data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "$361+2(19)(6)+36=x$. Solve for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 3300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\\sin A= \\sqrt {3}a\\cos C$.\n\n(Ⅰ) Find angle $C$; \n(Ⅱ) If $c= \\sqrt {21}$, and $\\sin C+\\sin (B-A)=5\\sin 2A$, fi... | math | {
"style": "rule",
"ground_truth": "\\frac {5 \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 3301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:\n1. $g(b)=3b+1$ if $b$ is odd\n2. $g(b)=\\frac{b}{2}$ if $b$ is even.\nWhat is the smallest possible number of integers in th... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 3302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of sides of a regular polygon that approximates the area of its circumscribed circle with an error of less than 1 per thousand (0.1%)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"split": "train",
"index": 3303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. On... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 3304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 3305,
"solution": "1. **Determine the diameter of each pepperoni circle**: \n Given that six pepperoni circles fit exactly across the diameter of a 12-inch pizza, the diameter of each pepperoni circle is calculated by dividing the pizza's diameter by the number of pepperoni circles ... |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\dfrac{\\sin(2\\pi-\\alpha)\\cos(\\pi+\\alpha)\\cos(\\frac{\\pi}{2}+\\alpha)\\cos(\\frac{11\\pi}{2}-\\alpha)}{\\cos(\\pi-\\alpha)\\sin(3\\pi-\\alpha)\\sin(-\\pi-\\alpha)\\sin(\\frac{9\\pi}{2}+\\alpha)\\tan(\\pi+\\alpha)}$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 3306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the angle bisector \\(AD\\) divides side \\(BC\\) in the ratio \\(BD : DC = 2 : 1\\). In what ratio does the median from vertex \\(C\\) divide this angle bisector? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 3307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = 4,$ then find\n\\[\\begin{vmatrix} a & 7a + 3b \\\\ c & 7c +3d \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $6$, and suppose we have ${AB_6}+{C_6}={C}0_6$ and ${AB_6}+{BA_6}={CC_6}$. Find the three-digit number ${ABC}$. (Interpret $AB_6$ as a base-6 number with digits $A$ and $B$, not as $A$ times $B$. The other ex... | math | {
"style": "rule",
"ground_truth": "415"
} | {
"split": "train",
"index": 3309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3310,
"solution": "To find the minimum number of packs needed to buy exactly 90 cans of soda, we consider the sizes of the packs available: 6, 12, and 24 cans. We aim to use the largest packs first to minimize the total number of packs.\n\n1. **Using the largest pack (24 cans)**:\n ... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $T$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\\frac{\\sqrt{3}}{2}$, a line passing through the right focus $F$ with slope $k (k > 0)$ intersects $T$ at points $A$ and $B$. If $\\overline{AF} = 3\\overline{FB}$, determine the val... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 3311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$ . What is the value of $p - 3q$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 3312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(a_{0}, a_{1}, a_{2}, \\cdots, a_{n}\\) satisfies \\(a_{0}=\\sqrt{3}\\), \\(a_{n+1}=\\left\\lfloor a_{n}\\right\\rfloor + \\frac{1}{\\left\\{a_{n}\\right\\}}\\), where \\(\\left\\lfloor a_{n}\\right\\rfloor\\) and \\(\\left\\{a_{n}\\right\\}\\) represent the inte... | math | {
"style": "rule",
"ground_truth": "3024 + \\sqrt{3}"
} | {
"split": "train",
"index": 3313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $\\frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 3314,
"solution": "1. **Define the variables**: Let the shorter side of the rectangle be $x$ and the longer side be $y$. We know that $x < y$.\n\n2. **Expression for the diagonal**: The diagonal of the rectangle can be calculated using the Pythagorean theorem:\n \\[\n \\text{Diago... |
deepscaler | [
{
"role": "user",
"content": "The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is comp... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A(-4,1), B(-1,4)$ and $C(-1,1)$ are the vertices of $\\triangle ABC$. What will be the coordinates of the image of point A if $\\triangle ABC$ is rotated 90 degrees clockwise about the origin? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "(1, 4)"
} | {
"split": "train",
