id string | question string | image image | subquestions list | final_question string | response string | ground_truth string |
|---|---|---|---|---|---|---|
1 | Is this function continuous? | [
{
"answer": "The input values are discrete points: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.",
"question": "What are the allowed input values along the x-axis (n-axis)?"
},
{
"answer": "Yes, there are breaks between each of the points because the function is not defined for values between the integer inputs.",... | What fundamental property of this function's domain, which results in an undefined output for any non-integer input, prevents it from being classified as continuous? | The graph shows that the function is only defined for integer inputs on the n-axis, with no values existing for non-integer inputs (e.g., n=2.5 or n=3.5). A domain consisting of isolated, distinct points is known as a discrete domain. The absence of a continuous interval for the domain is the fundamental property that ... | The domain is discrete | |
2 | For what x does f reach its local maximum? | [
{
"answer": "Approximately x = 3",
"question": "At what x-value does the graph of f(x) reach its highest point in the interval shown?"
},
{
"answer": "Approximately 0",
"question": "What is the value of f'(x) at x = 3 according to the provided graph?"
},
{
"answer": "f'(x) = 0",
"que... | Using the provided graph, determine the value of the orange curve at the x-position corresponding to the peak of the blue curve. | The peak of the blue curve (f(x)) occurs at the x-coordinate where the function reaches its maximum value. Visually, this is at x = 3. At this x-coordinate, the corresponding value on the orange curve (f'(x)) is where it intersects the x-axis, which is 0.
Final Answer: 0 | 0 | |
3 | What is the limit of the as x approaches 1 from the left side? | [
{
"answer": "f(x) = x + 3",
"question": "What is the function defined for x values less than 1?"
},
{
"answer": "4",
"question": "What is the limit of f(x) = x+3 as x approaches 1?"
},
{
"answer": "No, there is a different function defined for x >= 1",
"question": "At x=1, is the fun... | Based on the graph, what is the absolute difference between the function's value at x=1 and its limit as x approaches 1 from the left? | Reasoning: The function's value at x=1 is determined by the formula for x ≥ 1, `f(1) = (1-2)^2 = 1`, which corresponds to the closed circle. The limit as x approaches 1 from the left is determined by the formula for x < 1, `lim(x→1⁻) (x+3) = 4`, which corresponds to the open circle. The absolute difference is |1 - 4| =... | 3 | |
4 | What is f(0.5)? | [
{
"answer": "m/N",
"question": "What is represented on the x-axis of the graph?"
},
{
"answer": "1/N",
"question": "What is represented on the y-axis of the graph?"
},
{
"answer": "m/N = 0.5",
"question": "According to the x-axis, what value should be used for calculation?"
},
{
... | Calculate the product of the ratio between the graph's absolute maximum value and the value indicated by the horizontal dashed line, and the total count of data points located strictly above that same line. | Step 1: The absolute maximum value on the y-axis is at the central peak, which is 0.5.
Step 2: The value indicated by the horizontal dashed red line on the y-axis is 0.2.
Step 3: The ratio is calculated as (Maximum value / Dashed line value) = 0.5 / 0.2 = 2.5.
Step 4: Count the data points strictly above the y=0.2 line... | 12.5 | |
5 | Estimate the area between the x-axis and the graph of f(x) = x^2 + 1 over the interval [0, 3] by using the three
rectangles | [
{
"answer": "1",
"question": "What is the width of each of the three rectangles used to approximate the area under the curve over the interval [0, 3]?"
},
{
"answer": "1, 2, and 3",
"question": "What are the x-coordinates at which the height of each rectangle is determined?"
},
{
"answer... | Using the provided function, calculate the total area of the shaded rectangles shown in the graph. | The graph visually represents a right Riemann sum to approximate the area under the curve f(x) = x^2 + 1 from x=0 to x=3.
- **Step 1: Identify Rectangle Properties from the Image:**
- There are three rectangles.
- The width of each rectangle is 1 unit (e.g., the first is from x=0 to x=1).
- The height of each rec... | 17 | |
6 | What is the center of this circle? | [
{
"answer": "0 radius and 180 degrees",
"question": "What are the coordinates of the lowest point of the circle in terms of radius and angle?"
},
{
"answer": "1 radius and 90 degrees",
"question": "What are the coordinates of the highest point of the circle in terms of radius and angle?"
},
... | Based on the polar plot, what are the Cartesian coordinates (x, y) of the circle's center, using the standard conversion from polar coordinates (r, θ)? | The center of the circle is located on the 90° axis, halfway between its lowest point (r=0) and its highest point (r=1). Therefore, the polar coordinates (r, θ) of the center are (0.5, 90°).
To convert to Cartesian coordinates (x, y):
x = r * cos(θ) = 0.5 * cos(90°) = 0.5 * 0 = 0
y = r * sin(θ) = 0.5 * sin(90°) = 0.5 ... | (0, 0.5) | |
7 | Is this function differentiable at each point? | [
{
"answer": "The graph has a 'V' shape with sharp corners.",
"question": "What is the general shape of the graph, and are there any sharp corners or cusps?"
},
{
"answer": "x = 1.5",
"question": "At what x-value does the sharp corner of the 'V' shape occur?"
},
{
"answer": "2",
"ques... | What is the sum of the slopes of the two distinct linear sections shown in the graph? | The first linear section on the left has a negative slope. Using points (0, 4) and (1, 2), the slope is calculated as (2 - 4) / (1 - 0) = -2.
The second linear section on the right has a positive slope. Using points (2, 2) and (3, 4), the slope is calculated as (4 - 2) / (3 - 2) = 2.
The sum of the slopes is (-2) + 2 =... | 0 | |
8 | What is the limit as x approaches 4? | [
{
"answer": "0.25",
"question": "What is the y-value of the open circle at x=4 on the graph?"
},
{
"answer": "0.25",
"question": "As x approaches 4 from the left, what y-value does the function approach?"
},
{
"answer": "0.25",
"question": "As x approaches 4 from the right, what y-va... | What is the product of the x-coordinate of the point discontinuity and the value of the function's limit at that x-coordinate? | 1. Visually identify the point discontinuity, which is the open circle on the graph.
2. Read its corresponding x-coordinate from the horizontal axis, which is 4.
3. Determine the limit of the function at x=4 by observing the y-value the curve approaches from both the left and right sides. This value, read from the v... | 1.0 | |
9 | How many unique function values does this function take? | [
{
"answer": "0.5",
"question": "What is the function value for x values between 1 and 7?"
},
{
"answer": "1.0",
"question": "What is the function value for x values between 8 and 11?"
},
{
"answer": "No",
"question": "Does the function take any values other than 0.5 and 1.0?"
},
... | What is the product of the slope of the function's non-constant segment and the sum of the two distinct constant values it maintains? | Reasoning:
1. The two distinct constant values shown on the y-axis (t) are 0.5 and 1.0.
2. The sum of these two values is 0.5 + 1.0 = 1.5.
3. The non-constant (sloped) segment starts at the point (x=7, t=0.5) and ends at the point (x=8, t=1.0).
4. The slope (rise over run) is calculated as (t2 - t1) / (x2 - x1) = (... | 0.75 | |
10 | What is the limit of the blue function as x approaches negative infinity? | [
{
"answer": "Blue",
"question": "In the graph, what color is the function we are interested in?"
