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math500-0200
[ "40" ]
{ "row_pos": 200, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/2086.json", "solution": "Multiply $5\\text{ yaps}=3\\text{ baps}$ by 14 to find that 70 yaps are equal to 42 baps. Then multiply $4\\text{ daps}=7\\text{ yaps}$ by 10 to find that $\\boxed{40}$ daps are equal to 70 yaps." }
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math500-0201
[ "66" ]
{ "row_pos": 201, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/149.json", "solution": "We are given that $N\\equiv 5\\pmod{8}$ and $N\\equiv 3\\pmod{6}$. We begin checking numbers which are 5 more than a multiple of 8, and we find that 5 and 13 are not 3 more than a multiple of 6,...
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math500-0202
[ "\\frac{2}{3}" ]
{ "row_pos": 202, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/291.json", "solution": "Alice has a $1/2$ chance of winning the game on her first turn. If she doesn't, then the probability that she wins the game on her second turn is $1/8,$ since she must not win on her first flip ($1/2$ chance...
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math500-0203
[ "12" ]
{ "row_pos": 203, "level": 3, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1555.json", "solution": "By the Remainder Theorem, to find the remainder, we set $x = -2.$ This gives us\n\\[(-1)^{611} + 3^{11} + (-3)^{11} + 3(-2)^2 + 1 = \\boxed{12}.\\]" }
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math500-0204
[ "0" ]
{ "row_pos": 204, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1510.json", "solution": "Since $|a| = 1,$ $a \\overline{a} = |a|^2,$ so $\\overline{a} = \\frac{1}{a}.$ Similarly, $\\overline{b} = \\frac{1}{b},$ $\\overline{c} = \\frac{1}{c},$ and $\\overline{d} = \\fr...
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math500-0205
[ "\\begin{pmatrix} -2 \\\\ -14 \\\\ -7 \\end{pmatrix}" ]
{ "row_pos": 205, "level": 3, "subject": "Precalculus", "unique_id": "test/precalculus/263.json", "solution": "Subtracting the equations $\\mathbf{A} \\begin{pmatrix} 3 \\\\ 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 4 \\\\ -3 \\end{pmatrix}$ and $\\mathbf{A} \\begin{pmatrix} -5 \\\\ 2 \\\\ 2 \\end{pma...
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math500-0206
[ "\\left( \\frac{3}{2}, -13 \\right)" ]
{ "row_pos": 206, "level": 2, "subject": "Precalculus", "unique_id": "test/precalculus/541.json", "solution": "We have that $x = 8 + 2t$ and $y = -1 + 3t.$ Isolating $t$ in $x = 8 + 2t,$ we find\n\\[t = \\frac{x - 8}{2}.\\]Then\n\\begin{align*}\ny &= -1 + 3t \\\\\n&= -1 + 3 \\cdot \\frac{x - 8}{2} \\\\\n&= \...
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math500-0207
[ "28" ]
{ "row_pos": 207, "level": 2, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/389.json", "solution": "There are 8 choices of toppings, and we need to choose 2 distinct toppings. This is represented by the number of 2-element subsets of an 8 -element set. We use the binomial co...
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math500-0208
[ "1" ]
{ "row_pos": 208, "level": 3, "subject": "Number Theory", "unique_id": "test/number_theory/1201.json", "solution": "We could do this by the divisibility rules, but that would be quite tedious. It's easier to note that a number divisible by $3, 4,$ and $5$ must be divisible by their product, $3 \\times 4 \\tim...
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math500-0209
[ "16" ]
{ "row_pos": 209, "level": 2, "subject": "Number Theory", "unique_id": "test/number_theory/234.json", "solution": "Since $90^3=729,\\!000$, $\\text{AB}$ is greater than 90. Therefore, $\\text{A}=9$. Since the ones digit of $\\text{AB}^3$ is 3, $\\text{AB}$ must be odd. The ones digit of $\\text{AB}^3$ is t...
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math500-0210
[ "10" ]
{ "row_pos": 210, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/378.json", "solution": "The total area of the rectangle is $3 \\times 4 =12$.\n\nThe total area of the shaded regions equals the total area of the rectangle (12) minus the area of the unshaded region.\n\nThe unshaded region i...
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math500-0211
[ "5" ]
{ "row_pos": 211, "level": 2, "subject": "Prealgebra", "unique_id": "test/prealgebra/1555.json", "solution": "Recall that $a^m\\div a^n=a^{m-n}$ for positive integers $m>n$ and $a^m\\cdot a^n=a^{m+n}$. Now we can write $5^5\\div5^4-5^3+5^2\\cdot5^1$ as $5^1-5^3+5^3$. Using the definition of subtraction and th...
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math500-0212
[ "56" ]
{ "row_pos": 212, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/2102.json", "solution": "Let the number of questions Frank answered correctly be $a$ and the number of questions he answered incorrectly be $b$. We have the two equations \\begin{align*}\na+b&=80\\\\\n5a-2b&=232\n\\end{align*} From...
