data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The diagonal of a square is 10 inches and the diameter of a circle is also 10 inches. Calculate:\n1. By how many square inches is the area of the circle greater than the area of the square?\n2. By how many inches is the circumference of the circle greater than the perimeter of ... | math | {
"style": "rule",
"ground_truth": "3.1"
} | {
"split": "train",
"index": 5800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"split": "train",
"index": 5801,
"solution": "1. **Identify the Triangle Type**: Given the side lengths $3, 4, 5$, we recognize $\\triangle ABC$ as a right triangle with $AB = 3$, $BC = 4$, and $AC = 5$.\n\n2. **Determine the Slope of $AB$**: The slope of line $AB$ is calculated as $\\frac{\\text{rise}}{\\text{... |
deepscaler | [
{
"role": "user",
"content": "The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 5802,
"solution": "1. Let $a$ and $b$ be two positive numbers such that $a > b$. According to the problem, the sum of these two numbers is $5$ times their difference. We can express this relationship mathematically as:\n \\[\n a + b = 5(a - b)\n \\]\n\n2. Expanding and rearrangi... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $13^{51}$ is divided by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 5803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 5804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 5805,
"solution": "1. **Define the problem**: We need to find the count of odd positive $3$-digit integers divisible by $3$ but not containing the digit $3$.\n\n2. **Set the constraints**:\n - The number $\\underline{ABC}$ must be divisible by $3$, which implies $A+B+C \\equiv 0 \\p... |
deepscaler | [
{
"role": "user",
"content": "The area of a triangle is $4 \\sqrt{21}$, its perimeter is 24, and the segment of the angle bisector from one of the vertices to the center of the inscribed circle is $\\frac{\\sqrt{30}}{3}$. Find the longest side of the triangle. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 5806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a mathematics quiz, there were $6x$ problems. Lucky Lacy missed $2x$ of them. What percent of the problems did she get correct? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66.67\\%"
} | {
"split": "train",
"index": 5807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $m \\gt 0$, $n \\gt 0$, and $m+2n=1$, find the minimum value of $\\frac{(m+1)(n+1)}{mn}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8+4\\sqrt{3}"
} | {
"split": "train",
"index": 5808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$, $B$, $C$, and $D$ are located on $\\overline{AB}$ such that $AB = 3AD = 6BC$. If a point is selected at random on $\\overline{AB}$, what is the probability that it is between $C$ and $D$? Express your answer as a common fraction.\n\n[asy]\n\ndraw((0,0)--(12,.0000000... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 5809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B ag... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 5810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cylinder with height $OO_1 = 12$ and a base radius $r = 5$. There are points $A$ and $B$ on the circumferences of the top and bottom bases respectively, with $AB = 13$. Find the distance between the axis $OO_1$ and line segment $AB$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} \\sqrt{3}"
} | {
"split": "train",
"index": 5811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser? L... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 5812,
"solution": "1. **Calculate the total cost of buying the appliance from the television advertiser:**\n The advertisement mentions three payments of $\\textdollar 29.98$ each and an additional one-time shipping and handling charge of $\\textdollar 9.98$. First, calculate the to... |
deepscaler | [
{
"role": "user",
"content": "If only one quarter of the earth's surface is not covered by bodies of water, and only one half of that exposed land area is inhabitable for humans (because of poles, deserts, etc.), what fraction of the earth's surface can humans live on? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 5813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $9^{x + 2} = 240 + 9^x$, then the value of $x$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 5814,