},
{
"answer": "The y-value increases",
"question": "As x moves towards the left side of the graph (negative infinity), what happens to the y-value of the blue function?"
},
{
"answer": "No",... | Which colored function on the graph exhibits the property of its y-value increasing indefinitely as x approaches negative infinity? | To answer the question, one must examine the behavior of each colored function as the x-value moves to the far left of the graph (towards negative infinity).
- The **blue** curve flattens and approaches a y-value of 0.
- The **purple** line goes downwards, approaching negative infinity.
- The **green** curve rises upwa... | Green | |
11 | What is the value of y at x=0? | [
{
"answer": "0",
"question": "What is the y-value of the graph immediately to the left of x=0?"
},
{
"answer": "-4",
"question": "What is the y-value of the graph immediately to the right of x=0?"
},
{
"answer": "No",
"question": "Is the function continuous at x=0?"
},
{
"ans... | What is the absolute difference between the y-values the function approaches from the immediate left and immediate right of the vertical axis? | Step 1: Identify the y-value the function approaches from the immediate left of the vertical axis (x=0). This value is 0.
Step 2: Identify the y-value the function approaches from the immediate right of the vertical axis (x=0). This value is -4.
Step 3: Calculate the absolute difference between these two values: |0 - (... | 4 | |
12 | Is the function (f: R to R) injective? | [
{
"answer": "A function is injective if each element of the range is associated with at most one element of the domain, meaning no two elements in the domain map to the same element in the range.",
"question": "What is the definition of an injective function?"
},
{
"answer": "f(x) = x^3",
"quest... | Based on the visual test for injectivity demonstrated by the orange lines on the graph of the function f(x)=x³, what is the sum of the x-coordinates corresponding to the function's intersections with the lines y=2, y=0, and y=-2? | The graph displays the function y = x³. The question requires finding the x-coordinates for three given y-values and then summing them.
1. For y = 2, we solve 2 = x³, which gives x = ∛2.
2. For y = 0, we solve 0 = x³, which gives x = 0.
3. For y = -2, we solve -2 = x³, which gives x = ∛(-2) = -∛2.
4. The sum of the... | 0 | |
13 | Is this function continuous? | [
{
"answer": "m/N",
"question": "What is plotted on the x-axis of the graph?"
},
{
"answer": "1/N",
"question": "What is plotted on the y-axis of the graph?"
},
{
"answer": "Yes, the function has many discontinuities",
"question": "Does the function have any discontinuities or breaks?... | How many data points are located strictly above the horizontal dashed line but strictly below the plot's maximum peak? | The horizontal dashed line is at a 1/N value of 0.2. The plot's maximum peak is at a 1/N value of 0.5. The question requires counting the number of points for which 0.2 < 1/N < 0.5.
1. The two points with a 1/N value of 1/3 (approximately 0.333) meet this criterion.
2. The two points with a 1/N value of 1/4 (0.25) al... | 4 | |
14 | Which quadrant is this graph in? | [
{
"answer": "x is positive, y is positive",
"question": "What are the signs of x and y coordinates in Quadrant I?"
},
{
"answer": "x is negative, y is positive",
"question": "What are the signs of x and y coordinates in Quadrant II?"
},
{
"answer": "x is negative, y is negative",
"qu... | If the sign of a variable is represented by +1 for positive and -1 for negative, what is the product of the signs of `x`, `y`, and `dy/dx` for the function plotted in the graph? | The graph is plotted in the fourth quadrant, which is visually confirmed by the positive x-axis values (1, 2) and negative y-axis values (-1, -2).
1. In the fourth quadrant, `x` is positive, so its sign is +1.
2. In the fourth quadrant, `y` is negative, so its sign is -1.
3. The derivative is given as `dy/dx = -1/x^... | +1 | |
15 | What is the value of f(0)? | [
{
"answer": "g(x) = 0.5^x",
"question": "What is the function represented in the graph?"
},
{
"answer": "x = 0",
"question": "What is the input value of x for which we need to find the function's value?"
},
{
"answer": "1",
"question": "What is 0.5 raised to the power of 0?"
},
{... | Based on the function formula provided in the plot, what is the value of g(x) where the graphed curve crosses the vertical axis? | The function displayed on the graph is g(x) = 0.5^x. The vertical axis represents the line where x = 0. To find the intersection point, we need to calculate g(0), which is 0.5^0. Any non-zero number raised to the power of 0 is 1. This is also visually confirmed by observing that the green curve intersects the y-axis at... | 1 | |
16 | Is this an even function? | [
{
"answer": "f(x) = c",
"question": "What is the equation of the function shown in the image?"
},
{
"answer": "A constant value for all x",
"question": "What does 'c' represent in the context of the graph?"
},
{
"answer": "c",
"question": "What is the value of f(0) based on the graph... | Based on the visual information in the graph, what is the numerical value of f(-2) + c? | 1. From the graph, the horizontal line represents the function `f(x) = c`.
2. The line intersects the y-axis at the coordinate `y = 1.5`. Therefore, the constant `c` is equal to 1.5.
3. Since the function is constant for all values of `x`, `f(-2)` is also equal to `c`, which is 1.5.
4. The calculation required is `... | 3 | |
17 | What is the degree of this function? | [
{
"answer": "Parabola",
"question": "What is the general shape of the curve shown in the image?"
},
{
"answer": "Downwards",
"question": "Does the parabola open upwards or downwards?"
},
{
"answer": "Quadratic function",
"question": "What type of function is represented by a parabola... | Given that the graphed quadratic function passes through the origin, what is the value of its leading coefficient? | The image shows a parabola with its maximum (vertex) at (h, k) = (3, 18). The vertex form of a quadratic function is y = a(x - h)² + k. Substituting the vertex gives y = a(x - 3)² + 18. The function passes through the origin (0, 0). Substituting this point into the equation allows us to solve for the leading coefficien... | -2 | |
18 | What is the maximum value of y? | [
{
"answer": "x^2 + y^2 = 25",
"question": "What is the equation representing the circle in the image?"
},
{
"answer": "5",
"question": "What is the radius of the circle?"
},
{
"answer": "(0, 0)",
"question": "Where is the center of the circle located?"
},
{
"answer": "(0, 5)"... | Based on the graphical representation of the circle and its tangent line, what is the y-intercept of the tangent line? | The equation of the circle is x² + y² = 25 and the point of tangency is (3, -4). The slope of the radius to this point is -4/3. The slope of the tangent line is the negative reciprocal of the radius's slope, which is 3/4. Using the point-slope form, the equation of the tangent line is y - (-4) = (3/4)(x - 3), which sim... | -6.25 | |
19 | What is the value of f(-1)? | [
{
"answer": "f(x) = (x^3 + 3x^2 - 6x - 8) / 4",
"question": "What is the explicit formula for f(x) as given in the image?"
},
{
"answer": "-1",
"question": "What is the value of x that needs to be substituted into the formula to find f(-1)?"
},
{
"answer": "-1",
"question": "What is ... | Based on the mathematical formula shown in the image, what is the exact value of f(-1)? | Reasoning:
1. First, identify the formula for f(x) from the image: f(x) = (x³ + 3x² - 6x - 8) / 4.
2. Substitute x = -1 into the formula: f(-1) = ((-1)³ + 3(-1)² - 6(-1) - 8) / 4.
3. Calculate the terms in the numerator:
- (-1)³ = -1
- 3(-1)² = 3 * 1 = 3
- -6(-1) = 6
- -8
4. Sum the terms in the num... | 0 | |
20 | What is the maximum value of this function? | [
{
"answer": "1.0",
"question": "What is the highest y-value that the function reaches on the graph?"