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math500-0213
[ "2" ]
{ "row_pos": 213, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/102.json", "solution": "Since $F_{n + 1} = F_{n + 2} - F_n,$\n\\[\\frac{F_{n + 1}}{F_n F_{n + 2}} = \\frac{F_{n + 2} - F_n}{F_n F_{n + 2}} = \\frac{1}{F_n} - \\frac{1}{F_{n + 2}}.\\]Then\n\\begin{align*}\n...
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math500-0214
[ "110" ]
{ "row_pos": 214, "level": 5, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/765.json", "solution": "Let $x$ be the number of students taking physics, so the number in chemistry is $2x$. There are 15 students taking all three, and 30 students in both physics and calculus, me...
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math500-0215
[ "2" ]
{ "row_pos": 215, "level": 4, "subject": "Algebra", "unique_id": "test/algebra/907.json", "solution": "Rearranging $x^2 - x - 1= 0$ gives $x^2 = x + 1$. So, repeatedly substituting $x+1$ for $x^2$ gives us \\begin{align*}\nx^3 - 2x + 1 &= x(x^2)-2x + 1\\\\\n&=x(x+1) - 2x + 1\\\\\n&= x^2 + x -2x + 1\\\\\n&= x...
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math500-0216
[ "6" ]
{ "row_pos": 216, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/1436.json", "solution": "Multiply \\[\n3\\text{ Blinkets}=7\\text{ Drinkets}\n\\] by 8 to find that 24 Blinkets are equivalent to 56 Drinkets. Multiply \\[\n1\\text{ Trinkets}=4\\text{ Blinkets}\n\\] by 6 to find that $\\b...
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math500-0217
[ "11,\\! 111,\\! 111,\\! 100" ]
{ "row_pos": 217, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/417.json", "solution": "If a number is divisible by 450, then it must be divisible by all divisors of 450, including 9 and 50.\n\nFor a number to be divisible by 9, the sum of its digits must be divisible by 9. Since a ...
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math500-0218
[ "(-1,6)" ]
{ "row_pos": 218, "level": 2, "subject": "Algebra", "unique_id": "test/algebra/864.json", "solution": "Applying the midpoint formula gives us $$\\left(\\frac{-5+3}{2},\\frac{5+7}{2}\\right)=\\boxed{(-1,6)}.$$" }
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math500-0219
[ "\\frac{1}{2}" ]
{ "row_pos": 219, "level": 4, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/282.json", "solution": "The first remainder is even with probability $2/6=1/3$ and odd with probability 2/3. The second remainder is even with probability $3/6=1/2$ and odd with probability 1/2. Th...
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math500-0220
[ "\\frac{16}{27}" ]
{ "row_pos": 220, "level": 5, "subject": "Prealgebra", "unique_id": "test/prealgebra/1961.json", "solution": "At the end of the first mile, there will be $\\frac{2}{3}$ of the initial milk in the bucket. Each additional mile multiplies this amount by $\\frac{2}{3}$. Thus, when he arrives at home at the end of...
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math500-0221
[ "900" ]
{ "row_pos": 221, "level": 3, "subject": "Number Theory", "unique_id": "test/number_theory/89.json", "solution": "$240=2^4\\cdot3\\cdot5=2^3(2\\cdot3\\cdot5)$. For $240k$ to be a perfect cube (and not a perfect square), $k$ must be at least $2^2\\cdot3^2\\cdot5^2=\\boxed{900}$." }
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math500-0222
[ "54" ]
{ "row_pos": 222, "level": 4, "subject": "Geometry", "unique_id": "test/geometry/547.json", "solution": "Since $AB \\parallel DE,$ we know that $\\angle A = \\angle E$ and $\\angle B = \\angle D.$ That works out nicely, since that means $\\triangle ABC \\sim EDC.$ If $BD = 4BC,$ that means $CD = BD - BC = 3BC...
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math500-0223
[ "14" ]
{ "row_pos": 223, "level": 3, "subject": "Number Theory", "unique_id": "test/number_theory/183.json", "solution": "Rewriting $\\frac{4321}{5^7\\cdot2^8}$ as a decimal with a denominator of $5^8\\cdot2^8=10^8$, we have \\[ \\frac{4321}{5^7\\cdot2^8}\\cdot\\frac{5^{1}}{5^{1}}=\\frac{4321\\cdot5}{10^8}=\\frac{21...
