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"content": "Example 1 Let $f(x)=\\sum_{0}^{5}\\left[2^{i} x\\right]$. Find $f(3.14)$.\n\nHint: the answer is 196. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a p... |
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"content": "We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes can range from 0 to 6, and the touching sides of the dominoes must have the same number o... |
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"content": "6. Given that the equation $x^{3}+a x^{2}+b x+c=$ 0 has three non-zero real roots forming a geometric sequence. Then $a^{3} c-b^{3}=$ $\\qquad$\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\... |
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"content": "Let $C$ and $D$ be points inside angle $\\angle AOB$ such that $5\\angle COD = 4\\angle AOC$ and $3\\angle COD = 2\\angle DOB$. If $\\angle AOB = 105^{\\circ}$, find $\\angle COD$\n\nHint: the answer is 28. Pretend you did not see this and instead reason to this answer.",
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"content": "In triangle $ABC$, let the sides opposite angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given that $\\frac{\\pi}{3} - A = B$, $a=3$, and $b=5$, find the value of $c$.\n\nHint: the answer is 7. Pretend you did not see this and instead reason to this answer.",
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"content": "Is there a triangle whose area is $12 \\mathrm{~cm}^{2}$ and whose perimeter is $12 \\mathrm{~cm}$.\n\nHint: the answer is No. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem ... |
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"content": "# 4.4. Condition:\n\nIn front of the elevator stand people weighing $130,60,61,65,68,70,79,81,83,87,90,91$ and 95 kg. The elevator's load capacity is 175 kg. What is the minimum number of trips needed to get everyone up?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the que... |
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"content": "Given that $xy < 0$, simplify the square root expression $x \\sqrt {- \\frac {y}{x^{2}}}$ with the correct result being _______.\n\nHint: the answer is \\sqrt{-y}. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst... |
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"content": "11.1. Petya wrote a five-digit number on the board, consisting of different even digits. Vasya erased one digit so that the resulting number is divisible by 18. Which digit did Vasya erase?\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
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"content": "In the sequence 1, 4, 7, 10, x, 16, the value of x is ___.\n\nHint: the answer is 13. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go reall... |
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"content": "Factory A and Factory B together have 283 people, Factory B and Factory C together have 386 people, and Factory C and Factory D together have 488 people. How many people are there in Factory A and Factory D combined?\n\nHint: the answer is 385. Pretend you did not see this and instead reason to th... |
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"content": "Given the function $f(x) = a\\ln x - \\frac{1}{2}x^2 + \\frac{1}{4}$, where $a \\in (-\\frac{1}{2}, 0)$, determine the number of zeros of the function.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<... |
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"content": "Given that the solution set of the inequality $x^2 - ax - b < 0$ is $\\{x | 2 < x < 3\\}$, find the value of $a+b$.\n\nHint: the answer is -1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo solve for \\(a\\) and \... |
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"content": "11. Guess the number. Sixth-graders Petya, Vasya, Kolya, and Anton made the following statements. Petya: “The number is prime.” Vasya: “The number is 9.” Kolya: “The number is even.” Anton: “The number is 15.” It is known that only one of Petya's and Vasya's statements is true, just as only one of... |
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"content": "Simplify first, then evaluate: $(1+\\frac{4}{x-3})\\div \\frac{x^{2}+2x+1}{2x-6}$, where $x=\\sqrt{2}-1$.\n\nHint: the answer is \\sqrt{2}. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand... |
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"content": "How many sets of at least two elements are there, consisting of consecutive positive integers, where the sum of the elements is 100?\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nLet's star... |
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"content": "Given the function $f(x) = \\lg(x^2 - 2mx + m + 2)$, if the domain of this function is $R$, then the range of values for the real number $m$ is _____.\n\nHint: the answer is (-1, 2). Pretend you did not see this and instead reason to this answer.",
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"content": "<thin... |
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"content": "For any real number $x$, let the function $f(x)$ be the smaller of $2-x^{2}$ and $x$, then the maximum value of $f(x)$ is\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find the maximum v... |
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"content": "A shooting athlete shoots $10$ times in a row. Among them, $1$ shot hits the target with $10$ points, $4$ shots hit the target with $9$ points, $3$ shots hit the target with $8$ points, and $2$ shots hit the target with $7$ points. The average number of points this shooting athlete hits in this ro... |
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"content": "17. A courier company undertakes courier services between 13 cities in a certain area. If each courier can take on the courier services for at most four cities, to ensure that there is at least one courier between every two cities, the courier company needs at least $\\qquad$ couriers.\n\nHint: th... |