"solution": "1. **Rewrite the given equation using properties of exponents:**\n \\[\n 9^{x+2} = 240 + 9^x\n \\]\n We know that $9^{x+2} = 9^x \\cdot 9^2$. Since $9^2 = 81$, we can substitute:\n \\[\n 81 \\cdot 9^x = 240 + 9^x\n \\]\n\n2. **Isolate terms involving $... |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\\to0$ as $x\\to\\infty$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x)=\\frac1x"
} | {
"split": "train",
"index": 5815,
"solution": "\nTo tackle this problem, we want to find all functions \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) that satisfy:\n\n1. \\( f(xf(y)) = yf(x) \\) for all \\( x, y \\in \\mathbb{R}^+ \\).\n2. \\( \\lim_{x \\to \\infty} f(x) = 0 \\).\n\n### Step-by-step Solution:\n\n1.... |
deepscaler | [
{
"role": "user",
"content": "If $α$ and $β$ are acute angles, and they satisfy $\\cos α= \\frac{4}{5}$ and $\\cos (α+β)= \\frac{5}{13}$, calculate the value of $\\sin β$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"split": "train",
"index": 5816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number, find $(x+1)^2+2(x+1)(3-x)+(3-x)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$, the magnitude of $\\overrightarrow {a}$ is the positive root of the equation x^2+x-2=0, $|\\overrightarrow {b}|= \\sqrt {2}$, and $(\\overrightarrow {a}- \\overrightarrow {b})\\cdot \\overrightarrow {a}=0$, find t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 5818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14.7"
} | {
"split": "train",
"index": 5819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $PB$ given that $AP$ is a tangent to $\\Omega$, $\\angle PAB=\\angle PCA$, and $\\frac{PB}{PA}=\\frac{4}{7}=\\frac{PA}{PB+6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{11}"
} | {
"split": "train",
"index": 5820,
"solution": "Since $AP$ is a tangent to $\\Omega$, we know that $\\angle PAB=\\angle PCA$, so $\\triangle PAB \\sim \\triangle PCA$, so we get that $$\\frac{PB}{PA}=\\frac{4}{7}=\\frac{PA}{PB+6}$$ Solving, we get that $7PB=4PA$, so $$4(PB+6)=7PA=\\frac{49}{4}PB \\Rightarrow \\fr... |
deepscaler | [
{
"role": "user",
"content": " $a,b,c$ - are sides of triangle $T$ . It is known, that if we increase any one side by $1$ , we get new\na) triangle\nb)acute triangle\n\nFind minimal possible area of triangle $T$ in case of a) and in case b) Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 5821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A smaller regular tetrahedron is formed by joining the midpoints of the edges of a larger regular tetrahedron. Determine the ratio of the volume of the smaller tetrahedron to the volume of the larger tetrahedron. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 5822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 15. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 5823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A vessel with a capacity of 100 liters is filled with a brine solution containing 10 kg of dissolved salt. Every minute, 3 liters of water flows into it, and the same amount of the resulting mixture is pumped into another vessel of the same capacity, initially filled with water... | math | {
"style": "rule",
"ground_truth": "333.33"
} | {
"split": "train",
"index": 5824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never ex... | math | {
"style": "rule",
"ground_truth": "70 \\frac{1}{3}"
} | {
"split": "train",
"index": 5825,
"solution": "1. **Identify the range of possible distances traveled**: Ashley's maximum speed is 75 miles per hour, and she drove for 3 hours. Therefore, the maximum distance she could have traveled is:\n \\[\n 3 \\times 75 = 225 \\text{ miles}\n \\]\n\n2. **Determine the ... |
deepscaler | [
{
"role": "user",
"content": "Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:\n\n[list]\n\n [*]If at some stage the cards are in $m$ piles... | math | {
"style": "rule",
"ground_truth": "2^{n-2}"
} | {
"split": "train",
"index": 5826,
"solution": "\nFiona starts with \\( n \\) cards labeled from 1 to \\( n \\) lined up in increasing order. Her task is to perform a series of pile rearrangements until she consolidates them into a single pile. Given the specific operation she can perform, we explore how many dis... |
deepscaler | [
{
"role": "user",
"content": "An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Emplo... | math | {
"style": "rule",
"ground_truth": "191"
} | {
"split": "train",
"index": 5827,
"solution": "To solve this problem, we need to determine the maximum number of cables that can be used such that every employee can communicate with each other, under the constraint that cables can only connect a brand A computer to a brand B computer.\n\n#### Step 1: Understand... |