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math500-0224
[ "\\frac{9}{256}" ]
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math500-0225
[ "11" ]
{ "row_pos": 225, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/2159.json", "solution": "Since \\begin{align*}\n(3x-2)(4x+1)-(3x-2)4x+1 &=(3x-2)(4x+1-4x)+1 \\\\\n&=(3x-2) \\cdot 1 +1 =3x-1,\n\\end{align*} when $x=4$ we have the value $3 \\cdot 4 -1 =\\boxed{11}$." }
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math500-0226
[ "3" ]
{ "row_pos": 226, "level": 3, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/71.json", "solution": "Since both the balls and boxes are indistinguishable, we can arrange them with 5 in one and 0 in the other, 4 in one and 1 in the other, or 3 in one and 2 in the other, for a t...
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math500-0227
[ "\\text{(C)}" ]
{ "row_pos": 227, "level": 3, "subject": "Precalculus", "unique_id": "test/precalculus/819.json", "solution": "In cylindrical coordinates, $z$ simply denotes the $z$-coordinate of a point. Thus, for a fixed $z$-coordinate $c,$ all the points lie on a plane that is parallel to the $xy$-plane. The answer is $...
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math500-0228
[ "288 \\pi" ]
{ "row_pos": 228, "level": 4, "subject": "Precalculus", "unique_id": "test/precalculus/1056.json", "solution": "Let $P = (a,b,c).$ Then the distance from $P$ to the plane $x - z = 0$ is\n\\[d_1 = \\frac{|a - c|}{\\sqrt{1^2 + (-1)^2}} = \\frac{|a - c|}{\\sqrt{2}}.\\]The distance from $P$ to the plane $x - 2y ...
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math500-0229
[ "90" ]
{ "row_pos": 229, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/2057.json", "solution": "Yann can order 10 different dishes. After he has chosen a dish, Camille has 9 choices left for her dish, because she won't order the same dish as Yann did. Thus there are a total of $10\\cdot 9 = \\...
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math500-0230
[ "\\frac{16}{5}" ]
{ "row_pos": 230, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/986.json", "solution": "By Cauchy-Schwarz,\n\\[(a^2 + b^2 + c^2 + d^2)(1 + 1 + 1 + 1) \\ge (a + b + c + d)^2.\\]Thus, $(16 - e^2)(4) \\ge (8 - e)^2.$ This simplifies to $16e - 5e^2 \\ge 0,$ or $e(16 - 5e)...
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math500-0231
[ "10" ]
{ "row_pos": 231, "level": 4, "subject": "Algebra", "unique_id": "test/algebra/1578.json", "solution": "Let Amy's, Ben's, and Chris's ages be $a$, $b$, and $c$, respectively. We have the equations \\begin{align*} \\tag{1}\n\\frac{a+b+c}{3}=6 \\Rightarrow a+b+c&=18 \\\\ \\tag{2}\nc-4&=a\\\\ \\tag{3}\nb+4&=\\fr...
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math500-0232
[ "\\frac{1997}{2}" ]
{ "row_pos": 232, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1354.json", "solution": "Note that\n\\begin{align*}\n\\frac{1}{1 + \\omega^k} + \\frac{1}{1 + \\omega^{1997 - k}} &= \\frac{1}{1 + \\omega^k} + \\frac{\\omega^k}{\\omega^k + \\omega^{1997}} \\\\\n&= \\frac...
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math500-0233
[ "30^\\circ" ]
{ "row_pos": 233, "level": 2, "subject": "Precalculus", "unique_id": "test/precalculus/441.json", "solution": "Let $\\theta$ be the angle between $\\mathbf{a}$ and $\\mathbf{b}.$ Then\n\\[\\|\\mathbf{a} \\times \\mathbf{b}\\| = \\|\\mathbf{a}\\| \\|\\mathbf{b}\\| \\sin \\theta.\\]From the given information, ...
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math500-0234
[ "34" ]
{ "row_pos": 234, "level": 3, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1837.json", "solution": "Let $a,$ $b,$ $c$ be the roots of $x^3 - 49x^2 + 623x - 2015.$ Then by Vieta's formulas, $a + b + c = 49.$\n\nThe roots of $g(x) = f(x + 5)$ are $a - 5,$ $b - 5,$ and $c - 5,$ and...
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math500-0235
[ "\\frac 59" ]
{ "row_pos": 235, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/153.json", "solution": "In general, to express the number $0.\\overline{n}$ as a fraction, we call it $x$ and subtract it from $10x$: $$\\begin{array}{r r c r@{}l}\n&10x &=& n&.nnnnn\\ldots \\\\\n- &x &=& 0&.nnnnn\\ldots \\\\...
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math500-0236
[ "12" ]
{ "row_pos": 236, "level": 4, "subject": "Precalculus", "unique_id": "test/precalculus/989.json", "solution": "We can write\n\\begin{align*}\n\\frac{1}{\\cos^2 10^\\circ} &= \\frac{2}{1 + \\cos 20^\\circ} \\\\\n&= \\frac{2 (1 - \\cos 20^\\circ)}{(1 + \\cos 20^\\circ)(1 - \\cos 20^\\circ)} \\\\\n&= \\frac{2 (1...