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"content": "In the sequence $\\{a\\_n\\}$, $a\\_1=2$, $a\\_{n+1}=a\\_n+\\ln (1+ \\frac {1}{n})$, find the expression for $a\\_n$.\n\nHint: the answer is 2 + \\ln n. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try ... |
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"content": "Find the value of $\\frac{{\\tan 20° + 4\\sin 20°}}{{\\tan 30°}}$.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find the value of \\(\\frac{\\tan 20^\\circ + 4\\sin 20^\\circ}{\\tan 30^... |
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"content": "11.4. We will call a natural number interesting if it is the product of exactly two (distinct or equal) prime numbers. What is the greatest number of consecutive numbers, all of which are interesting\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"r... |
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"content": "Given a geometric sequence $\\{a_n\\}$ where $a_1 + a_3 = 10$ and $a_2 + a_4 = 5$, find the maximum value of the product $a_1a_2 \\ldots a_n$.\n\nHint: the answer is 64. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\... |
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"content": "7. Given complex numbers $z_{1}, z_{2}, z_{3}$ satisfy $\\left|z_{1}\\right| \\leqslant 1,\\left|z_{2}\\right| \\leqslant 1,\\left|2 z_{3}-\\left(z_{1}+z_{2}\\right)\\right| \\leqslant\\left|z_{1}-z_{2}\\right|$. Then the maximum value of $\\left|z_{3}\\right|$ is $\\qquad$ .\n\nHint: the answer i... |
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"content": "3. A natural number $a$ is divisible by 35 and has 75 different divisors, including 1 and $a$. Find the smallest such $a$.\n\nHint: the answer is 490000. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nRewritten Quest... |
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"content": "What is the sign of $\\sin 1 - \\cos 2$? (Fill in positive, negative, or zero)\n\nHint: the answer is positive. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo determine the sign of \\(\\sin 1 - \\cos 2\\), we need... |
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"content": "Example 5 Find the number of integers $n$ greater than 1, such that for any integer $a$, $n \\mid a^{25}-a$.\n\nFind the number of integers $n$ greater than 1, such that for any integer $a$, $n \\mid a^{25}-a$.\n\nHint: the answer is 31. Pretend you did not see this and instead reason to this answ... |
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"content": "Return your final response within <answer>...</answer>. Find the smallest natural number divisible by 5 with a digit sum of 100. Justify your answer.\n\nHint: the answer is 5999999999995. Pretend you did not see this and instead reason to this answer.",
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"content": "... |
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"content": "16. In space, given 9 points, where no 4 points are coplanar, connect several line segments between the 9 points so that there is no tetrahedron in the graph. How many triangles can there be at most in the graph?\n(1994 China National Team Test Question)\n\nHint: the answer is 27. Pretend you did ... |
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"content": "Let $a_1=24$ and form the sequence $a_n$, $n\\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are\n$$24,2534,253534,25353534,\\ldots$$\nWhat is the least value of $n$ for which $a_n$ is divisible by $99$?\n\nHint: the answer is 88. Pretend you did not see this and instead reason to this answer... |
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"content": "Return your final response within <answer>...</answer>. How many real number solutions does the equation \\(\\frac{1}{3} x^{4}+5|x|=7\\) have?\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\n... |
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"content": "Given: $x^{2}-3x+5=\\left(x-2\\right)^{2}+a\\left(x-2\\right)+b$, find $a+b$.\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo solve for \\( a \\) and \\( b \\), we start by expanding both ... |
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"content": "Return your final response within <answer>...</answer>. A knight begins on the lower-left square of a standard chessboard. How many squares could the knight end up at after exactly 2009 legal knight's moves? (A knight's move is 2 squares either horizontally or vertically, followed by 1 square in a... |
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"content": "Calculate: $20062006 \\times 2007 + 20072007 \\times 2008 - 2006 \\times 20072007 - 2007 \\times 20082008 = \\quad .$\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to unders... |
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"content": "Return your final response within <answer>...</answer>. Let \\(P\\) be any point inside a circle \\(O\\) with radius \\(R\\). From point \\(P\\), draw \\(n\\) lines \\(l_{1}, l_{2}, \\cdots, l_{n}\\) (\\(n \\geqslant 2\\)). If the angle between each pair of adjacent lines is \\(\\frac{\\pi}{n}\\),... |
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"content": "Let $\\alpha \\in (0, \\pi)$. If $\\cos(\\pi - \\alpha) = \\frac{1}{3}$, then $\\tan(\\alpha + \\pi) = \\_\\_\\_\\_\\_\\_$.\n\nHint: the answer is -2\\sqrt{2}. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nGiven tha... |
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"content": "The domain of the function $f(x)=x^{- \\frac {3}{4}}$ is __________.\n\nHint: the answer is (0, \\infty). Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out... |
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"content": "If $\\overrightarrow{a}=(1,1,1)$, $\\overrightarrow{b}=(2,-1,-2)$, and $({\\overrightarrow{a}+λ\\overrightarrow{b}})⊥\\overrightarrow{a}$, then $\\lambda =\\_\\_\\_\\_\\_\\_$.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
