic function y = -k/x (k ≠ 0). A perpendicular line is drawn from point P to the x-axis at point A. Point B is on the negative half of the y-axis, and OB = 2OA. Connecting BP, if the area of △ABP is 3/2, then the value of k is ________.
|
1
|
NULL
|
free_form
|
func401
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/401.png
|
As shown in the figure, on a grid paper where each unit is 1, the triangles △A1A4A7, △A2A5A8, △A3A6A9, ... are all isosceles right triangles with their hypotenuses lying on the x-axis, and the hypotenuse lengths are 2, 4, 6, ... respectively. If the coordinates of A2, A4, A6, A8 are given as A1(2, 0), A2(1, 1), A3(0, 2), A4(-1, 3), then according to the pattern shown in the figure, the coordinates of A2024 are ______.
|
(2, -1012)
|
NULL
|
free_form
|
func402
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/402.png
|
In the rectangular coordinate system, square ABCD is placed as shown in the figure. Point A is on the line y = -3x, and point D is on the line y = kx. Then the value of k is ________.
|
-3/4
|
NULL
|
free_form
|
func403
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/403.png
|
As shown in the diagram, given the linear function y = kx + b, the solution set for kx + b < 0 is ______.
|
x < 2.5
|
NULL
|
free_form
|
func404
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/404.png
|
As shown in the figure, there is an inverse proportional function y = k/x (x < 0) with points A and B on its graph. The x-coordinates of points A and B are a and b respectively, and b = 2a. A horizontal line is drawn through point A parallel to the x-axis, and another horizontal line is drawn through point B parallel to the x-axis, intersecting at point C. Line OC is connected. If the area of △OBC is 2, then k = ________.
|
-8
|
NULL
|
free_form
|
func405
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/405.png
|
As shown in the figure, in the rectangular coordinate system, point A lies on the graph of the function y = k (x < 0, k ≠ 0). The line passing through point A and perpendicular to the y-axis intersects the x-axis at point B and the y-axis at point E. Points C and D lie on the x-axis, and BC is parallel to AD. If the area of quadrilateral ABCD is 3, then the value of k is ________.
|
-3
|
NULL
|
free_form
|
func407
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/407.png
|
The functions y=mx+n and y=kx have graphs in the same coordinate system as shown in the figure. Then, the solution set of the inequality mx+n>kx is ______.
|
x<-1
|
NULL
|
free_form
|
func408
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/408.png
|
As shown in the figure, a cargo ship encountered distress at point P and sent a distress signal to the Chinese Navy escort ship located at point Q, 60 nautical miles away. Use direction and distance to describe the position of the distressed cargo ship relative to the Chinese Navy escort ship ________.
|
25° south of west, 60 nautical miles away.
|
NULL
|
free_form
|
func409
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/409.png
|
In a plane rectangular coordinate system, the graphs of the linear functions y = ax and y = kx + 7 are as shown in the figure. The solution set of the linear inequality ax > kx + 7 is ______________________.
|
x > 2
|
NULL
|
free_form
|
func41
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/41.png
|
As shown in the figure, in rectangle ABCD, A(2,3), C(a,1), and BC is parallel to the x-axis.
(1) If one branch of the reciprocal function y = k / (x + 0) passes through points A and C, then a = ____.
(2) If one branch of the reciprocal function y = k / (x + 0) intersects rectangle ABCD, then the range of k is ________.
|
6 ≤ k ≤ 18
|
NULL
|
free_form
|
func410
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/410.png
|
As shown in the figure, the graph of the direct proportional function y = kx (k is a constant, and k ≠ 0) intersects with the linear function y = ax + b (a and b are constants, and a ≠ 0) at point P. Based on the graph, the solution set of the inequality kx < ax + b is __________.
|
x > -3
|
NULL
|
free_form
|
func411
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/411.png
|
We know that quadrilaterals are unstable. As shown in the figure, in a Cartesian coordinate plane, the square ABCD has its edge AB lying on the x-axis, and the coordinates of point A are (-1, 0). Fixing points A and B, the square is pushed along the indicated direction such that point D falls on the positive y-axis at point D. At this time, ∠ABC = 120°. Then the coordinates of point C are ________.
|
(2, √3)
|
NULL
|
free_form
|
func412
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/412.png
|
As shown in the figure, after encountering an accident at point A, a ship reports its position relative to the rescue ship located at point B, 100 nautical miles away, using direction and distance ____________.