},
{
"answer": "No",
"question": "Does the function ever exceed the y-value of 1.0 at any point on the graph?"
},
{
"answer": "Yes",
"question": "Is the function continuous and well-d... | At how many distinct points visible on the graph does the function's value equal the upper bound of the vertical axis? | The upper bound of the vertical axis is labeled as 1.0. The function, which is a continuous wave, reaches its maximum value, or peak, at exactly this y-value (1.0). By visually inspecting the graph, we can count three distinct peaks where the function touches the y=1.0 line.
Final Answer: 3 | 3 | |
21 | What is the period of the function? | [
{
"answer": "t",
"question": "What variable is represented on the horizontal axis of the graph?"
},
{
"answer": "S(t)",
"question": "What variable is represented on the vertical axis of the graph?"
},
{
"answer": "The period of the function",
"question": "What does the variable 'T' r... | What expression, using the labels on the horizontal axis, defines the period T for the function S(t)? | The image displays a graph of the function S(t) versus t. The label 'T' represents the period of the function. This period is visually aligned with the interval on the horizontal axis that starts at t0 and ends at t1. The length of this interval, which corresponds to the period T, is calculated by subtracting the start... | t1 - t0 | |
22 | N_m stands for number of monkeys. Which of the following describes the function correctly? | [
{
"answer": "The variable 'm' represents an independent variable.",
"question": "What does the variable 'm' represent on the x-axis?"
},
{
"answer": "Approximately 10.",
"question": "What is the approximate value of N_m when m is close to 0?"
},
{
"answer": "The rate of change decreases ... | At what approximate value of 'm' does the function N_m reach the midpoint between its value near m=0 and its asymptotic limit? | The initial value of N_m (at m ≈ 0) is approximately 10. The function's curve appears to approach a horizontal asymptote, with a limiting value of approximately 80. The midpoint between these two values is (10 + 80) / 2 = 45. By locating N_m = 45 on the y-axis and tracing it to the curve, the corresponding value on the... | 9 | |
23 | What is the biggest zero of this function? | [
{
"answer": "Three",
"question": "How many times does the function, as shown in the graph, intersect the x-axis?"
},
{
"answer": "Approximately -4, -2, and 2",
"question": "What are the approximate x-coordinates where the function intersects the x-axis?"
},
{
"answer": "2",
"question... | What is the product of all x-intercepts shown on the graph? | The graph intersects the x-axis at three x-coordinates: -4, -2, and 2. The product of these values is calculated as (-4) × (-2) × (2), which equals 16.
Final Answer: 16 | 16 | |
24 | Which functions are monotonic? | [
{
"answer": "A monotonic function is a function that is either entirely non-increasing or entirely non-decreasing.",
"question": "What is the definition of a monotonic function?"
},
{
"answer": "x^2",
"question": "What function does the blue curve represent?"
},
{
"answer": "x",
"que... | At the positive x-coordinate where the red and blue curves intersect, what is the y-value of the function that is monotonic and has a domain restricted to positive values? | 1. First, identify the intersection point of the red curve (y=x) and the blue curve (y=x^2) for a positive x-value. Visually, this occurs at x=1, y=1.
2. Next, identify the function that is both monotonic (always increasing or decreasing) over its domain and whose domain is restricted to positive x-values. This descr... | 0 | |
25 | The slope of the function at x=8 is ____ that at x=4 | [
{
"answer": "Increasing",
"question": "Is the slope of the function increasing, decreasing, or constant as x increases?"
},
{
"answer": "Accelerating",
"question": "Does the function's rate of change appear to be accelerating or decelerating as x increases?"
},
{
"answer": "Concave up",
... | Characterize the function's behavior by stating the sign (positive, negative, or zero) of its first derivative and its second derivative, assuming the horizontal axis represents the input variable. | The function's curve moves upwards from left to right, indicating an increasing function, which means its first derivative is positive. The curve bends upwards, indicating that it is concave up and its rate of change is accelerating, which means its second derivative is also positive.
Final Answer: First derivative: po... | First derivative: positive, Second derivative: positive | |
26 | Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray-and-white tiles shown below on the right.<image1>
What is the probability that the tiling will contain a large gray diamond in one of the smaller $2\times 2$ grids? Below is an example of one such tiling.
<image2> | [
{
"answer": "4",
"question": "How many $2 \times 2$ grids are there in a $3 \times 3$ grid?"
},
{
"answer": "Top-left, top-right, bottom-left, and bottom-right triangles all shaded.",
"question": "What combination of the four given tiles creates a gray diamond in a $2\times 2$ grid?"
},
{
... | Assuming each of the four basic tile orientations is equally probable, what is the ratio of the observed number of central gray diamond patterns in the lower 3x3 grid to its statistical expectation? | 1. **Observed Count:** First, identify the target pattern: a 2x2 sub-grid forming a central gray diamond. By visually inspecting the lower 3x3 grid, we find this pattern occurs exactly once, in the bottom-right 2x2 section. So, the observed count is 1.
2. **Statistical Expectation:**
- There are 4 possible tile o... | 64 | |
27 | Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is
<image1> | [
{
"answer": "3",
"question": "What is the side length of the original equilateral triangle ABC?"
},
{
"answer": "PA' = PA",
"question": "Since A is folded to A', what is the length of PA'?"
},
{
"answer": "QA' = QA",
"question": "Similarly, what is the length of QA'?"
},
{
"a... | If point P is the midpoint of the side on which it is located within the original equilateral triangle of side length 3, what is the length of the resulting crease segment PQ? | 1. The original triangle ABC is equilateral with side length 3. Point P is the midpoint of side AB, so AP = BP = 1.5.
2. The fold maps vertex A to A', with the crease being PQ. This implies that the length from any point on the crease to A is equal to its length to A'. Thus, PA' = PA = 1.5.
3. Consider the triangle ... | 1.5 | |
28 | The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.
<image1> | [
{
"answer": "-1/2",
"question": "What is the slope of the line segment AB?"
},
{
"answer": "sqrt(5)",
"question": "What is the length of AE in terms of grid units?"
},
{
"answer": "2",
"question": "What is the slope of the line segment CD?"
},
{
"answer": "1",
"question":... | What is the area of the triangle formed by the points A, C, and D, assuming the distance between adjacent grid points is one unit? | To determine the area of triangle ACD, we can assign coordinates to the vertices based on the grid.
1. Let's set the vertex A at the coordinate (0, 2).
2. Observing the grid, vertex C is located 3 units to the right of A on the same horizontal line. Therefore, the coordinates of C are (3, 2).
3. Vertex D is located ... | 6 square units | |
29 | A square of area $125 \mathrm{~cm}^{2}$ was divided into five parts of equal area - four squares and one L-shaped figure as shown in the picture. Find the length of the shortest side of the L-shaped figure.
<image1> | [
{
"answer": "25 cm^2",
"question": "What is the area of each of the five equal parts?"
},
{
"answer": "5 cm",
"question": "What is the side length of each of the four squares?"