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math500-0237
[ "22" ]
{ "row_pos": 237, "level": 1, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/337.json", "solution": "Multiplying all three equations, we get $x^2 y^2 z^2 = 82944.$ Since $x,$$y,$ and $z$ are positive, $xyz = \\sqrt{82944} = 288.$ Then\n\\begin{align*}\nx &= \\frac{xyz}{yz} = \\fr...
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math500-0238
[ "-4" ]
{ "row_pos": 238, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/975.json", "solution": "$x$ is not in the domain of $f$ if the denominator is zero. Since both absolute values are nonnegative, both must be zero for the denominator to be zero. So\n\n\\begin{align*}\n0=x^2+3x-4=(x+4)(x-1)&\\Righ...
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math500-0239
[ "16" ]
{ "row_pos": 239, "level": 5, "subject": "Geometry", "unique_id": "test/geometry/229.json", "solution": "Two vertices of the first parallelogram are at $(0,c)$ and $(0,d)$.\n\n[asy]\nunitsize(0.5 cm);\n\npair P, Q, R, S;\n\nP = (0,9);\nQ = (3,12);\nR = (0,3);\nS = (-3,0);\n\ndraw(interp(P,Q,-0.4)--interp(P,Q,...
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math500-0240
[ "116" ]
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math500-0241
[ "17" ]
{ "row_pos": 241, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/1143.json", "solution": "By the distance formula, we are trying to minimize $\\sqrt{x^2+y^2}=\\sqrt{x^2+\\frac{1}{4}x^4-9x^2+81}$. In general, minimization problems like this require calculus, but one optimization method that somet...
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math500-0242
[ "\\$32,\\!348" ]
{ "row_pos": 242, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/2626.json", "solution": "Five years make sixty months, so interest will have been compounded 30 times. That means that the investment will have grown to $\\$24,\\!000 \\cdot 1.01^{30} \\approx \\boxed{\\$32,\\!348}$, to the nearest...
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math500-0243
[ "3" ]
{ "row_pos": 243, "level": 2, "subject": "Geometry", "unique_id": "test/geometry/254.json", "solution": "Call the length of the third side $n$. By the triangle inequality, $1+3>n$ and $1+n>3$, or $2<n<4$. The only integer $n$ which satisfies this is $\\boxed{3}$." }
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math500-0244
[ "8" ]
{ "row_pos": 244, "level": 5, "subject": "Prealgebra", "unique_id": "test/prealgebra/1251.json", "solution": "The sum of the interior angles in any $n$-sided polygon is $180(n-2)$ degrees, so the angle measures in a polygon with 7 sides sum to $180(7-2) = 900$ degrees, which means that the desired polygon has...
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math500-0245
[ "-13x+3" ]
{ "row_pos": 245, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1210.json", "solution": "We have\n$$f(x) = q(x)(x^2-1) + r(x),$$where $q(x)$ is the quotient and $r(x)$ is the remainder. Since $x^2-1$ is quadratic, the remainder is at most linear; let us write $r(x) = a...
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math500-0246
[ "8" ]
{ "row_pos": 246, "level": 2, "subject": "Algebra", "unique_id": "test/algebra/1787.json", "solution": "If 12 people order $\\frac{18}{12}=1\\frac{1}{2}$ times too much food, they should have ordered $\\frac{12}{\\frac{3}{2}}=\\frac{2}{3}\\times 12=\\boxed{8}$ meals." }
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math500-0247
[ "\\frac{10}{11}" ]
{ "row_pos": 247, "level": 3, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1123.json", "solution": "Because $p(x)$ has integer coefficients (in particular, because it has rational coefficients), the other root of $p(x)$ must be the radical conjugate of $4-\\sqrt{11},$ which is $4...
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math500-0248
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{ "row_pos": 248, "level": 4, "subject": "Precalculus", "unique_id": "test/precalculus/920.json", "solution": "Note that\n\\[\\mathbf{A}^2 = \\begin{pmatrix} 2 & 7 \\\\ -1 & -3 \\end{pmatrix}\\]and\n\\[\\mathbf{A}^3 = \\mathbf{A} \\mathbf{A}^2 = \\begin{pmatrix} -1 & 0 \\\\ 0 & -1 \\end{pmatrix} = -\\mathbf{I...
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math500-0249
[ "17" ]
{ "row_pos": 249, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/1065.json", "solution": "We could find the answer by trial and error -- testing each candidate $t$ to see if $t\\cdot (t+2)\\equiv 1\\pmod{23}$. However, here is another way:\n\nWe can easily see that $4\\cdot 6=24\\equ...
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math500-0250
[ "13" ]
{ "row_pos": 250, "level": 4, "subject": "Algebra", "unique_id": "test/algebra/1934.json", "solution": "We can find the $x$-intercept by setting $y=0$ in the equation. This gives us $\\frac{x}{4} = 1$, so $x =4$, which means $a=4$. Similarly, letting $x=0$ gives $\\frac{y}{12} = 1$, so $y=12$, which means $...