|
In the south 25° west direction and the two ships are 100 nautical miles apart
|
NULL
|
free_form
|
func413
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/413.png
|
The growth of infants aged 1-6 months is rapid. If the birth weight of an infant is 3300g, the relationship between their weight y (g) and age x (months) can be approximately expressed as y = 3300 + 700x. When x is 3, the corresponding value of y is ________.
|
5400
|
NULL
|
free_form
|
func414
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/414.png
|
As shown in the figure, in the plane Cartesian coordinate system, the straight line y = kx + b (k ≠ 0, k and b are constants) passes through the point (4, 2). Then the solution set of the inequality kx + b < 2 with respect to x is ________.
|
x < 4
|
NULL
|
free_form
|
func416
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/416.png
|
As shown in the figure, the graphs of the linear functions y = -2x and y = kx + b intersect at point A(m, 6). Then the solution set of the inequality kx + b ≤ -2 is __________.
|
x ≤ -3
|
NULL
|
free_form
|
func417
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/417.png
|
As shown in the figure, the coordinates of vertex C of rhombus OABC are (4,3), vertex A is on the positive x-axis, and the graph of the inverse proportional function y=k/x passes through vertex B. The value of k is ________.
|
27
|
NULL
|
free_form
|
func418
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/418.png
|
As shown in the figure, point B is on the hyperbolic function y=8/x (x>0), point C is on the hyperbolic function y=k/x (x>0), and BC∥x-axis, AC⊥BC, with the foot of the perpendicular at C, meeting the x-axis at point A. If the area of △ABC is 5, then k = _________.
|
-2
|
NULL
|
free_form
|
func419
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/419.png
|
As shown in the figure, the straight line y=kx+b intersects the coordinate axes at points A(-3/2, 0) and B(0, 1). Then the solution set of the inequality kx-b<0 is __________.
|
x > -3/2
|
NULL
|
free_form
|
func42
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/42.png
|
The graphs of the inverse proportional functions y = k1/x, y = k2/x, y = k3/x are shown in the same coordinate system. What is the relationship of the sizes of k1, k2, and k3?
|
k1 < k2 < k3
|
NULL
|
free_form
|
func420
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/420.png
|
As shown in the figure, there is a maple leaf specimen. The entire leaf is broadly ovate with a slightly pointed tip and serrated edges. When placed in a rectangular coordinate system, if the coordinates of point A (the tip of the leaf) and point B (a point on the edge of the leaf) are (-1, -2) and (1, -1), respectively, then the coordinates of point C (the end of the petiole) are ________.
|
(3, 5)
|
NULL
|
free_form
|
func421
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/421.png
|
As shown in the figure, in the rectangular coordinate plane, △ABC is an isosceles right triangle, ∠ABC = 90°, AB = BC, and point C lies on the graph of the inverse proportional function y = k/x (x > 0). Given A(0,2) and B(1,0), the value of k is ______.
|
3
|
NULL
|
free_form
|
func422
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/422.png
|
As shown in the figure, it is known that the linear function y = k₁x + b and the reciprocal function y = k₂ / x intersect at point A(1/2, 8) and point B(4, m) in the first quadrant. Through point A, a line AP is drawn perpendicular to the x-axis at point P, and through point B, a line BQ is drawn perpendicular to the x-axis at point Q. If the area of ΔAOP is denoted as S₁ and the area of ΔBOQ is denoted as S₂, then S₁ __ S₂ (fill in '＞', '＜', or '=').
|
=
|
NULL
|
free_form
|
func423
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/423.png
|
As shown in the figure, the graph of the inverse proportional function y1 = k/x (k ≠ 0) intersects with the linear function y2 = mx + b (m ≠ 0) at points A(1,3) and B(3,n). Then:
(1) m = ____.
(2) A point C lies on the positive y-axis (not coinciding with the origin O), and AB = AC. The coordinates of point C are ______.
|
-2 (0, 3+√7) or (0, 3-√7)
|
NULL
|
free_form
|
func424
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/424.png
|
In a sealed container, a certain mass of a gas is enclosed. When its volume V changes, the gas's density ρ also changes. Within a certain range, ρ and V satisfy the relationship ρ = m/V (where m is a constant and m ≠ 0). The graph of this relationship is shown in the figure. When ρ is 2.4 kg/m³, the value of V is ______.
|
3
|
NULL
|
free_form
|
func425
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/425.png
|
The straight line y1 = kx(k ≠ 0) and the straight line y2 = ax + 4(a ≠ 0) are shown in the same Cartesian coordinate system. The solution set of the inequality kx < ax + 4 is ________.
|
x > -1
|
NULL
|
free_form
|
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