},
{
"answer": "10 cm",
"question": "What is the total length of the outside of the L shape?"
},
{
... | Given that the figure is composed of five regions with equal area, and the side length of each small square is 5 cm, calculate the total length of all the black line segments shown. | 1. **Determine the area of each part:** The area of one small square is 5 cm * 5 cm = 25 cm². Since all five regions (the four squares and the L-shape) have equal area, the area of each part is 25 cm².
2. **Calculate the total area:** The total area of the entire large square is 5 parts × 25 cm²/part = 125 cm².
3. *... | 40 + 20√5 cm | |
30 | Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
<image1> | [
{
"answer": "2",
"question": "What is the distance between the centers of circles A and B?"
},
{
"answer": "1",
"question": "What is the distance between the center of circle C and the midpoint of the line segment connecting the centers of circles A and B?"
},
{
"answer": "$\\pi$",
"... | What is the ratio of the grey area to the total white area that falls inside the boundaries of circle C? | Reasoning:
1. Let SA be the grey shaded area and WA_C be the total white area inside circle C. The question asks for the ratio SA / WA_C.
2. From the sub-questions, we deduce the area of circle C is π. The area of the intersection of circle A and circle C is (π/3 - √3/4). By symmetry, the intersection of B and C has ... | (2π + 3√3) / (4π - 3√3) | |
31 | This line graph represents the price of a trading card during the first $6$ months of $1993$.
<image1>
The greatest monthly drop in price occurred during | [
{
"answer": "$2",
"question": "What was the price of the trading card in February?"
},
{
"answer": "$4",
"question": "What was the price of the trading card in March?"
},
{
"answer": "$1.50",
"question": "What was the price of the trading card in April?"
},
{
"answer": "$4.50... | Calculate the sum of the largest single month-over-month price increase and the magnitude of the largest single month-over-month price decrease depicted on the graph. | Reasoning:
1. Identify the month-over-month price changes from the graph:
* Jan ($2.50) to Feb ($2.00): Decrease of $0.50
* Feb ($2.00) to Mar ($4.00): Increase of $2.00
* Mar ($4.00) to Apr ($1.50): Decrease of $2.50
* Apr ($1.50) to May ($4.50): Increase of $3.00
* May ($4.50) to Jun ($... | $6.50 | |
32 | The diagram shows a cube with edges of length $12 \mathrm{~cm}$. An ant crawls from the point $P$ to the point $Q$ along the route shown. What is the length of the ant's path? <image1> | [
{
"answer": "5 edges",
"question": "How many edges does the ant crawl along?"
},
{
"answer": "12 cm",
"question": "What is the length of each edge of the cube?"
},
{
"answer": "12 cm * 5 = 60 cm",
"question": "What is the total length of the ant's path?"
}
] | Assuming each edge of the cube shown is 12 cm long, what is the total length, in cm, of the path traced by the arrows from P to Q? | The path traced by the ant from point P to point Q covers 5 distinct edges of the cube. Given that each edge has a length of 12 cm, the total distance is calculated by multiplying the number of edges by the length of one edge.
Calculation: 5 edges * 12 cm/edge = 60 cm.
Final Answer: 60 cm | 60 cm | |
33 | Which beetle has to fly away so that the remaining beetles have 20 dots altogether?
<image1> | [
{
"answer": "5",
"question": "How many beetles are in the image?"
},
{
"answer": "5",
"question": "What is the number of dots on the first beetle from the left?"
},
{
"answer": "5, 7, 4, 6, 3",
"question": "What is the number of dots on each of the beetles?"
},
{
"answer": "2... | What is the product of the number of dots on the beetle with the most dots and the number of dots on the beetle with the fewest dots? | The number of dots on each beetle from left to right must first be identified: 5, 7, 4, 6, and 3. The beetle with the most dots has 7, and the beetle with the fewest dots has 3. The product of these two values is 7 * 3 = 21.
Final Answer: 21 | 21 | |
34 | Caroline wants to write the numbers $1,2,3,4$ in the square $4 \times 4$ in such a way that every row and every column has each of the numbers. You see how she started. How many of the 4 numbers can be written in place of $x$?
<image1> | [
{
"answer": "1 and x",
"question": "What numbers are already in the first row?"
},
{
"answer": "x, 1, 3, and 2",
"question": "What numbers are already in the third column?"
},
{
"answer": "2, 3, and 4",
"question": "What numbers are missing from the first row?"
},
{
"answer":... | If the third column is composed of the set of numbers {x, 1, 3, 2} and follows the rule that each unique digit from 1 to 4 must appear in that column, what is the product of the value of 'x', the number in the second row's first column, and the number in the fourth row's second column? | Reasoning:
1. The question provides the constraint that the third column contains the numbers {x, 1, 3, 2} and must use each digit from 1 to 4 exactly once. The only digit missing from the set {1, 3, 2} is 4. Therefore, the value of 'x' must be 4.
2. The number in the second row's first column is visually identified ... | 32 | |
35 | In rectangle $ ABCD$, we have $ AB=8$, $ BC=9$, $ H$ is on $ \overline{BC}$ with $ BH=6$, $ E$ is on $ \overline{AD}$ with $ DE=4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \overline{GF}\perp\overline{AF}$. Find the length $ GF$.
<image1> | [
{
"answer": "AE = AD - DE = 9 - 4 = 5",
"question": "What is the length of AE?"
},
{
"answer": "Slope of AH = (BH - AB) / (0 - 8) = (6 - 0)/(0-8) = -6/8 = -3/4",
"question": "What is the slope of line AH?"
},
{
"answer": "Slope of CE = (DE - DC) / (0 - BC) = (4 - 0)/(0-9) = -4/9",
"q... | Utilizing all geometric constraints and numerical data from the image and its associated sub-questions, and assuming the larger bounding shape GFA is a right triangle with the right angle at F, calculate the area of triangle GCH. | **Reasoning:**
1. From the context of sub-question 1 ("AE = AD - DE = 9 - 4 = 5"), we deduce that the side length of the square ABCD, denoted as `s = AD`, is 9 units.
2. By establishing a coordinate system with F at the origin (0,0), and using the given lengths `BH=6` and `DE=4`, along with the side length `s=9`, the... | ** 20.25 | |
36 | In the diagram, lines $Q T$ and $R S$ are parallel and $P Q$ and $Q T$ are equal. Angle $S T Q$ is $154^{\circ}$. What is the size of angle $S R Q$ ? <image1> | [
{
"answer": "26 degrees",
"question": "What is the measure of angle PQT?"
},
{
"answer": "26 degrees",
"question": "What is the measure of angle TPQ?"
},
{
"answer": "128 degrees",
"question": "What is the measure of angle PTQ?"
},
{
"answer": "128 degrees",
"question": "... | Based on the geometric properties implied by the diagram's markings and the fact that angle PQT is 26 degrees, what is the ratio of the largest interior angle of the quadrilateral QRST to the smallest interior angle of the triangle PQT, rounded to two decimal places? | **Reasoning:**
1. **Analyze Triangle PQT:** The problem states `angle PQT = 26°`. Since QT is parallel to RS (indicated by arrows), the corresponding angles ∠PQT and ∠PRS are equal. Also, ∠TPQ and ∠RPS are the same angle. For triangle PQT, if we assume it is isosceles based on the larger triangle's properties (as impl... | 5.92 | |
37 | Monika wants to find a path through the labyrinth from 'Start' to 'Ziel'. She has to stick to the following rules: She is only allowed to move horizontally and vertically respectively. She has to enter every white circle exactly once but is not allowed to enter a black circle. In which direction does Monika have to mov... | [
{
"answer": "Only downwards.",
"question": "What are the possible directions Monika can move from 'Start' without entering a black circle?"
