prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to th... | 21 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA simplified model of a bicycle of mass \\(M\\) has two tires that each comes into contact with the ground at a point. The whe... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet the bisectors of the exterior angles at \\(B\\) and \\(C\\) of triangle \\(ABC\\) meet at \\(D\\). Then, if all measuremen... | 183 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt most, how many elements can a set have such that all elements are less than $102$ and the set does not contain the sum of a... | 51 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a bag of marbles, \\(\\frac{3}{5}\\) of the marbles are blue and the rest are red. If the number of red marbles is doubled ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per seco... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\overline{AB}$ be a diameter in a circle of radius $5\\sqrt{2}$. Let $\\overline{CD}$ be a chord in the circle that inte... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of the $2$ smallest prime factors of $2^{1024} - 1$.\n\nRemember to put your answer on its own line after \"Answe... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the least positive integer $k$ such that $n+k \\in S$ for every $n \\in S$, where $S = \\{ n : n3^n + (2n+1)5^n \\equi... | 21 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $x_1=5, x_2=401$, and \\[ x_n=x_{n-2}-\\frac{1}{x_{n-1}} \\] for every $3\\leq n \\leq m$, find the largest value of $m$.\n... | 2007 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCalculate the product \\(\\frac{1}{3} \\times \\frac{2}{4} \\times \\frac{3}{5} \\times \\cdots \\times \\frac{18}{20} \\times... | 211 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe parallelogram bounded by the lines $y = ax + c$, $y = ax + d$, $y = bx + c$, and $y = bx + d$ has an area of 18. The paral... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\\widehat... | 150 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of prime numbers $p$ such that $2p^4 - 7p^2 + 1$ is equal to the square of an integer.\n\nRemember to put your... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a membe... | 20 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEquilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\\overline{DE} \\perp \\overline{BC}$. Find t... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix points on a circle are given. Four of the chords joining pairs of the six points are selected at random. Find the probabil... | 92 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree mutually tangent spheres of radius \\(1\\) rest on a horizontal plane. A sphere of radius \\(2\\) rests on them. What is... | 72 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.\n\nRemember to put your answer ... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the radius of the inscribed circle of a triangle with sides of length \\(13\\), \\(30\\), and \\(37\\). Provide your answ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. What is the... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall a number \"prime-looking\" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers a... | 100 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide ... | 97 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ les... | 118 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe function $f$ satisfies the functional equation \\[ f(x) + f(y) = f(x + y) - xy - 1 \\] for every pair $x, y$ of real numbe... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nReal numbers \\(x\\) and \\(y\\) are chosen independently and uniformly at random from the interval \\((0,1)\\). Find the prob... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCharles has $5q + 1$ quarters and Richard has $q + 5$ quarters. Find the difference in their money in dimes. The answer should... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a map with a scale of \\( 400 \\) miles to an inch and a half, a certain estate is represented by a rhombus with a \\( 60^{... | 1253 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在集合$\\{0, 1, 2, \\ldots, 2014\\}$中，有多少个整数满足条件：$C^x_{2014} \\ge C^{999}_{2014}$？请计算满足该条件的整数个数。\n\nRemember to put your answer o... | 17 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLabel one disk \"1\", two disks \"2\", three disks \"3\", ..., fifty disks \"50\". Put these $1+2+3+ \\cdots+50=1275$ labeled ... | 415 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$ABC$ is a triangle with vertices $A=(0,0)$, $B=(36,15)$, and $C=(x,y)$ where both $x$ and $y$ are integers. What is the minim... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest integer $n$ such that $2^n + 65$ is equal to the square of an integer? Find the value of $n$.\n\nRemember ... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\triangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. $M$ is the midpoint of side $AB$, and $H$ is the foot of the altitud... | 65 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA cyclist travels at a constant speed of $22.0\\,\\text{km/hr}$ except for a $20$ minute stop. The cyclist’s average speed was... | 285 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest positive odd integer $n$ such that the product \n\\[2^{1/7} \\cdot 2^{3/7} \\cdot \\ldots \\cdot 2^{(2n+1... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In a... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers are there such that $\\left \\lfloor \\frac{m}{11} \\right \\rfloor = \\left \\lfloor \\frac{m}{10}... | 54 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many pairs \\((a,b)\\) from \\((1,2)\\), \\((3,5)\\), \\((5,7)\\), \\((7,11)\\), does the polynomial \\(P(x)=x^5+ax^4+... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHorses X, Y, and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are $3\\tex... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many positive integers $m$ is \\[ \\frac{2002}{m^2-2} \\] a positive integer? Write your answer as a number.\n\nRememb... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many primes \\( p \\), there exist unique integers \\( r \\) and \\( s \\) such that for every integer \\( x \\), the ... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(n\\) is a multiple of \\(4\\), find the sum \\(s=1+2i+3i^2+ \\ldots +(n+1)i^{n}\\), where \\(i=\\sqrt{-1}\\). The origin... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\triangle PAT$, $\\angle P=36^{\\circ}$, $\\angle A=56^{\\circ}$, and $PA=10$. Points $U$ and $G$ lie on sides $\\overlin... | 80 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJanet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at... | 265 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(x, y \\in (-2, 2)\\) and \\(xy = -1\\), find the minimum value of \\(u = \\frac{4}{4-x^2} + \\frac{9}{9-y^2}\\). The ori... | 17 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat are the last two digits of the decimal representation of \\(9^{8^{7^{\\cdot^{\\cdot^{\\cdot^{2}}}}}}\\)?\n\nRemember to p... | 21 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the equation system \\[\\begin{array}{rcl} f(x) + g(x) &=& 0 \\\\ f(x)-(g(x))^3 &=& 0 \\end{array}\\] has more than one rea... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(a+b=b-c=c-a=3\\), find the value of \\(a+b+c\\) in the form of \\(\\frac{k}{m}\\), where \\(\\frac{k}{m}\\) is a simplif... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe polynomial \\(P(x) = x^3 + ax^2 + bx + c\\) has the property that the mean of its zeros, the product of its zeros, and the... | -11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe base of an isosceles triangle is $\\sqrt{2}$. The medians to the legs intersect each other at right angles. If the area of... | 15 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn their base $10$ representation, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a s... | 17865 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enc... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn *iterative average* of the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), and \\(5\\) is computed in the following way. Arrang... | 25 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the area of \\(\\triangle ABC\\) is \\(64\\) square units and the geometric mean (mean proportional) between sides \\(AB\\)... | 17 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDoug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take ... | 15 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKeiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the end... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe distance light travels in one year is approximately \\( 5,870,000,000,000 \\) miles. If the distance light travels in \\( ... | 587 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA park is in the shape of a regular hexagon with sides of \\(2\\) km each. Starting at a corner, Alice walks along the perimet... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMarie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishe... | 150 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a parallelogram with $\\angle ABC=120^\\circ$, $AB=16$, and $BC=10$. Extend $\\overline{CD}$ through $D$ to $E$ ... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $y = \\sqrt{x^2 - 2x + 1} + \\sqrt{x^2 + 2x + 1}$, find the expression for $y$ in terms of $x$. The original answer is in t... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts... | 337 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the squ... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nVertex \\(E\\) of equilateral triangle \\(ABE\\) is in the interior of square \\(ABCD\\), and \\(F\\) is the point of intersec... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe fraction \\(\\frac{\\sqrt{a^2+x^2}-(x^2-a^2)/\\sqrt{a^2+x^2}}{a^2+x^2}\\) reduces to a simpler form. The reduced expressio... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n$ not exceeding $100$ is chosen such that if $n \\leq 50$, the probability of choosing $n$ is $p$, and if... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c, d, e, f, g,$ and $h$ be distinct elements in the set:\n\\[\n\\{-7, -5, -3, -2, 2, 4, 6, 13\\}.\n\\]\nFind the mi... | 34 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle and $AD$ be the bisector of the triangle ($D \\in (BC)$). Assume that $AB = 14$ cm, $AC = 35$ cm, and ... | 1181 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $4 \\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn engineer said he could finish a highway section in $3$ days with his present supply of a certain type of machine. However, ... | 18 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and \\[5x^2 + kx + 12 = 0\\] has at least one intege... | 78 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls?... | 393 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a function from the non-negative integers to the non-negative integers such that $f(nm) = n f(m) + m f(n)$, with th... | 36 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLines $L_1, L_2, \\dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each othe... | 4351 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the value of 100 times the expression $\\sqrt{65} - \\sqrt{63}$ rounded to the nearest integer?\n\nRemember to put you... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many pairs of real numbers $(x, y)$ are there such that \\( x^{4} - 2^{-y^{2}} x^{2} - \\lfloor x^{2} \\rfloor + 1 = 0 \\)... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many polynomial functions $f$ of degree $\\ge 1$ satisfy the following conditions?\n\\[\nf(x^2) = [f(x)]^2 = f(f(x))\n\\]\... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many ordered pairs of positive integers \\((m,n)\\), does \\(m \\cdot n\\) divide \\(2008 \\cdot 2009 \\cdot 2010\\)? ... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400... | 700 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is c... | 94 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRight triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and... | 47 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rock is dropped off a cliff of height \\(h\\). As it falls, a camera takes several photographs at random intervals. In each ... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$.\n\nRem... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSmall lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, an... | 225 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(a\\) and \\(b\\) be positive reals such that:\n\\[ a = 1 + \\frac{a}{b} \\]\n\\[ b = 3 + \\frac{4 + a}{b - 2} \\]\nThe ... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many integers $n$, are there four distinct real numbers satisfying the equation $|x^2 - 4x - 7| = n$?\n\nRemember to p... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two num... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which digit is **... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose \\(S\\) cans of soda can be purchased from a vending machine for \\(Q\\) quarters. Find the expression that describes ... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a trapezoid with $AB$ and $CD$ parallel, $\\angle D = 2 \\angle B$, $AD = 5$, and $CD = 2$. Find the length of $... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many non-negative integer triples $(m,n,k)$ are there such that $m^3 - n^3 = 9^k + 123$?\n\nRemember to put your answer on... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider sequences of positive real numbers of the form $x, 2000, y, \\ldots,$ in which every term after the first is 1 less t... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not nece... | 12 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe polynomial \\((x+y)^9\\) is expanded in decreasing powers of \\(x\\). The second and third terms have equal values when ev... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number $n$ is called a composite number if it can be written in the form $n = a \\times b$, where $a, b$ are positive inte... | 502 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\\triangle ABC$ parallel to $... | 30 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, a... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the average length in letters of a word in this question? Provide your answer as an integer.\n\nRemember to put your a... | 4 |
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