"index": 3316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral pyramid \\( P-ABCD \\), it is given that \\( AB \\parallel CD \\), \\( AB \\perp AD \\), \\( AB = 4 \\), \\( AD = 2\\sqrt{2} \\), \\( CD = 2 \\), and \\( PA \\perp \\) plane \\( ABCD \\) with \\( PA = 4 \\). Let \\( Q \\) be a point on the segment \\( PB \\... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 3317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 3318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest real number $a$ such that for any non-negative real numbers $x, y, z$ whose sum is 1, the inequality $a\\left(x^2 + y^2 + z^2\\right) + xyz \\geq \\frac{9}{3} + \\frac{1}{27}$ holds. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 3319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many nonzero terms are in the expansion of $$(x+4)(2x^2+3x+9)-3(x^3-2x^2+7x)?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 3320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $23=x^4+\\frac{1}{x^4}$, then what is the value of $x^2+\\frac{1}{x^2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Player A and player B are two basketball players shooting from the same position independently, with shooting accuracies of $\\dfrac{1}{2}$ and $p$ respectively, and the probability of player B missing both shots is $\\dfrac{1}{16}$.\n- (I) Calculate the probability that player... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{8}"
} | {
"split": "train",
"index": 3322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Adam and Bettie are playing a game. They take turns generating a random number between $0$ and $127$ inclusive. The numbers they generate are scored as follows: $\\bullet$ If the number is zero, it receives no points. $\\bullet$ If the number is odd, it receives one more ... | math | {
"style": "rule",
"ground_truth": "429"
} | {
"split": "train",
"index": 3323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a $6 \\times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw? Le... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 3324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A semicircular sheet of iron with a radius of 6 is rolled into the lateral surface of a cone. The volume of this cone is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{3}\\pi"
} | {
"split": "train",
"index": 3325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $C(1,1)$ and $D(8,6)$ are the endpoints of a diameter of a circle on a coordinate plane. Calculate both the area and the circumference of the circle. Express your answers in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{74}\\pi"
} | {
"split": "train",
"index": 3326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, find the ratio of the shorter side to the longe... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"split": "train",
"index": 3327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{2+4+6+\\cdots + 34}{3+6+9+\\cdots+51}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 3328,
"solution": "1. **Identify the sequences and their properties:**\n - The numerator is an arithmetic sequence with the first term $a = 2$, common difference $d = 2$, and last term $l = 34$.\n - The denominator is an arithmetic sequence with the first term $a = 3$, common diff... |
deepscaler | [
{
"role": "user",
"content": "A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, th... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 3329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all irreducible positive fractions which increase threefold if both the numerator and the denominator are increased by 12. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 3330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3331,
"solution": "1. **Calculate the total distance of the round trip**: \n The student drove $120$ miles to his home and $120$ miles back, so the total distance of the round trip is:\n \\[\n 120 \\text{ miles} + 120 \\text{ miles} = 240 \\text{ miles}\n \\]\n\n2. **Calculate... |
deepscaler | [
{
"role": "user",
"content": "Alan, Beth, Carla, and Dave weigh themselves in pairs. Together, Alan and Beth weigh 280 pounds, Beth and Carla weigh 230 pounds, Carla and Dave weigh 250 pounds, and Alan and Dave weigh 300 pounds. How many pounds do Alan and Carla weigh together? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 3332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On every kilometer marker along the highway between the villages of Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is indicated, and on the other side, the distance to Palkino is indicated. Borya noticed that the sum of all th... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 3333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Driving at a constant speed, Sharon usually takes $180$ minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving $\\frac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by $20$ miles per hour.... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 3334,
"solution": "1. **Define Variables:**\n Let the total distance from Sharon's house to her mother's house be $x$ miles. \n\n2. **Calculate Speeds:**\n Sharon's usual speed is $\\frac{x}{180}$ miles per minute, since she normally takes 180 minutes to cover $x$ miles.\n\n3. **D... |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, $PQ = 12$, $QR = 16$, and $PR = 20$. Point $X$ is on $\\overline{PQ}$, $Y$ is on $\\overline{QR}$, and $Z$ is on $\\overline{PR}$. Let $PX = u \\cdot PQ$, $QY = v \\cdot QR$, and $RZ = w \\cdot PR$, where $u$, $v$, and $w$ are positive and satisfy $u+v+w=3/4$... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 3335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral with $\\angle DAB =\\angle B DC = 90^o$ . Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$ , respectively, with $P$ lying in between $B$ and $Q$ . If $AD = 999$ and $PQ = 200$ then what is the sum... | math | {