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math500-0251
[ "540" ]
{ "row_pos": 251, "level": 5, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/894.json", "solution": "We count the number of ways that some lane can be left empty, and subtract from the total number, $3^6=729$ because each driver has three choices. Suppose the left-turn lane i...
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math500-0252
[ "81" ]
{ "row_pos": 252, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/2064.json", "solution": "The graph of $y = 4(x + 7)(2 - x)$ is a parabola. Since $y = 0$ when $x = -7$ and $x = 2$, the $x$-intercepts of the parabola are $(-7,0)$ and $(2,0)$. If the vertex of the parabola is $(h,k)$, then the $...
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math500-0253
[ "\\frac{1}{8}" ]
{ "row_pos": 253, "level": 3, "subject": "Prealgebra", "unique_id": "test/prealgebra/1458.json", "solution": "Since 2 feet is $2\\cdot12=24$ inches, 3 inches is $3/24=\\boxed{\\frac{1}{8}}$ of 2 feet." }
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math500-0254
[ "28" ]
{ "row_pos": 254, "level": 5, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/1009.json", "solution": "The numbers of the three types of cookies must have a sum of six. Possible sets of whole numbers whose sum is six are \\[\n0,0,6;\\ 0,1,5;\\ 0,2,4;\\ 0,3,3;\\ 1,1,4;\\ 1,2,3;...
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math500-0255
[ "\\text{(E)}" ]
{ "row_pos": 255, "level": 2, "subject": "Precalculus", "unique_id": "test/precalculus/452.json", "solution": "From $r^2 \\cos 2 \\theta = 4,$\n\\[r^2 (\\cos^2 \\theta - \\sin^2 \\theta) = r^2 \\cos^2 \\theta - r^2 \\sin^2 \\theta = 4.\\]Then $x^2 - y^2 = 4,$ or\n\\[\\frac{x^2}{4} - \\frac{y^2}{4} = 1.\\]Thus...
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math500-0256
[ "\\frac{8}{21}" ]
{ "row_pos": 256, "level": 3, "subject": "Prealgebra", "unique_id": "test/prealgebra/1995.json", "solution": "First, we can use the division rule so that we have an expression with only multiplication of fractions. We get $$\\frac{1}{5}\\cdot \\frac{8}{7}\\div \\frac{12}{20}=\\frac{1}{5}\\cdot \\frac{8}{7}\\...
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math500-0257
[ "864 \\mbox{ inches}^2" ]
{ "row_pos": 257, "level": 4, "subject": "Geometry", "unique_id": "test/geometry/473.json", "solution": "Recall that $$1 \\mbox{ foot} = 12 \\mbox{ inches}$$\n\nHence $$1 \\mbox{ foot}^2 = 12^2 \\mbox{ inches}^2 = 144 \\mbox{ inches}^2$$\n\nFinally, remember the formula $V = l \\times w \\times h$, that is, ...
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math500-0258
[ "16" ]
{ "row_pos": 258, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/694.json", "solution": "Let $x$ be the number of blue marbles and $y$ the number of yellow marbles before I added more. We are given that the ratio of blue to yellow is 4:3, so $\\dfrac{x}{y}=\\dfrac{4}{3}$. Additionally, after we ...
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math500-0259
[ "22" ]
{ "row_pos": 259, "level": 1, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/149.json", "solution": "The radical conjugate of this number is $5-\\sqrt{3},$ so the product of the two numbers is \\[(5+\\sqrt3)(5-\\sqrt3) = 5^2 - (\\sqrt3)^2 = 25 - 3 = \\boxed{22}.\\]" }
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math500-0260
[ "120^\\circ" ]
{ "row_pos": 260, "level": 3, "subject": "Precalculus", "unique_id": "test/precalculus/580.json", "solution": "From the distance formula, we compute that $AB = 3 \\sqrt{6},$ $AC = 9 \\sqrt{2},$ and $BC = 3 \\sqrt{6}.$ Then from the Law of Cosines,\n\\[\\cos \\angle ABC = \\frac{(3 \\sqrt{6})^2 + (3 \\sqrt{6}...
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math500-0261
[ "76^\\circ" ]
{ "row_pos": 261, "level": 3, "subject": "Geometry", "unique_id": "test/geometry/347.json", "solution": "If $AB$ is a diameter, that means the triangle must have a right angle at $C.$ Therefore, we have that \\begin{align*}\n\\angle B &= 180^\\circ - (\\angle A + \\angle C) \\\\\n&= 180^\\circ - (14^\\circ + ...