deepscaler | [
{
"role": "user",
"content": "Each of the $2500$ students at a university studies either Physics or Chemistry, and some study both. The number who study Physics is between $70\\%$ and $75\\%$ of the university population, and the number who study Chemistry is between $40\\%$ and $45\\%$. Let $m$ be the smal... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 5828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\\tfrac{S}{20}$.\n Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"split": "train",
"index": 5829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the co... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 5830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Joy has 50 thin rods, one each of every integer length from 1 cm through 50 cm, and rods with lengths 8 cm, 12 cm, and 25 cm are already placed on a table, determine the number of the remaining rods that can be chosen as the fourth rod to form a quadrilateral with positiv... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 5831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the CleverCat Academy, there are three skills that the cats can learn: jump, climb, and hunt. Out of the cats enrolled in the school:\n- 40 cats can jump.\n- 25 cats can climb.\n- 30 cats can hunt.\n- 10 cats can jump and climb.\n- 15 cats can climb and hunt.\n- 12 cats can ... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 5832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is an opaque bag containing 4 identical balls labeled with the numbers 1, 2, 3, and 4.\n(Ⅰ) If balls are drawn one by one without replacement twice, calculate the probability that the first ball drawn has an even number and the sum of the two balls’ numbers is divisible b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 5833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\dfrac{\\sqrt[5]{11}}{\\sqrt[7]{11}}$ expressed as 11 raised to what power? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{35}"
} | {
"split": "train",
"index": 5834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$X, Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets \n$X, Y, Z, X \\cup Y, X \\cup Z$ and $Y \\cup Z$ are $37, 23, 41, 29, 39.5$ and $33$ respectively. \nFind the average age of the people in set $X \\cup Y \\cup Z$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 5835,
"solution": "1. **Define Variables:**\n Let $X$, $Y$, and $Z$ represent the sums of the ages of the people in sets $X$, $Y$, and $Z$ respectively. Let $x$, $y$, and $z$ represent the numbers of people in sets $X$, $Y$, and $Z$ respectively.\n\n2. **Use Given Averages:**\n - ... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\sqrt {2}csinAcosB=asinC$.\n(I) Find the measure of $\\angle B$;\n(II) If the area of $\\triangle ABC$ is $a^2$, find the value of $cosA$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {10}}{10}"
} | {
"split": "train",
"index": 5836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the fraction with the smallest possible denominator that lies between \\(\\frac{7}{25}\\) and \\(\\frac{8}{25}\\). Provide all possible solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 5837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store received apples of the first grade worth 228 rubles and apples of the second grade worth 180 rubles. During unloading, the apples got mixed up. Calculations showed that if all the apples are now sold at a price 90 kopeks lower than the price per kilogram of first grade ... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 5838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible value of the sum $\\lvert x + 2\\rvert + \\lvert x + 4\\rvert + \\lvert x + 5\\rvert$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\min \\{a, b\\}$ denote the smaller value between $a$ and $b$. When the positive numbers $x$ and $y$ vary, $t = \\min \\left\\{ x, \\frac{y}{x^{2}+y^{2}} \\right\\}$ also varies. What is the maximum value of $t$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 5840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, \\) \\( b, \\) \\( c \\) be positive real numbers such that\n\\[\n\\left( \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a} \\right) + \\left( \\frac{b}{a} + \\frac{c}{b} + \\frac{a}{c} \\right) = 9.\n\\]\nFind the minimum value of\n\\[\n\\left( \\frac{a}{b} + \\frac{b}{c} ... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 5841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A computer screen shows a $98 \\times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white ... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 5842,