"style": "rule",
"ground_truth": "799"
} | {
"split": "train",
"index": 3336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \\cap E| = |E \\cap F| = |F \\cap D| = 1$ and $D \\cap E \\cap F = \\emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\\{1,2,3,4,5,6,7,8\\}$. Find $M$ modulo ... | math | {
"style": "rule",
"ground_truth": "064"
} | {
"split": "train",
"index": 3337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x) \\) is a function defined on \\(\\mathbf{R}\\), for any \\( x, y \\in \\mathbf{R} \\), it always holds that\n\\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\\]\n\nFind \\( f(x) \\) and calculate the value of \\( f(\\sqrt{2014}) \\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "-1006"
} | {
"split": "train",
"index": 3338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many nonempty subsets of $\\{1,2,3, \\ldots, 12\\}$ have the property that the sum of the largest element and the smallest element is 13? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1365"
} | {
"split": "train",
"index": 3339,
"solution": "If $a$ is the smallest element of such a set, then $13-a$ is the largest element, and for the remaining elements we may choose any (or none) of the $12-2 a$ elements $a+1, a+2, \\ldots,(13-a)-1$. Thus there are $2^{12-2 a}$ such sets whose smallest element is $a$. A... |
deepscaler | [
{
"role": "user",
"content": "A function $f$ is defined recursively by $f(1)=f(2)=1$ and \\[f(n)=f(n-1)-f(n-2)+n\\]for all integers $n \\geq 3$. What is $f(2018)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 3340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $2^x$ be the greatest power of $2$ that is a factor of $144$, and let $3^y$ be the greatest power of $3$ that is a factor of $144$. Evaluate the following expression: $$\\left(\\frac15\\right)^{y - x}$$ Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 3341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: y^{2}=2px\\left(p \\lt 0\\right)$ passing through the point $A\\left(-2,-4\\right)$. \n$(1)$ Find the equation of the parabola $C$ and its directrix equation. \n$(2)$ A line passing through the focus of the parabola, making an angle of $60^{\\circ}$ wit... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}"
} | {
"split": "train",
"index": 3342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + 2}{2}"
} | {
"split": "train",
"index": 3343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88352"
} | {
"split": "train",
"index": 3344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\cos(α + \\frac{π}{3}) = -\\frac{\\sqrt{3}}{3}$, find the value of $\\sin α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + 3}{6}"
} | {
"split": "train",
"index": 3345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers \\(1, 2, 3, \\dots, 100, x\\) is \\(50x\\). What is \\(x\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5050}{5049}"
} | {
"split": "train",
"index": 3346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ellie's drawer of hair clips contains 4 red clips, 5 blue clips, and 7 green clips. Each morning, she randomly chooses one hair clip to wear for the day. She returns this clip to the drawer each evening. One morning, Kyne removes $k$ hair clips before Ellie can make her daily s... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3347,
"solution": "Before Kyne removes hair clips, Ellie has 4 red clips and $4+5+7=16$ clips in total, so the probability that she randomly chooses a red clip is $\frac{4}{16}$ which equals $\frac{1}{4}$. After Kyne removes the clips, the probability that Ellie chooses a red clip is ... |
deepscaler | [
{
"role": "user",
"content": "Read the following material: Expressing a fraction as the sum of two fractions is called expressing the fraction as \"partial fractions.\"<br/>Example: Express the fraction $\\frac{{1-3x}}{{{x^2}-1}}$ as partial fractions. Solution: Let $\\frac{{1-3x}}{{{x^2}-1}}=\\frac{M}{{x+1... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{21}"
} | {
"split": "train",
"index": 3348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?\n\nThe order of the selected cards does not matter: for example, selecting cards with numbers ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 3349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the coordinate plane, points $D(3, 15)$, $E(15, 0)$, and $F(0, t)$ form a triangle $\\triangle DEF$. If the area of $\\triangle DEF$ is 50, what is the value of $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{12}"
} | {
"split": "train",