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math500-0262
[ "4" ]
{ "row_pos": 262, "level": 1, "subject": "Prealgebra", "unique_id": "test/prealgebra/1317.json", "solution": "Since $10^x - 10 = 9990,$ we have $$10^x = 9990+10=10000.$$If $10^x = 10000,$ then $x=\\boxed{4},$ since $10000$ ends in four zeroes." }
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math500-0263
[ "15" ]
{ "row_pos": 263, "level": 4, "subject": "Geometry", "unique_id": "test/geometry/483.json", "solution": "The measure of an interior angle of a regular $n$-gon is $\\frac{180(n-2)}{n}$ degrees and the measure of an exterior angle is $\\frac{360}{n}$ degrees. Solving \\[\n\\frac{180(n-2)}{n}=6.5\\cdot\\left(\...
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math500-0264
[ "1\\frac{4}{5}" ]
{ "row_pos": 264, "level": 5, "subject": "Geometry", "unique_id": "test/geometry/826.json", "solution": "Let $J$ be the intersection of $\\overline{BE}$ and $\\overline{GC}$. [asy]\nunitsize(5mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\npair A=(0,0), B=(3,0), C=(6,0), D=(9,0), Ep=(9,3), G=(6,3), K=(33/5,...
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math500-0265
[ "4005" ]
{ "row_pos": 265, "level": 2, "subject": "Algebra", "unique_id": "test/algebra/524.json", "solution": "The sequence of odd numbers 1, 3, 5, 7, and so on, is an arithmetic sequence, with common difference 2. Therefore, the $2003^{\\text{rd}}$ term is $1+2002\\cdot2=\\boxed{4005}$." }
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math500-0266
[ "\\dfrac{33}{100}" ]
{ "row_pos": 266, "level": 3, "subject": "Prealgebra", "unique_id": "test/prealgebra/1742.json", "solution": "There are 100 numbers possible between 1 and 100. There are 33 multiples of 3 between 1 and 100: $(3,6,9,\\ldots,99)=(1\\times 3,2\\times 3,3\\times 3,\\ldots,33\\times 3)$. So the probability that a...
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math500-0267
[ "180^\\circ" ]
{ "row_pos": 267, "level": 1, "subject": "Geometry", "unique_id": "test/geometry/226.json", "solution": "Since $\\overline{OH}$ and $\\overline{NH}$ are tangent to radii of the circle at $O$ and $N$, we have $\\angle O =\\angle N = 90^\\circ$. The sum of the measures of the interior angles of quadrilateral $...
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math500-0268
[ "1" ]
{ "row_pos": 268, "level": 1, "subject": "Number Theory", "unique_id": "test/number_theory/466.json", "solution": "Looking at our sum, we can see that the numbers $1$ through $8$ can be paired off to form $9,$ so we may eliminate them. That is, $1 + 8 = 2 + 7 = 3 + 6 = 4 + 5 = 9.$ Therefore, the only remainin...
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math500-0269
[ "6" ]
{ "row_pos": 269, "level": 3, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/913.json", "solution": "There are 4 slots in the row. He can choose two of them for his IMO medals in $\\binom{4}{2} = \\boxed{6}$ ways." }
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math500-0270
[ "10" ]
{ "row_pos": 270, "level": 1, "subject": "Algebra", "unique_id": "test/algebra/2551.json", "solution": "We need to calculate the total purchase price that Susan paid and that Pam paid.\n\nSusan bought $4$ tickets with $25\\%$ discount: $$4 \\times \\$20 = \\$80.$$With a $25$ percent discount, she paid $\\$80 ...
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math500-0271
[ "333" ]
{ "row_pos": 271, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/634.json", "solution": "We note that $$997\\equiv -3\\pmod{1000},$$and $$(-3)\\cdot 333 = -999 = -1000+1\\equiv 1\\pmod{1000}.$$Therefore, $$997\\cdot 333\\equiv 1\\pmod{1000},$$and the inverse of $997$ modulo $1000$ is...
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math500-0272
[ "1030" ]
{ "row_pos": 272, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/533.json", "solution": "Let $x = 0.\\overline{1331}$, so $10000x = 1331.\\overline{1331}$. As a result, $9999x = 1331$, so $x = \\frac{1331}{9999}$. We can factor out 11 out of both the numerator and denominator, so $x ...
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math500-0273
[ "1250" ]
{ "row_pos": 273, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/993.json", "solution": "Multiply both sides of $1200\\text{ lire}=\\$1.50$ by $1,\\!000,\\!000/1200$ to find that one million lire equals $\\frac{3}{2}\\cdot10,\\!000/12=\\boxed{1250}$ dollars." }
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math500-0274
[ "18+2\\pi" ]
{ "row_pos": 274, "level": 5, "subject": "Geometry", "unique_id": "test/geometry/686.json", "solution": "$S$ looks like a nonagon with slightly rounded corners. We draw adjacent sides of the nonagon and look at the boundary of $S$:\n\n[asy]\nsize(200);\ndraw((-7.66,-6.43)--(0,0)--(10,0)--(17.66,-6.43));\ndra...