},
{
"answer": "Monika needs to avoid the black circles.",
"question": "After the first move, what are Monika's possible routes, considering the black circl... | Following the only possible path from 'Start' to 'Ziel' that visits every white circle exactly once while avoiding all black ones, what are the grid coordinates of the fifth circle visited on this route, assuming the top-left circle is at coordinate (1,1)? | The path must start at (1,1). The circle at (2,1) is a forced pass-through node, as it only has two white neighbors: (1,1) and (3,1). Since the path begins at (1,1), it must proceed (1,1) -> (2,1) -> (3,1). From (3,1), the only valid move is to (3,2). From (3,2), the only valid move is to (4,2).
The first five circles... | (4,2) | |
38 | The area of the wooden square equals $a$. The area of each wooden circle equals $b$. Three circles are lined up as shown in the picture. If we tie together the three circles with a thread as short as possible, without moving them, what is the area inside the thread?
<image1> | [
{
"answer": "Rounded rectangle",
"question": "What shape is formed if a thread is tied tightly around the three circles?"
},
{
"answer": "Diameter of two circles",
"question": "What is the length of the straight portion of the thread, based on the diagram?"
},
{
"answer": "It is equal to... | If S represents the area of the square and C represents the area of the circle, what is the total area of the shape on the right in terms of S and C? | The diagram illustrates a visual theorem about area. The total area of the composite shape on the right is the sum of the areas of the three circles it contains, plus the area of the space between them created by the wrapping. The diagram's premise establishes that this "in-between" area is equivalent to the area of th... | S + 3C | |
39 | In the diagram, $P Q R S$ is a square of side $10 \mathrm{~cm}$. The distance $M N$ is $6 \mathrm{~cm}$. The square is divided into four congruent isosceles triangles, four congruent squares and the shaded region.
<image1>
What is the area of the shaded region? | [
{
"answer": "2 cm",
"question": "What is the side length of each of the four congruent small squares?"
},
{
"answer": "16 cm^2",
"question": "What is the combined area of the four congruent small squares?"
},
{
"answer": "100 cm^2",
"question": "What is the area of the larger square ... | What is the difference, in square centimeters, between the total area of all unshaded shapes and the area of the central shaded shape? | The total area of the large square is 100 cm^2. The total area of the four small corner squares is 16 cm^2, and the total area of the four isosceles triangles is 36 cm^2.
- Total unshaded area = Area of small squares + Area of triangles = 16 cm^2 + 36 cm^2 = 52 cm^2.
- Shaded area = Area of large square - Total unshade... | 4 cm^2 | |
40 | Else has two machines R and S. If she puts a square piece of paper into machine $R$ it is rotated: <image1>
If she puts the piece of paper in machine $S$ it is printed on: <image2>
She wants to produce the following picture: <image3>
In which order does Else use the two machines so that she gets this picture?
<image4> | [
{
"answer": "Rotation",
"question": "What transformation does machine R apply to a square?"
},
{
"answer": "Prints a design",
"question": "What transformation does machine S apply to a square?"
},
{
"answer": "Rotated with dot in bottom-left",
"question": "What is the orientation of ... | Based on the demonstrated operations in the top two diagrams, what is the correct three-letter sequence for the machines with question marks to produce the final output from the given input? | Reasoning: Machine R rotates the square 90 degrees counter-clockwise. Machine S prints a standard, upright club symbol. To achieve the final state, the sequence must be RSR.
1. **R**: The initial square with the dot at the bottom-left is rotated, moving the dot to the top-left.
2. **S**: A standard upright club symbo... | RSR | |
41 | The Kangaroo Hotel has 30 floors numbered from 1 to 30 and each floor has 20 rooms numbered from 1 to 20. The code to enter the room is formed by joining the floor number with the room number, in that order. But this code can be confusing, as shown in the picture. Note that the code 101 is not confusing, as it can only... | [
{
"answer": "111",
"question": "What is the room code shown on the sign?"
},
{
"answer": "Floor 1, Room 11 and Floor 11, Room 1",
"question": "What are the possible floor and room number combinations that result in the code 111?"
},
{
"answer": "1 to 30",
"question": "What is the ran... | Considering the principle of ambiguity illustrated in the image, if a hotel has floors numbered 1 through 30 and rooms 1 through 20 on each floor, what is the sum of all possible three-digit codes that can be interpreted as two different valid floor-room combinations? | The image illustrates that a three-digit code `d1d2d3` is ambiguous if it can be interpreted as both (Floor `d1`, Room `d2d3`) and (Floor `d1d2`, Room `d3`). For a code to be ambiguous, both interpretations must be valid based on the hotel's constraints (Floors: 1-30, Rooms: 1-20).
1. **Analyze the conditions for amb... | 2970 | |
42 | Sepideh is making a magic multiplication square using the numbers 1, 2, 4, 5, 10, 20, 25, 50 and 100 . The products of the numbers in each row, in each column and in the two diagonals should all be the same. In the figure you can see how she has started. Which number should Sepideh place in the cell with the question m... | [
{
"answer": "20",
"question": "What is the product of the numbers in the first row?"
},
{
"answer": "50",
"question": "What number can be multiplied by 20 and 1 to give a product of 1000?"
},
{
"answer": "10",
"question": "If the product of each row, column and diagonal is 1000, what... | If the product of the three numbers in the top row is 1000, and this product remains constant for every row, column, and main diagonal, what value replaces the question mark? | 1. **Identify the magic product and complete the top row:** The problem states the product for each row is 1000. The top row contains the numbers 20 and 1. Therefore, the top-right cell must be 1000 / (20 * 1) = 50.
2. **Determine the center cell:** In a 3x3 multiplication magic square, the center cell is the cube ro... | 4 | |
43 | One of the five coins $A, B, C, D$ or $E$ shall be placed in an empty square so that there are exactly two coins in each row and in each column. Which coin should be moved?
<image1> | [
{
"answer": "Rows: 1, 1, 2, 2, 1. Columns: 2, 0, 1, 2, 1.",
"question": "How many coins are currently in each row and column?"
},
{
"answer": "Second row.",
"question": "Which row has no coin and needs two coins?"
},
{
"answer": "Fifth row.",
"question": "If coin 'C' is moved from th... | To achieve a state where every row and column contains exactly two coins, which two lettered coins must be moved, and what are their respective final coordinates (row, col)? | Reasoning:
1. First, we count the coins in each row and column. Rows have counts {2, 1, 3, 1, 3} from top to bottom. Columns have counts {3, 2, 2, 1, 2} from left to right. The goal is for all counts to be 2.
2. Two moves are required.
3. The first move must resolve the column with 3 coins (C1) and a column with 1 c... | Coin C to (4,4), and coin D to (2,3). | |
44 | The diagram shows a triangle and three circles whose centres are at the vertices of the triangle. The area of the triangle is $80 \mathrm{~cm}^{2}$ and each of the circles has radius $2 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded area? <image1> | [
{
"answer": "180 degrees",
"question": "What is the sum of the interior angles of the triangle?"