"index": 3350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of $5,8$ and $17$ is equal to the mean of $12$ and $y$. What is the value of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 3351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex pentagon $ABCDE$, $\\angle A = \\angle B = 120^{\\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7\\sqrt{3}"
} | {
"split": "train",
"index": 3352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=2x^3-6x^2+a$ (where $a$ is a constant), the function has a maximum value of 3 on the interval $[-2, 2]$. Find the minimum value of $f(x)$ on the interval $[-2, 2]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-29"
} | {
"split": "train",
"index": 3353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral \\(ABCD\\), \\(\\angle DAC = 98^\\circ\\), \\(\\angle DBC = 82^\\circ\\), \\(\\angle BCD = 70^\\circ\\), and \\(BC = AD\\). Find \\(\\angle ACD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 45^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 3355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag, there are several balls, including red, black, yellow, and white balls. The probability of drawing a red ball is $\\frac{1}{3}$, the probability of drawing either a black or a yellow ball is $\\frac{5}{12}$, and the probability of drawing either a yellow or a white ba... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 3356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$ . $AN$ and $MB$ intersect at $X$ . Given that $\\an... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 3357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to 100 (so $S$ has $100^{2}$ elements), and let $\\mathcal{L}$ be the set of all lines $\\ell$ such that $\\ell$ passes through at least two points in $S$. Find, with proof,... | math | {
"style": "rule",
"ground_truth": "4950"
} | {
"split": "train",
"index": 3358,
"solution": "Let the lines all have slope $\\frac{p}{q}$ where $p$ and $q$ are relatively prime. Without loss of generality, let this slope be positive. Consider the set of points that consists of the point of $S$ with the smallest coordinates on each individual line in the set ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = \\frac{x + a}{x^2 + \\frac{1}{2}} \\), where \\( x \\) is a real number and the maximum value of \\( f(x) \\) is \\( \\frac{1}{2} \\) and the minimum value of \\( f(x) \\) is \\( -1 \\). If \\( t = f(0) \\), find the value of \\( t \\). Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 3359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $2001 \\cdot 2002 \\cdot 2003 \\cdot 2004 \\cdot 2005$ is divided by 19? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 3360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p(x) = x^4 + ax^3 + bx^2 + cx + d \\), where \\( a, b, c, d \\) are constants, and given that \\( p(1) = 1993 \\), \\( p(2) = 3986 \\), and \\( p(3) = 5979 \\). Calculate \\( \\frac{1}{4} [p(11) + p(-7)] \\). Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "5233"
} | {
"split": "train",
"index": 3361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $b$ is an even multiple of $7786$, find the greatest common divisor of $8b^2 + 85b + 200$ and $2b + 10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{1695}{4096}"
} | {
"split": "train",
"index": 3363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos (2x+\\varphi)$, where $|\\varphi| \\leqslant \\frac{\\pi}{2}$, if $f\\left( \\frac{8\\pi}{3}-x\\right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\\sin 2x$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 3364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAA$, while $BBAB$ is not such a seque... | math | {
"style": "rule",
"ground_truth": "172"
} | {
"split": "train",
"index": 3365,
"solution": "Let $a_n$ and $b_n$ denote, respectively, the number of sequences of length $n$ ending in $A$ and $B$. If a sequence ends in an $A$, then it must have been formed by appending two $A$s to the end of a string of length $n-2$. If a sequence ends in a $B,$ it must have... |
deepscaler | [
{
"role": "user",
"content": "The median of the set of numbers $\\{$12, 38, 45, $x$, 14$\\}$ is five less than the mean. If $x$ is a negative integer, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 3366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive four-digit integers are divisible by both 13 and 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 3367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?\n$\\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \\cdots, a_n$ is defined to be \\[\\left[\\frac{a_1^2 + a_2^2 + \\cdots + a_n^... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 3368,
"solution": "Let's first obtain an algebraic expression for the root mean square of the first $n$ integers, which we denote $I_n$ . By repeatedly using the identity $(x+1)^3 = x^3 + 3x^2 + 3x + 1$ , we can write \\[1^3 + 3\\cdot 1^2 + 3 \\cdot 1 + 1 = 2^3,\\] \\[1^3 + 3 \\cdot(... |
deepscaler | [
{
"role": "user",