"solution": "Answer: $98$ .\nThere are $4\\cdot97$ adjacent pairs of squares in the border and each pair has one black and one white square. Each move can fix at most $4$ pairs, so we need at least $97$ moves. However, we start with two corners one color and two another, so at l... |
deepscaler | [
{
"role": "user",
"content": "Max picks two different cards without replacement from a standard 52-card deck. What is the probability that the cards are of different suits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{17}"
} | {
"split": "train",
"index": 5843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class, there are 30 students: honor students, average students, and poor students. Honor students always answer questions correctly, poor students always answer incorrectly, and average students alternate between correct and incorrect answers in a strict sequence. Each stu... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in quadrilateral $ABCD$, tangent to $\\overline{AB}$ at $P$ and to $\\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "647"
} | {
"split": "train",
"index": 5845,
"solution": "Just use the area formula for tangential quadrilaterals. The numbers are really big. A terrible problem to work on ($a, b, c,$ and $d$ are the tangent lengths, not the side lengths). \\[A = \\sqrt{(a+b+c+d)(abc+bcd+cda+dab)} = 105\\sqrt{647}\\] $r^2=\\frac{A^2}{a+b+... |
deepscaler | [
{
"role": "user",
"content": "There is a box containing red, blue, green, and yellow balls. It is known that the number of red balls is twice the number of blue balls, the number of blue balls is twice the number of green balls, and the number of yellow balls is more than seven. How many yellow balls are in... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$ , what is the smallest possible value of $k$ greater than $100$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 5847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \\mapsto 3x \\pmod{m}$ on the set $\\{ 1,2,\\dotsc ,m-1\\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\\mapsto 3\\to 9,\\mapsto ... | math | {
"style": "rule",
"ground_truth": "m \\equiv 2, 5, 7, 10 \\pmod{12}"
} | {
"split": "train",
"index": 5848,
"solution": "\nGiven a positive integer \\( m \\), not divisible by 3, we are interested in the permutation \\( x \\mapsto 3x \\pmod{m} \\) on the set \\(\\{ 1, 2, \\dotsc, m-1 \\}\\). The task is to determine for which integers \\( m \\) the number of cycles in this permutation... |
deepscaler | [
{
"role": "user",
"content": "Point \\( D \\) lies on side \\( CB \\) of right triangle \\( ABC \\left(\\angle C = 90^{\\circ} \\right) \\), such that \\( AB = 5 \\), \\(\\angle ADC = \\arccos \\frac{1}{\\sqrt{10}}, DB = \\frac{4 \\sqrt{10}}{3} \\). Find the area of triangle \\( ABC \\). Let's think step by... | math | {
"style": "rule",
"ground_truth": "15/4"
} | {
"split": "train",
"index": 5849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domains of functions $f(x)$ and $g(x)$ are both $\\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, find the sum of the values of $f(k)$ from $k=1$ to $k=22$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 5850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ p... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 5851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x,y),$ with $y<x\\le 100,$ are both $\\frac xy$ and $\\frac{x+1}{y+1}$ integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 5852,
"solution": "We know that $x \\equiv 0 \\mod y$ and $x+1 \\equiv 0 \\mod y+1$.\n Write $x$ and $ky$ for some integer $k$. Then, $ky+1 \\equiv 0\\mod y+1$. We can add $k$ to each side in order to factor out a $y+1$. So, $ky+k+1 \\equiv k \\mod y+1$ or $k(y+1)+1 \\equiv k \\mod y+... |
deepscaler | [
{
"role": "user",
"content": "On the extension of side $AD$ of rhombus $ABCD$, point $K$ is taken beyond point $D$. The lines $AC$ and $BK$ intersect at point $Q$. It is known that $AK=14$ and that points $A$, $B$, and $Q$ lie on a circle with a radius of 6, the center of which belongs to segment $AA$. Find... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of the polynomial $$(x+ \\frac {1}{ \\sqrt {x}})^{6}( \\sqrt {x}-1)^{10}$$, the constant term is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-495"