},
{
"answer": "The fraction corresponds to the angle at that vertex divided by 360 degrees",
"question": "What fraction of each circle is inside the triangle?"
},
{
"answer": "1/2",
"que... | If the area of the shaded region is equal to the combined area of the three circular sectors inside the triangle, what is the ratio of the triangle's total area to the area of one of the complete circles? | Let T be the area of the triangle and C be the area of one of the full circles. The three circles have the same radius.
The three portions of the circles inside the triangle are sectors whose angles are the interior angles of the triangle.
The sum of the interior angles of any triangle is 180°.
The combined area of the... | 1 | |
45 | A rectangular sheet of paper which measures $6 \mathrm{~cm} \times 12 \mathrm{~cm}$ is folded along its diagonal (Diagram A). The shaded areas in Diagram B are then cut off and the paper is unfolded leaving the rhombus shown in Diagram C. What is the length of the side of the rhombus? <image1> | [
{
"answer": "6 cm x 12 cm",
"question": "What are the dimensions of the rectangular sheet of paper?"
},
{
"answer": "sqrt(6^2 + 12^2) = sqrt(180) = 6*sqrt(5)",
"question": "When the rectangular sheet is folded along its diagonal, what is the length of the diagonal?"
},
{
"answer": "Two c... | Given an initial 6 cm by 12 cm rectangular paper undergoes the process shown, calculate the ratio of the perimeter of the final shape in Diagram C to the diagonal length of the original rectangle, providing the answer as a number rounded to two decimal places. | The process shown involves folding a rectangle along its diagonal, trimming the non-overlapping sections, and unfolding the remainder to form a rhombus (Diagram C).
1. **Calculate the side length (s) of the rhombus:** For a rectangle with length l=12 cm and width w=6 cm, the side length of the rhombus formed by this m... | 2.24 | |
46 | Consider these two geoboard quadrilaterals. Which of the following statements is true?
<image1> | [
{
"answer": "2 square units",
"question": "What is the area of quadrilateral I in square units?"
},
{
"answer": "2 square units",
"question": "What is the area of quadrilateral II in square units?"
},
{
"answer": "2 + 2sqrt(2)",
"question": "What is the perimeter of quadrilateral I i... | Given that both polygons have identical areas, calculate the value of ((Perimeter of polygon II) - 2)^2 - ((Perimeter of polygon I) - 2)^2. | Reasoning:
1. From the sub-questions, the perimeter of polygon I is P(I) = 2 + 2√2.
2. From the sub-questions, the perimeter of polygon II is P(II) = 2 + 2√5.
3. The first term of the expression is ((Perimeter of polygon II) - 2)^2. Substituting the value: ((2 + 2√5) - 2)^2 = (2√5)^2 = 4 * 5 = 20.
4. The second ter... | 12 | |
47 | The next window is a square of area $1 \mathrm{~m}^{2}$ and is composed of four triangles, which areas, indicated in the figure, follow the ratios $3 A=4 B$ and $2 C=3 D$. A fly is placed exactly at the point where these four triangles touch each other. The fly flies directly to the side closest to the window. How much... | [
{
"answer": "1 meter",
"question": "What is the side length of the square window in meters?"
},
{
"answer": "0.5 meters",
"question": "What is the distance from the center point to any side of the square?"
},
{
"answer": "3A/3 + 4B/4 + 2C/2 + 3D/3 = 1",
"question": "If 3A = 4B and 2C... | If the side length of the entire square is 1 meter and the central point is equidistant from each side, what is the combined area of regions A and C in square centimeters? | The total area of the square is 1 m * 1 m = 1 m². Since the central point is equidistant from each side, the square is divided into four triangles of equal area. Therefore, the area of region A is 0.25 m² and the area of region C is 0.25 m². Their combined area is 0.5 m². To convert this to square centimeters, we multi... | 5000 cm² | |
48 | The diagram shows a circle with centre $O$ as well as a tangent that touches the circle in point $P$. The arc $A P$ has length 20, the arc $B P$ has length 16. What is the size of the angle $\angle A X P$?
<image1> | [
{
"answer": "20:16 or 5:4",
"question": "What is the ratio of the arc lengths AP to BP?"
},
{
"answer": "5:4",
"question": "Since the ratio of arc lengths is proportional to the central angles, what is the ratio of angle AOP to angle BOP?"
},
{
"answer": "90-x",
"question": "If we le... | Using the provided values for the arc lengths, calculate the measure of angle BXO in degrees. | The ratio of arc lengths AP to BP is 20:16, which simplifies to 5:4. Consequently, the ratio of their corresponding central angles, ∠AOP to ∠BOP, is also 5:4. Since A, O, and B are collinear, they form a straight angle, meaning ∠AOP + ∠BOP = 180°. Solving this system of equations reveals that ∠BOP = 80°. The line segme... | 10 | |
49 | A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
<image1> | [
{
"answer": "(4/3)πr^3",
"question": "What is the formula for the volume of a sphere with radius r?"
},
{
"answer": "(1/3)πh(R^2 + Rr + r^2)",
"question": "What is the formula for the volume of a truncated cone with bottom radius R, top radius r, and height h?"
},
{
"answer": "(1/3)πh(R^... | Given the geometric configuration shown where a sphere is perfectly enclosed within a truncated cone, if the volume of the translucent cone is exactly double the volume of the sphere, what is the numerical value of the ratio of the cone's larger base radius to its smaller base radius? | Reasoning:
1. **Identify Shapes and Geometric Constraints from the Image:** The image shows a sphere perfectly inscribed within a truncated cone. The sphere touches the top and bottom circular bases. This visual configuration implies two key relationships:
* The height of the truncated cone (`h`) is equal to the... | (sqrt(13) - 1) / 2 | |
50 | $ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$
<image1> | [
{
"answer": "Square",
"question": "What is the shape of $ABCD$?"
},
{
"answer": "1:1",
"question": "What are the ratios of BM:MC and DN:NC?"
},
{
"answer": "AM = $a\\sqrt{5}/2$, AN = $a\\sqrt{5}/2$",
"question": "If the side length of the square is 'a', what are the lengths of AM and... | Assuming the vertices form a square and points M and N are the midpoints of their respective sides, what is the numerical value of 20 times the tangent of the angle θ? | Let the side length of the square be 'a'.
1. In right triangle ABM, by the Pythagorean theorem, $AM^2 = AB^2 + BM^2 = a^2 + (a/2)^2 = 5a^2/4$. So, $AM = a\sqrt{5}/2$. By symmetry, $AN = a\sqrt{5}/2$.
2. In right triangle MCN, $MN^2 = MC^2 + CN^2 = (a/2)^2 + (a/2)^2 = 2a^2/4 = a^2/2$.
3. Using the Law of Cosines in t... | 15 | |
51 | [
{
"answer": "1",
"question": "Subtract all purple cylinders. How many cylinders are left?"
},
{
"answer": "1",
"question": "Subtract all purple spheres. Subtract all green cylinders. How many spheres are left?"