"content": "Given that Adam has a triangular field ABC with AB = 5, BC = 8, and CA = 11, and he intends to separate the field into two parts by building a straight fence from A to a point D on side BC such that AD bisects ∠BAC, find the area of the part of the field ABD. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{21}}{4}"
} | {
"split": "train",
"index": 3369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $tanA=\\frac{2}{3}$, find the value of $\\cos A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{13}}{13}"
} | {
"split": "train",
"index": 3370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the triple integral \\( I = \\iiint_{G} \\frac{d x d y}{1-x-y} \\), where the region \\( G \\) is bounded by the planes:\n1) \\( x + y + z = 1 \\), \\( x = 0 \\), \\( y = 0 \\), \\( z = 0 \\)\n2) \\( x = 0 \\), \\( x = 1 \\), \\( y = 2 \\), \\( y = 5 \\), \\( z = 2 \\),... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 3371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a large library storage room, there are $1584$ boxes, each containing $45$ books. The library dean asks for these books to be repacked so that each new box contains $47$ books. How many books will be left over after repacking the books into as many full boxes of $47$ books e... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $[x]$ denote the greatest integer not exceeding the real number $x$. If\n\\[ A = \\left[\\frac{7}{8}\\right] + \\left[\\frac{7^2}{8}\\right] + \\cdots + \\left[\\frac{7^{2019}}{8}\\right] + \\left[\\frac{7^{2020}}{8}\\right], \\]\nwhat is the remainder when $A$ is divided b... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 3373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the interior numbers of the eighth row of Pascal's Triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 3374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different collections of 9 letters are there? A letter can appear multiple times in a collection. Two collections are equal if each letter appears the same number of times in both collections. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\binom{34}{9}"
} | {
"split": "train",
"index": 3375,
"solution": "We put these collections in bijections with binary strings of length 34 containing 9 zeroes and 25 ones. Take any such string - the 9 zeroes will correspond to the 9 letters in the collection. If there are $n$ ones before a zero, then that zero corresponds to the $(... |
deepscaler | [
{
"role": "user",
"content": "How many even divisors does $7!$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 3376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the following expression (accurate to 8 decimal places):\n$$\n16\\left(\\frac{1}{5}-\\frac{1}{3} \\times \\frac{1}{5^{3}}+\\frac{1}{5} \\times \\frac{1}{5^{5}}-\\frac{1}{7} \\times \\frac{1}{5^{7}}+\\frac{1}{9} \\times \\frac{1}{5^{9}}-\\frac{1}{11} \\times \\frac{1}{... | math | {
"style": "rule",
"ground_truth": "3.14159265"
} | {
"split": "train",
"index": 3377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c, x, y$, and $z$ be complex numbers such that $a=\\frac{b+c}{x-2}, \\quad b=\\frac{c+a}{y-2}, \\quad c=\\frac{a+b}{z-2}$. If $x y+y z+z x=67$ and $x+y+z=2010$, find the value of $x y z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5892"
} | {
"split": "train",
"index": 3378,
"solution": "Manipulate the equations to get a common denominator: $a=\\frac{b+c}{x-2} \\Longrightarrow x-2=$ $\\frac{b+c}{a} \\Longrightarrow x-1=\\frac{a+b+c}{a} \\Longrightarrow \\frac{1}{x-1}=\\frac{a}{a+b+c}$; similarly, $\\frac{1}{y-1}=\\frac{b}{a+b+c}$ and $\\frac{1}{z-1}... |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of triples of positive integers $(a, b, c)$ satisfying $a \\leq b \\leq c, \\quad \\operatorname{gcd}(a, b, c)=1, \\quad a b c=6^{2020}$. Compute the remainder when $N$ is divided by 1000. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "602"
} | {
"split": "train",
"index": 3379,
"solution": "Let $n=2020$. If we let $a=2^{p_{1}} \\cdot 3^{q_{1}}, b=2^{p_{2}} \\cdot 3^{q_{2}}, c=2^{p_{3}} \\cdot 3^{q_{3}}$, then the number of ordered triples $(a, b, c)$ that satisfy the second and third conditions is the number of nonnegative solutions to $p_{1}+p_{2}+p_{... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha \\in \\left(\\frac{\\pi}{2}, \\pi\\right)$ and $\\sin \\alpha = \\frac{4}{5}$, calculate the value of $\\sin 2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{25}"
} | {
"split": "train",
"index": 3380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the absolute value of the difference between the squares of 101 and 99? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 3381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following is equal to $110 \\%$ of 500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "550"
} | {
"split": "train",
"index": 3382,
"solution": "Solution 1: $10 \\%$ of 500 is $\\frac{1}{10}$ of 500, which equals 50. Thus, $110 \\%$ of 500 equals $500+50$, which equals 550. Solution 2: $110 \\%$ of 500 is equal to $\\frac{110}{100} \\times 500=110 \\times 5=550$."