} | {
"split": "train",
"index": 5854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of pairs of integers $(x, y)$ such that $x^{2}+2y^{2}<25$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 5855,
"solution": "We do casework on $y$. If $y=0$, we have $x^{2}<25$, so we get 9 values of $x$. If $y=\\pm 1$, then $x^{2}<23$, so we still have 9 values of $x$. If $y=\\pm 2$, we have $x^{2}<17$, so we have 9 values of $x$. If $y=\\pm 3$, we have $x^{2}<7$, we get 5 values of $x$.... |
deepscaler | [
{
"role": "user",
"content": "Given that $13^{-1} \\equiv 29 \\pmod{47}$, find $34^{-1} \\pmod{47}$, as a residue modulo 47. (Give a number between 0 and 46, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 5857,
"solution": "1. **Calculate the area of the pan**: \n The pan has dimensions $20$ inches by $18$ inches. Therefore, the area of the pan is calculated by multiplying these dimensions:\n \\[\n \\text{Area of the pan} = 20 \\times 18 = 360 \\text{ square inches}\n \\]\n\n2.... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 210^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 5858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of 0-1 binary sequences of ten 0's and ten 1's which do not contain three 0's together. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24068"
} | {
"split": "train",
"index": 5859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the first year, ABC's stock price starts at $ \\$100 $ and increases $ 100\\% $. During the second year, its stock price goes down $ 25\\% $ from its price at the end of the first year. What is the price of the stock, in dollars, at the end of the second year? Let's thin... | math | {
"style": "rule",
"ground_truth": "\\$150"
} | {
"split": "train",
"index": 5860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximal size of a set of positive integers with the following properties:\n\n(1) The integers consist of digits from the set \\(\\{1,2,3,4,5,6\\}\\).\n\n(2) No digit occurs more than once in the same integer.\n\n(3) The digits in each integer are in increasing ord... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tatiana's teacher drew a $3 \\times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \\times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the ri... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Translate the graph of $y= \\sqrt {2}\\sin (2x+ \\frac {\\pi}{3})$ to the right by $\\phi(0 < \\phi < \\pi)$ units to obtain the graph of the function $y=2\\sin x(\\sin x-\\cos x)-1$. Then, $\\phi=$ ______. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac {13\\pi}{24}"
} | {
"split": "train",
"index": 5863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, \\dots, k_n$ for which\n\\[k_1^2 + k_2^2 + \\dots + k_n^2 = 2002?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5864,
"solution": "The problem asks for the maximum number of distinct positive integers $k_1, k_2, \\ldots, k_n$ such that the sum of their squares equals 2002. We need to find the largest possible $n$ for which this is possible.\n\n1. **Understanding the constraint**: The sum of squ... |
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9448"
} | {
"split": "train",
"index": 5865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\cos(\\omega x+\\frac{π}{4})$ $(\\omega>0)$ is transformed to an odd function by shifting it to the left by $\\frac{π}{3}$ units. Determine the minimum value of the real number $\\omega$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 5867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The largest divisor of a natural number \\( N \\), smaller than \\( N \\), was added to \\( N \\), producing a power of ten. Find all such \\( N \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 5868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $A B C$ is a triangle such that $A B=13, B C=15$, and $C A=14$. Say $D$ is the midpoint of $\\overline{B C}, E$ is the midpoint of $\\overline{A D}, F$ is the midpoint of $\\overline{B E}$, and $G$ is the midpoint of $\\overline{D F}$. Compute the area of triangle $E F ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{4}"
} | {
"split": "train",
"index": 5869,
"solution": "By Heron's formula, $[A B C]=\\sqrt{21(21-15)(21-14)(21-13)}=84$. Now, unwinding the midpoint conditions yields $[E F G]=\\frac{[D E F]}{2}=\\frac{[B D E]}{4}=\\frac{[A B D]}{8}=\\frac{[A B C]}{16}=\\frac{84}{16}=\\frac{21}{4}$."