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math500-0275
[ "7" ]
{ "row_pos": 275, "level": 2, "subject": "Algebra", "unique_id": "test/algebra/346.json", "solution": "We have $f(5) = 2(5) -3 = 7$, so $g(f(5)-1) = g(7-1) = g(6) = 6+1 = \\boxed{7}$." }
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math500-0276
[ "11" ]
{ "row_pos": 276, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/1282.json", "solution": "Suppose that $k = \\sqrt{120 - \\sqrt{x}}$ is an integer. Then $0\\le k \\le \\sqrt{120}$, and because $k$ is an integer, we have $0\\le k \\le 10$. Thus there are 11 possible integer values of $k$. For ea...
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math500-0277
[ "\\sqrt{53}" ]
{ "row_pos": 277, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/1834.json", "solution": "Since 53 is prime, $\\boxed{\\sqrt{53}}$ is already in simplest radical form." }
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[ { "bucket": "math:\\sqrt{53}", "label": "\\sqrt{53}", "count": 100, "match": "exact", "judge": null, "members": [ { "raw": "\\sqrt{53}", "count": 100 } ] } ]
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math500-0278
[ "255" ]
{ "row_pos": 278, "level": 3, "subject": "Number Theory", "unique_id": "test/number_theory/691.json", "solution": "The largest eight-digit base 2 integer is 1 less than the smallest nine-digit base 2 integer, which is $$ 100000000_{2} = 1 \\cdot 2^8 = 256. $$Thus, the largest eight-digit base 2 integer is $25...
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math500-0279
[ "7\\pi" ]
{ "row_pos": 279, "level": 5, "subject": "Prealgebra", "unique_id": "test/prealgebra/1512.json", "solution": "The area of the inside circle (region $X$) is $\\pi\\cdot 4^2=16\\pi.$\n\nUsing a similar technique, the area of the middle ring (region $Y$) is $$\\pi\\cdot 6^2-\\pi\\cdot 4^2=36\\pi-16\\pi = 20\\pi....
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math500-0280
[ "8" ]
{ "row_pos": 280, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/1287.json", "solution": "Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\\cdot a^1\\cdot b^1$ has $(1+1)(1+1)...
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math500-0281
[ "36^\\circ" ]
{ "row_pos": 281, "level": 4, "subject": "Prealgebra", "unique_id": "test/prealgebra/260.json", "solution": "The sum of the angle measures in a pentagon is $180(5-2) = 540$ degrees, so each of the interior angles of regular pentagon $FGHIJ$ has measure $540^\\circ / 5 = 108^\\circ$. Specifically, $\\angle JF...
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math500-0282
[ "129" ]
{ "row_pos": 282, "level": 4, "subject": "Algebra", "unique_id": "test/algebra/1184.json", "solution": "Continuing this sequence from the 27, we add four to make 31, then multiply 31 by four to make 124, then add five to 124 to make 129. Thus, $\\boxed{129}$ is the first term that is greater than 125." }
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math500-0283
[ "1" ]
{ "row_pos": 283, "level": 3, "subject": "Prealgebra", "unique_id": "test/prealgebra/1787.json", "solution": "First we convert $4\\dfrac{5}{8}$ into an improper fraction: \\[4\\dfrac{5}{8} = 4 + \\dfrac{5}{8} = \\dfrac{32}{8} + \\dfrac{5}{8} = \\dfrac{37}{8}.\\]We discover that $4\\dfrac{5}{8}$ and $\\dfrac{8...
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math500-0284
[ "3 \\pm 2 \\sqrt{2}" ]
{ "row_pos": 284, "level": 5, "subject": "Precalculus", "unique_id": "test/precalculus/768.json", "solution": "Since $\\cot^{-1} \\left( \\frac{1}{x} \\right) = \\tan^{-1} x$ for all $x,$ we can write\n\\[\\sin \\left( 2 \\tan^{-1} x \\right) = \\frac{1}{3}.\\]Let $\\theta = \\tan^{-1} x,$ so $x = \\tan \\the...
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math500-0285
[ "440" ]
{ "row_pos": 285, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1411.json", "solution": "Let $r, s, t$ be the three positive integer roots of $P(x).$ Then by Vieta's formulas, \\[\\begin{aligned} r+s+t &= a, \\\\ rs+st+rt &= \\frac{a^2-81}{2}, \\\\ rst &= \\frac{c}{2}....
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math500-0286
[ "\\frac{17}{21}" ]
{ "row_pos": 286, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/960.json", "solution": "For a triangle with side lengths $a,$ $b,$ $c,$ let $s = \\frac{a + b + c}{2},$ and let\n\\begin{align*}\nx &= s - a = \\frac{-a + b + c}{2}, \\\\\ny &= s - b = \\frac{a - b + c}{2}...