},
{
"answer": "6",
"question": "Subtract all small brown shiny things. S... | What is the product of the count of matte objects that are not cubes and the count of objects that are neither purple nor shiny? | Reasoning:
1. First, identify and count the "matte objects that are not cubes". The single sphere is shiny, all other 8 objects are matte. Among these 8 matte objects, the ones that are not cubes are the 3 cylinders (two purple, one cyan). So, the first count is 3.
2. Second, identify and count the "objects that are ... | 18 | ||
52 | [
{
"answer": "0",
"question": "Add 2 big things. How many large reds exist?"
},
{
"answer": "4",
"question": "Subtract all big green cylinders. Subtract all big blue cylinders. How many objects are left?"
},
{
"answer": "9",
"question": "Add 3 tiny green matte cubes. How many objects ... | What is the result of taking the total number of cylinders, subtracting the count of large red objects, and then adding the count of small non-red objects? | - First, the total number of cylinders is identified: 2 red, 1 blue, 1 green, and 1 yellow, totaling 5 cylinders.
- Second, the count of large red objects is determined: There are 2 large red cylinders and 1 large red cube, totaling 3 objects.
- Third, the count of small non-red objects is identified: There is 1 small ... | 3 | ||
53 | [
{
"answer": "3",
"question": "Subtract 2 cylinders. How many cylinders are left?"
},
{
"answer": "5",
"question": "Subtract all brown cylinders. Subtract all green spheres. How many cylinders are left?"
},
{
"answer": "9",
"question": "Subtract all big yellow metal blocks. Subtract a... | Calculate the total number of metallic objects, subtract the number of small, matte-finished cylinders, and then add the count of any purple objects. What is the final result? | - First, count the metallic (shiny) objects: There is a large yellow cube, a large silver cube, a small golden cube, a large cyan cylinder, a green sphere, and a purple sphere. Total = 6.
- Second, count the small, matte-finished (non-shiny) cylinders: There is a red one, a yellow one, a small cyan one, a purple one, a... | 3 | ||
54 | [
{
"answer": "6",
"question": "Add 3 tiny brown matte cylinders. How many objects exist?"
},
{
"answer": "3",
"question": "Subtract 0 brown cubes. How many objects are left?"
},
{
"answer": "2",
"question": "Subtract all big cyan metal blocks. Subtract all green shiny things. How many... | If the quantity of matte cubes is doubled, every green object is removed, and a new tiny shiny cylinder is introduced, what is the total number of objects? | - **Step 1: Initial Scene Analysis:** The image contains 3 objects: one large purple matte cube, one small blue shiny cube, and one small green shiny cylinder.
- **Step 2: Double the quantity of matte cubes:** There is currently 1 matte cube. Doubling this means adding 1 more, bringing the total object count to 3 + 1 =... | 4 | ||
55 | [
{
"answer": "2",
"question": "Subtract 1 cubes. How many cubes are left?"
},
{
"answer": "4",
"question": "Add 1 small cubes. How many small cubes are left?"
},
{
"answer": "1",
"question": "Subtract all yellow spheres. How many brown cylinders are left?"
},
{
"answer": "1",
... | If all metallic items that are not cylinders are removed, along with the single matte sphere, how many objects remain in the scene? | The initial scene contains 8 objects.
1. Identify metallic items that are not cylinders: This includes the blue metallic sphere, the purple metallic cube, the yellow metallic cube, and the blue metallic cube (4 objects).
2. Identify the single matte sphere: This is the small gray sphere (1 object).
3. Calculate the ... | 3 | ||
56 | [
{
"answer": "9",
"question": "Add 1 yellow shiny spheres. How many objects exist?"
},
{
"answer": "6",
"question": "Subtract all spheres. How many objects are left?"
},
{
"answer": "7",
"question": "Subtract all small yellow shiny spheres. Subtract all tiny blue cubes. How many objec... | What is the final object count if we first subtract all shiny objects that are colored either yellow or cyan, and then add a number of new objects equal to the initial count of all green objects? | The initial scene contains 8 objects.
1. **Identify objects to subtract**: The shiny objects colored yellow or cyan are the shiny yellow sphere, the shiny yellow cylinder, and the shiny cyan cylinder. This is a total of 3 objects.
2. **Perform subtraction**: 8 (initial count) - 3 = 5 objects remaining.
3. **Identify... | 8 | ||
57 | [
{
"answer": "4",
"question": "Subtract 0 red cubes. How many objects are left?"
},
{
"answer": "3",
"question": "Add 2 large metal cubes. How many large metal cubes are left?"
},
{
"answer": "1",
"question": "Subtract all yellow spheres. Subtract all blue cylinders. How many spheres ... | If all non-cubic objects and all objects that are neither brown nor shiny are removed, and then a number of new purple spheres equal to the initial count of large metallic items is added, what is the final number of objects? | 1. **Initial state:** There are 4 objects: a large cyan shiny cube, a small brown cube, a small purple cube, and a brown sphere.
2. **First removal condition ("non-cubic objects"):** The brown sphere is removed. This leaves the 3 cubes.
3. **Second removal condition ("neither brown nor shiny"):** From the remaining ... | 3 | ||
58 | [
{
"answer": "4",
"question": "Add 4 large purple rubber balls. How many large purple rubber balls exist?"
},
{
"answer": "3",
"question": "Subtract 1 purple blocks. How many objects are left?"
},
{
"answer": "2",
"question": "Subtract all tiny brown rubber objects. Subtract all cylin... | What is the final object count after removing all purple items and all cylinders, and then adding one new sphere for each remaining metallic item? | 1. **Identify initial objects:** The scene contains 4 objects: a yellow sphere, a gold cube, a purple cube, and a brown cylinder.
2. **Perform removals:** Remove all purple items (1, the purple cube) and all cylinders (1, the brown cylinder). This removes a total of 2 unique objects.
3. **Calculate remaining objects... | 3 | ||
59 | [
{
"answer": "4",
"question": "Subtract all brown balls. Subtract all purple cylinders. How many balls are left?"
},
{
"answer": "6",
"question": "Add 1 rubber things. How many rubber things exist?"
},
{
"answer": "5",
"question": "Subtract all blocks. How many objects are left?"
},... | What is the final result when the total number of cubes is subtracted from the sum of large blue items and small shiny items? | First, identify and count the large blue items: there are two large blue spheres and one large blue cylinder, for a total of 3.
Next, identify and count the small shiny items: there is one small green sphere and one small brown cube, for a total of 2.
The sum of these two groups is 3 + 2 = 5.
Then, count the total numb... | 2 | ||
60 | [
{
"answer": "6",
"question": "Subtract all yellow cylinders. Subtract all red cubes. How many cylinders are left?"
},
{
"answer": "6",
"question": "Subtract all cubes. How many objects are left?"
},
{
"answer": "4",
"question": "Add 3 small rubber cylinders. How many small rubber cyl... | What is the result of subtracting the number of shiny cylinders from the total count of objects that are either large or blue? | First, identify all objects that are either large or blue:
- Large objects: 1 large green cylinder, 1 large red cylinder, 1 large blue cylinder. (Total: 3)
- Blue objects (including the large one already counted): 1 large cylinder, 1 medium cylinder, 1 small cylinder, 1 small cube. (Total: 4)
- The total unique count o... | 2 | ||
61 | [
{
"answer": "3",
"question": "Subtract 1 spheres. How many spheres are left?"