} |
deepscaler | [
{
"role": "user",
"content": "The graph of \\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]has its foci at $(0,\\pm 4),$ while the graph of \\[\\frac{x^2}{a^2}-\\frac{y^2}{b^2} = 1\\]has its foci at $(\\pm 6,0).$ Compute the value of $|ab|.$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{65}"
} | {
"split": "train",
"index": 5870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $T$ has a circumference of $12\\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\\circ}$, what is the length, in inches, of segment $XZ$?\n\n[asy]\nsize(150);\ndraw(Circle((0,0),13),linewidth(1));\ndraw((-12,-5)--(-5,-12)--(12,5)--cycle,... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 5871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 8.5-meter ladder is leaning against a vertical wall. How many meters is its base from the wall if the ladder reaches 7.5 meters up the wall? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 5872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three Graces each had the same number of fruits and met 9 Muses. Each Grace gave an equal number of fruits to each Muse. After that, each Muse and each Grace had the same number of fruits. How many fruits did each Grace have before meeting the Muses? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the answer to this problem. Hexagon $ABCDEF$ is inscribed in a circle of radius 90. The area of $ABCDEF$ is $8n$, $AB=BC=DE=EF$, and $CD=FA$. Find the area of triangle $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2592"
} | {
"split": "train",
"index": 5874,
"solution": "Let $O$ be the center of the circle, and let $OB$ intersect $AC$ at point $M$; note $OB$ is the perpendicular bisector of $AC$. Since triangles $ABC$ and $DEF$ are congruent, $ACDF$ has area $6n$, meaning that $AOC$ has area $3n/2$. It follows that $\\frac{BM}{OM}=\... |
deepscaler | [
{
"role": "user",
"content": "If $a>0$ and $b>0,$ a new operation $\\Delta$ is defined as follows: $$a \\Delta b = \\frac{a^2 + b^2}{1 + ab}.$$ Calculate $(2 \\Delta 3) \\Delta 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6661}{2891}"
} | {
"split": "train",
"index": 5875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a fraction $\\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 5876,
"solution": "1. **Identify Special Fractions**: A fraction $\\frac{a}{b}$ is special if $a+b=15$ and $a, b$ are positive integers. The possible pairs $(a, b)$ are:\n - $(1, 14), (2, 13), (3, 12), (4, 11), (5, 10), (6, 9), (7, 8), (8, 7), (9, 6), (10, 5), (11, 4), (12, 3), (13,... |
deepscaler | [
{
"role": "user",
"content": "Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards in a row. Each leopard has a unique collar color. The two shortest leopards have inferiority complexes and demand to be placed at the ends of the row. Moreover, the two tallest leopards, w... | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"split": "train",
"index": 5877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $1-x+x^2-x^3+\\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and the $a_i$'s are constants. Find the value of $a_2$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "816"
} | {
"split": "train",
"index": 5878,
"solution": "Using the geometric series formula, $1 - x + x^2 + \\cdots - x^{17} = \\frac {1 - x^{18}}{1 + x} = \\frac {1-x^{18}}{y}$. Since $x = y - 1$, this becomes $\\frac {1-(y - 1)^{18}}{y}$. We want $a_2$, which is the coefficient of the $y^3$ term in $-(y - 1)^{18}$ (beca... |
deepscaler | [
{
"role": "user",
"content": "In the movie \"The Wandering Earth 2,\" there are many UEG (United Earth Government) mechanical devices that are drivable, operable, and deformable, all of which are from the leading Chinese engineering machinery brand - XCMG. Many of the hardcore equipment in the movie are not... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable but the boxes are distinguishable, with the condition that no box remains empty? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 5880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of rectangle \\(ABCD\\) is 48, and the diagonal is 10. On the plane where the rectangle is located, a point \\(O\\) is chosen such that \\(OB = OD = 13\\). Find the distance from point \\(O\\) to the vertex of the rectangle farthest from it. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 5881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store owner bought $1500$ pencils at $\\$ 0.10$ each. If he sells them for $\\$ 0.25$ each, how many of them must he sell to make a profit of exactly $\\$ 100.00$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 5882,
"solution": "1. **Calculate the total cost of the pencils**: The store owner bought $1500$ pencils at $0.10$ each. Therefore, the total cost is:\n \\[\n 1500 \\times 0.10 = 150\n \\]\n So, the total cost is $150$ dollars.\n\n2. **Determine the profit required**: The stor... |
deepscaler | [
{
"role": "user",