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math500-0287
[ "36" ]
{ "row_pos": 287, "level": 5, "subject": "Number Theory", "unique_id": "test/number_theory/631.json", "solution": "One gear turns $33\\frac{1}{3}=100/3$ times in 60 seconds, so it turns 5/9 times in one second, or 5 times in 9 seconds. The other gear turns 45 times in 60 seconds, so it turns 3/4 times in one ...
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math500-0288
[ "7" ]
{ "row_pos": 288, "level": 5, "subject": "Precalculus", "unique_id": "test/precalculus/1172.json", "solution": "We see that $(3,-10,1)$ is a point on the first line.\n\nA point on the second line is given by\n\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} -5 \\\\ -3 \\\\ 6 \\end{pmatrix...
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math500-0289
[ "16" ]
{ "row_pos": 289, "level": 3, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1300.json", "solution": "By AM-GM,\n\\[a + b + c + d \\ge 4 \\sqrt[4]{abcd},\\]and\n\\[\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{d} \\ge 4 \\sqrt[4]{\\frac{1}{abcd}},\\]so\n\\[(a + b + c + d) ...
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math500-0290
[ "4" ]
{ "row_pos": 290, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/634.json", "solution": "Factoring the quadratic on the left gives $$\\frac{(2r-7)(3r+1)}{2r-7} = 4r-3.$$Canceling the common factor on the left gives $3r + 1 = 4r - 3$. Solving this equation gives $r = \\boxed{4}$." }
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math500-0291
[ "63" ]
{ "row_pos": 291, "level": 4, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/25149.json", "solution": "The rectangular region is 10 units by 8 units, resulting in an 8 by 6 rectangular region in the interior, which forms a 9-by-7 array of lattice points. That's $\\boxed{63}$ ...
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math500-0292
[ "898" ]
{ "row_pos": 292, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/90.json", "solution": "Shifting the recurrence over by one and adding, we have: \\[\\begin{aligned} x_n &= x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} \\\\ x_{n-1} &= x_{n-2} - x_{n-3} + x_{n-4} - x_{n-5} \\\\ \...
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math500-0293
[ "\\left(\\frac{3}{5},\\frac{8}{3}\\right]" ]
{ "row_pos": 293, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/2486.json", "solution": "We take the inequalities one at a time. Adding $144$ to both sides of the first inequality, we get $$144\\ge 54p,$$implying $$\\frac{144}{54}\\ge p.$$Reducing the fraction and switching the sides (along wit...
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math500-0294
[ "58" ]
{ "row_pos": 294, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/488.json", "solution": "The sum of the set $\\{n-1, n, n+1\\}$ of three consecutive integers is $3n$. Therefore, we are looking for the largest three-digit palindromic multiple of $3$ less than $220$. Checking through...
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math500-0295
[ "11" ]
{ "row_pos": 295, "level": 3, "subject": "Precalculus", "unique_id": "test/precalculus/1201.json", "solution": "By the Binomial Theorem,\n\\begin{align*}\n(z + i)^4 &= z^4 + 4z^3 i + 6z^2 i^2 + 4zi^3 + 1 \\\\\n&= z^4 + 4iz^3 - 6z^2 - 4iz + 1.\n\\end{align*}So, if $z^4 + 4z^3 i - 6z^2 - 4zi - i = 0,$ then\n\\[...
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math500-0296
[ "\\text{(B)}" ]
{ "row_pos": 296, "level": 2, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/754.json", "solution": "Intuitively, $x$ will be largest for the option for which the value in the parentheses is smallest.\n\nFormally, first note that each of the values in parentheses is larger than $1$...
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math500-0297
[ "103" ]
{ "row_pos": 297, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/1172.json", "solution": "Let $m$ be the least possible number of leaves. Then $2m$ is the least possible number of pages. We know that $2m\\equiv 3\\pmod 7\\implies 8m \\equiv 3\\cdot 4\\pmod 7\\implies m\\equiv 12\\equ...
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math500-0298
[ "\\begin{pmatrix} -18 \\\\ -49 \\\\ 96 \\end{pmatrix}" ]
{ "row_pos": 298, "level": 4, "subject": "Precalculus", "unique_id": "test/precalculus/881.json", "solution": "Expanding, we get\n\\begin{align*}\n(2 \\mathbf{b} - \\mathbf{a}) \\times (3 \\mathbf{c} + \\mathbf{a}) &= 6 \\mathbf{b} \\times \\mathbf{c} + 2 \\mathbf{b} \\times \\mathbf{a} - 3 \\mathbf{a} \\time...
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math500-0299
[ "63" ]
{ "row_pos": 299, "level": 4, "subject": "Geometry", "unique_id": "test/geometry/1097.json", "solution": "Points with integer coordinates are called lattice points. The length of the rectangle is $5 - (-5) = 10$ units. There will be 9 lattice positions between the two vertical sides of the rectangle. The heig...
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