},
{
"answer": "6",
"question": "Add 5 tiny red shiny blocks. How many tiny red shiny blocks are left?"
},
{
"answer": "4",
"question": "Subtract all tiny blocks. Subtract all big red shiny spheres. How man... | Subtract the number of matte blocks from the number of shiny spheres, and then multiply the result by the total count of spheres. | 1. **Identify and count shiny spheres:** There are 3 shiny spheres (red, green, blue).
2. **Identify and count matte blocks:** There is 1 matte block (the dull red cube).
3. **Perform subtraction:** 3 (shiny spheres) - 1 (matte block) = 2.
4. **Identify and count all spheres:** There are 4 spheres in total (red, gr... | 8 | ||
62 | [
{
"answer": "3",
"question": "Add 2 small metal cubes. How many small metal cubes exist?"
},
{
"answer": "8",
"question": "Subtract 0 cyan cubes. How many objects are left?"
},
{
"answer": "4",
"question": "Subtract all tiny yellow shiny balls. Subtract all purple things. How many ob... | What is the product of the count of shiny, non-cubic objects and the count of matte, non-spherical objects? | Reasoning:
1. Identify the shiny objects: there are 5 (silver cube, blue cylinder, large blue sphere, small purple sphere, small yellow sphere).
2. Count the shiny, non-cubic objects by excluding the silver cube from this group: 5 - 1 = 4.
3. Identify the matte objects: there are 3 (large purple cube, small brown cu... | 12 | ||
63 | [
{
"answer": "1",
"question": "Subtract all large cylinders. How many cylinders are left?"
},
{
"answer": "4",
"question": "Add 3 purple metallic blocks. How many purple metallic blocks are left?"
},
{
"answer": "3",
"question": "Subtract all cyan cylinders. Subtract all purple cubes.... | Calculate the total number of matte objects that are not spheres, and divide this value by the total number of objects that are not cylinders. | Reasoning:
1. First, identify the total number of matte objects. There are 6 objects in total, and the purple cube is metallic. The other 5 objects (red cylinder, blue cylinder, brown cylinder, green sphere, brown sphere) are matte.
2. Next, identify the spheres among the matte objects. The green and brown spheres ar... | 1 | ||
64 | [
{
"answer": "8",
"question": "Add 3 brown blocks. How many objects exist?"
},
{
"answer": "5",
"question": "Subtract 0 purple spheres. How many objects are left?"
},
{
"answer": "4",
"question": "Subtract all tiny green shiny cylinders. Subtract all small red metallic cubes. How many... | If all non-blue cylinders are removed, and then a number of new items equal to the product of the existing cubes and spheres is added, what is the final object count? | 1. **Initial Count:** The image contains 5 objects in total.
2. **Removal Step:** Identify "non-blue cylinders". There is one green cylinder and one cyan cylinder. This makes a total of 2 objects to be removed.
- 5 (initial objects) - 2 (removed cylinders) = 3 objects remaining.
3. **Addition Step:** Calculate t... | 4 | ||
65 | [
{
"answer": "4",
"question": "Subtract 1 blue cylinders. How many objects are left?"
},
{
"answer": "1",
"question": "Subtract all blue spheres. Subtract all red cylinders. How many spheres are left?"
},
{
"answer": "7",
"question": "Add 2 large red shiny spheres. How many objects ex... | If all shiny objects were removed and a large matte blue cube was added, what would be the result of subtracting the total number of cubes from the total number of matte objects? | Reasoning:
1. **Identify shiny objects:** In the image, there is one red shiny cube and one blue shiny cylinder. These two objects are removed.
2. **Identify remaining objects:** The remaining objects are the gray matte sphere, the small red matte cube, and the large yellow matte cube.
3. **Add the new object:** A l... | 1 | ||
66 | [
{
"answer": "7",
"question": "Subtract 1 brown cylinders. How many objects are left?"
},
{
"answer": "5",
"question": "Subtract all cyan cylinders. Subtract all yellow spheres. How many cylinders are left?"
},
{
"answer": "6",
"question": "Add 5 big yellow rubber objects. How many bi... | What is the result of subtracting the number of metallic cylinders from the number of matte spheres? | First, identify the number of matte spheres. There is a large grey matte sphere and a large blue matte sphere, making a total of 2 matte spheres.
Second, identify the number of metallic cylinders. There is a small brown metallic cylinder and a small silver metallic cylinder, making a total of 2 metallic cylinders.
Fina... | 0 | ||
67 | [
{
"answer": "9",
"question": "Add 7 big shiny blocks. How many big shiny blocks are left?"
},
{
"answer": "1",
"question": "Subtract all tiny metal balls. How many balls are left?"
},
{
"answer": "6",
"question": "Subtract all red metal blocks. Subtract all large yellow cylinders. Ho... | What is the result of multiplying the number of unique shapes by the count of matte-finished objects, and then subtracting the total number of metallic objects? | 1. **Identify unique shapes:** The image contains cubes, a cylinder, and spheres. This totals 3 unique shapes.
2. **Count matte-finished objects:** The green cube, yellow cylinder, and cyan sphere have a matte finish. This totals 3 matte objects.
3. **Count metallic objects:** The silver cube, gold cube, purple sphe... | 5 | ||
68 | [
{
"answer": "1",
"question": "Subtract all large cylinders. How many cylinders are left?"
},
{
"answer": "5",
"question": "Add 4 blue cylinders. How many blue cylinders are left?"
},
{
"answer": "4",
"question": "Subtract all red cylinders. Subtract all blue spheres. How many cylinde... | After mentally removing all large metallic objects and all non-cylindrical shapes from the scene, how many matte cylinders remain? | 1. **Identify objects for removal based on the first condition ("large metallic objects"):** The large blue cylinder, the large green sphere, and the large purple sphere are all large and have a metallic finish.
2. **Identify objects for removal based on the second condition ("non-cylindrical shapes"):** The large gr... | 2 | ||
69 | [
{
"answer": "5",
"question": "Subtract all spheres. How many objects are left?"
},
{
"answer": "4",
"question": "Subtract all red cubes. Subtract all brown cylinders. How many cubes are left?"
},
{
"answer": "8",
"question": "Add 7 gray objects. How many gray objects are left?"
},
... | What is the product of the number of large non-cubic items and the difference between the total number of cubes and the total number of metallic spheres? | Reasoning: First, identify the total number of cubes (4: one large green, one small cyan, one small yellow, one small purple). Second, identify the total number of metallic spheres (2: one large green, one small cyan). The difference is 4 - 2 = 2. Third, identify the large, non-cubic objects (2: the large green sphere ... | 4 | ||
70 | [
{
"answer": "2",
"question": "Subtract all purple blocks. How many blocks are left?"
},
{
"answer": "1",
"question": "Subtract all blue balls. Subtract all yellow cylinders. How many balls are left?"
},
{
"answer": "8",
"question": "Add 2 purple cylinders. How many objects exist?"
... | After removing all rubber objects and then adding one matte cylinder for each remaining purple object, what is the final number of total objects? | - **Step 1:** Identify the initial number of objects. There are 6 objects in total.
- **Step 2:** Identify and count the rubber (matte) objects. There are 3: the large blue cube, the small blue cube, and the cyan sphere.
- **Step 3:** Remove the rubber objects from the total count. 6 - 3 = 3 objects remaining.
- **Step... | 5 |
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