"content": "John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 5883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move you randomly select one of the three switches and click it. Let $m... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 5884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $3(72+76+80+84+88+92+96+100+104+108)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2700"
} | {
"split": "train",
"index": 5885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a linear function \\( f(x) \\). It is known that the distance between the points of intersection of the graphs \\( y = x^2 + 1 \\) and \\( y = f(x) \\) is \\( 3\\sqrt{2} \\), and the distance between the points of intersection of the graphs \\( y = x^2 \\) and \\( y = f(x... | math | {
"style": "rule",
"ground_truth": "\\sqrt{26}"
} | {
"split": "train",
"index": 5886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a pyramid S-ABCD, each vertex is colored with one color, and the two ends of the same edge are colored differently. If there are exactly 5 colors available, calculate the number of different coloring methods. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 5887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the areas of the three squares in the figure, what is the area of the interior triangle? [asy]\n/* AMC8 2003 #6 Problem */\ndraw((0,0)--(12,0)--(12,5)--cycle);\ndraw((12,0)--(17,0)--(17,5)--(12,5));\ndraw((0,0)--(0,-12)--(12,-12)--(12,0));\ndraw((0,0)--(-5,12)--(7,17)--(1... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $P(x)=(1+x+x^2+\\cdots+x^{17})^2-x^{17}$ has $34$ complex roots of the form $z_k = r_k[\\cos(2\\pi a_k)+i\\sin(2\\pi a_k)], k=1, 2, 3,\\ldots, 34,$ with $0 < a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_{34} < 1$ and $r_k>0.$ Given that $a_1 + a_2 + a_3 + a_4 + a_5 ... | math | {
"style": "rule",
"ground_truth": "482"
} | {
"split": "train",
"index": 5889,
"solution": "We see that the expression for the polynomial $P$ is very difficult to work with directly, but there is one obvious transformation to make: sum the geometric series:\n\\begin{align*} P(x) &= \\left(\\frac{x^{18} - 1}{x - 1}\\right)^2 - x^{17} = \\frac{x^{36} - 2x^{1... |
deepscaler | [
{
"role": "user",
"content": "On a straight segment of a one-way, single-lane highway, cars travel at the same speed and follow a safety rule where the distance from the back of one car to the front of the next is equal to the car’s speed divided by 10 kilometers per hour, rounded up to the nearest whole nu... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 5890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of the product \\( \\cos \\frac{\\pi}{15} \\cos \\frac{2 \\pi}{15} \\cos \\frac{3 \\pi}{15} \\cdots \\cos \\frac{7 \\pi}{15} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{128}"
} | {
"split": "train",
"index": 5891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 5892,
"solution": "1. **Identify the Polynomial**: We are given the polynomial $(x-2y)^{18}$ and asked to find the sum of the numerical coefficients of all the terms in its expansion.\n\n2. **Understanding the Sum of Coefficients**: The sum of the coefficients of a polynomial when all... |
deepscaler | [
{
"role": "user",
"content": "Determine both the ratio of the volume of the cone to the volume of the cylinder and the ratio of their lateral surface areas. A cone and a cylinder have the same height of 10 cm. However, the cone's base radius is half that of the cylinder's. The radius of the cylinder is 8 cm... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{116}}{40}"
} | {
"split": "train",
"index": 5893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 5894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $x,y,z\\geq 1$ satisfying \\[\\min(\\sqrt{x+xyz},\\sqrt{y+xyz},\\sqrt{z+xyz})=\\sqrt{x-1}+\\sqrt{y-1}+\\sqrt{z-1}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{c^2+c-1}{c^2}, \\frac{c}{c-1}, c\\right)"
} | {
"split": "train",
"index": 5895,
"solution": "The key Lemma is: \\[\\sqrt{a-1}+\\sqrt{b-1} \\le \\sqrt{ab}\\] for all $a,b \\ge 1$ . Equality holds when $(a-1)(b-1)=1$ .\nThis is proven easily. \\[\\sqrt{a-1}+\\sqrt{b-1} = \\sqrt{a-1}\\sqrt{1}+\\sqrt{1}\\sqrt{b-1} \\le \\sqrt{(a-1+1)(b-1+1)} = \\sqrt{ab}\\] by ... |
deepscaler | [
{
"role": "user",
"content": "A line passing through point P(1, 2) is tangent to the circle $x^2+y^2=4$ and perpendicular to the line $ax-y+1=0$. Find the value of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 5896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $EFGH$ has area $4032$. An ellipse with area $4032\\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of the rectangle $EFGH$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2016}"
} | {
"split": "train",
"index": 5897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\\text{4}}) = 10 = 12_{\\text{8}}$, and $g(2020) = \\t... | math | {
"style": "rule",
"ground_truth": "151"
} | {
"split": "train",
"index": 5898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 5899,
"solution": ""
} |
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