} |
deepscaler | [
{
"role": "user",
"content": "A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll al... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 3383,
"solution": "First, we determine how far the small circle goes. For the small circle to rotate completely around the circumference, it must rotate $5$ times (the circumference of the small circle is $2\\pi$ while the larger one has a circumference of $10\\pi$) plus the extra rot... |
deepscaler | [
{
"role": "user",
"content": "Given a rectangle \\(ABCD\\), a circle intersects the side \\(AB\\) at points \\(K\\) and \\(L\\), and the side \\(CD\\) at points \\(M\\) and \\(N\\). Find the length of segment \\(MN\\) if \\(AK = 10\\), \\(KL = 17\\), and \\(DN = 7\\). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 3384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\angle A = 60^\\circ$, $\\angle E = 40^\\circ$ and $\\angle C = 30^\\circ$, then $\\angle BDC =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50^\\circ"
} | {
"split": "train",
"index": 3385,
"solution": "1. **Identify the Known Angles**: Given $\\angle A = 60^\\circ$, $\\angle E = 40^\\circ$, and $\\angle C = 30^\\circ$. We assume that points $A$, $B$, $C$, $D$, and $E$ are positioned such that these angles are part of triangles or other geometric figures, but the e... |
deepscaler | [
{
"role": "user",
"content": "The prime factorization of 2007 is $3^{2}\\times223$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy=2007$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \\( n \\) energy, he may expend \\( k \\) energy \\((0 \\leq k \\leq n)\\) to fill up to \\( n(k+1) \\) barrels with cider. What is the minima... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 3387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $g :\\mathbb{N} \\rightarrow \\mathbb{R}$ Such that $g(x)=\\sqrt{4^x+\\sqrt {4^{x+1}+\\sqrt{4^{x+2}+...}}}$ .\nFind the last 2 digits in the decimal representation of $g(2021)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 3388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the infinite series: $1 - \\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} - \\frac{1}{243} + \\frac{1}{729} - \\cdots$. Let $T$ be the limiting sum of this series. Find $T$.\n**A)** $\\frac{3}{26}$ \n**B)** $\\frac{15}{26}$ \n**C)** $\\frac{27}{26}$ \n**... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{26}"
} | {
"split": "train",
"index": 3389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\\cos C=3a\\cos B-c\\cos B$, $\\overrightarrow{BA}\\cdot \\overrightarrow{BC}=2$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 3390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2010a + 2014b = 2018$ and $2012a + 2016b = 2020$, what is the value of $a - b$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 3391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area of the Crescent Gemini. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17\\pi}{4}"
} | {
"split": "train",
"index": 3392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110880"
} | {
"split": "train",
"index": 3393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the following infinite product: $3^{\\frac{1}{3}} \\cdot 9^{\\frac{1}{9}} \\cdot 27^{\\frac{1}{27}} \\cdot 81^{\\frac{1}{81}} \\dotsm.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3^{\\frac{3}{4}}"
} | {
"split": "train",
"index": 3394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum possible value for $y$ in the equation $y = x^2 + 12x + 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-31"
} | {
"split": "train",
"index": 3395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $wz = 12-8i$, and $|w| = \\sqrt{13}$. What is $|z|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 3396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane of equilateral triangle $PQR$, points $S$, $T$, and $U$ are such that triangle $PQS$, $QRT$, and $RUP$ are also equilateral triangles. Given the side length of $PQR$ is 4 units, find the area of hexagon $PQURTS$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "16\\sqrt{3}"
} | {
"split": "train",
"index": 3397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $8$-cm-by-$8$-cm square is partitioned as shown. Points $A$ and $B$ are the midpoints of two opposite sides of the square. What is the area of the shaded region?\n\n[asy]\ndraw((0,0)--(10,0));\ndraw((10,0)--(10,10));\ndraw((10,10)--(0,10));\ndraw((0,0)--(0,10));\ndraw((0,0)-... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 3398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, denote by $\\tau(n)$ the number of positive integer divisors of $n$, and denote by $\\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\\varphi(n)+4 \\tau(n)=... | math | {
"style": "rule",
"ground_truth": "172"
} | {
"split": "train",
"index": 3399,
"solution": "We claim that $44,56,72$ are the only good numbers. It is easy to check that these numbers work. Now we prove none others work. First, remark that as $n=1,2$ fail so we have $\\varphi(n)$ is even, thus $n$ is even. This gives us $\\varphi(n) \\leq n / 2$. Now remark... |
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