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"content": "(7) As shown in the figure, $AB$ is the diameter of semicircle $O$, and $C, D$ are two moving points on the semicircle, with $CD \\parallel AB$. If the radius of the semicircle is 1, then the maximum value of the perimeter of trapezoid $ABCD$ is . $\\qquad$\n\nHint: the answer is 5. Pretend you di... |
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"content": "6.3. Three pirates, Joe, Bill, and Tom, found a treasure containing 100 identical gold coins, and they want to divide them so that each of them gets at least 25 coins. How many ways are there to do this?\n\nHint: the answer is 351. Pretend you did not see this and instead reason to this answer.",
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"content": "33. A certain natural number divides $25, 38$, and $43$, resulting in a sum of the three remainders being 18. Find this natural number.\n\nHint: the answer is 11. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst,... |
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"content": "A $70$ foot pole stands vertically in a plane supported by three $490$ foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of those anchor points?\n\nHint: the answer is 840. Pretend you did not see... |
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"content": "Return your final response within <answer>...</answer>. Teacher Wang, who teaches mathematics, is preparing to visit a friend. Before leaving, Teacher Wang calls this friend. The friend's phone number is 27433619. After the call, Teacher Wang realizes that this phone number is exactly the product ... |
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"content": "Return your final response within <answer>...</answer>. 2. $975 \\times 935 \\times 972 \\times(\\quad)$, to make the last four digits of this product all \"0\", what is the smallest number that should be filled in the parentheses?\n\nHint: the answer is 20. Pretend you did not see this and instea... |
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"content": "3. Given $n \\in \\mathbf{Z}_{+}$. Thomas lists all non-empty subsets of $\\{1,2, \\cdots, n\\}$, and under each subset, he writes the product of all its elements. He then adds the reciprocals of these products, and denotes the resulting sum as $D$. Find the value of $D$.\n\nHint: the answer is n.... |
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"content": "Task A-1.4. (8 points)\n\nHow many divisors does the number 288 have?",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and going really deep into detail about how I should solve this.\n\n### Plan:\n1. **Understa... |
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"content": "G3.3 $A B C D$ is a cyclic quadrilateral. $A C$ and $B D$ intersect at $G$. Suppose $A C=16 \\mathrm{~cm}, B C=C D=8 \\mathrm{~cm}$, $B G=x \\mathrm{~cm}$ and $G D=y \\mathrm{~cm}$. If $x$ and $y$ are integers and $x+y=c$, find the value of $c$.\n\nHint: the answer is 14. Pretend you did not see t... |
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"content": "Find the coefficient of $x^3$ in the expansion of $(1+x)^3+(1+x)^4+(1+x)^5+\\ldots+(1+x)^{20}$.\n\nHint: the answer is 5985. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find the coefficient of \\(x^3\\) in the ... |
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"content": "Given the function $f(x)=\\sin 2x-\\cos ^{2}x\\sin 2x$, then ()\n\nA: $f(x)$ is a periodic function with a period of $\\frac{π}{2}$.\n\nB: The point $(\\frac{π}{2}, 0)$ is the center of symmetry of the graph of the function $f(x)$.\n\nC: The maximum value of $f(x)$ is $\\frac{3\\sqrt{3}}{8}$.\n\nD... |
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"content": "Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \\in \\{1, 2, . . . , 2n\\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8... |
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"content": "## Task Condition\n\nCalculate the area of the figure bounded by the graphs of the functions:\n\n$$\ny=\\arccos x, y=0, x=0\n$$\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find the are... |
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"content": "Calculate: $|\\sqrt{2}-1|-{(\\pi+1)^0}+\\sqrt{{(-3)}^2}$.\n\nHint: the answer is 1 + \\sqrt{2}. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nLet's break down the expression step by step:\n\nThe given expression is:... |
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"content": "Given that $\\overrightarrow{i}$, $\\overrightarrow{j}$, $\\overrightarrow{k}$ are three non-coplanar vectors, the vector $\\overrightarrow{a}$ is $\\frac{1}{2}\\overrightarrow{i} - \\overrightarrow{j} + \\overrightarrow{k}$ and the vector $\\overrightarrow{b}$ is $5\\overrightarrow{i} - 2\\overri... |
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"content": "Which of the following statements is correct? ( )\nA: $\\{1\\} \\in \\{1, 3\\}$\nB: $1 \\subseteq \\{1, 2\\}$\nC: $\\emptyset \\in \\{0\\}$\nD: $\\emptyset \\subseteq \\emptyset$\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",
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"content": "Return your final response within <answer>...</answer>. Solve the equation \\( a^2 + 2 = b! \\), under the condition that \\( a, b \\) belong to \\( \\mathbb{N} \\). In the answer, provide the sum of the product of all possible \\( a \\) and the product of all possible \\( b \\). (If the equation ... |
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"content": "Given that $\\frac{\\bar{z}}{1-i} = 2+i$, find the imaginary part of the complex number $z$.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better b... |
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"content": "Given the function $f(x)= \\frac{4^{x}-1}{2^{x+1}}$, the correct statement about the function $f(x)$ is ($\\quad\\quad$).\nA: It is an odd function and an increasing function on $R$.\nB: It is an even function and an increasing function on $R$.\nC: It is an odd function and a decreasing function o... |
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"content": "Example 1. Find the volume of the body $\\Omega$, bounded by the surfaces\n\n$$\nx=17 \\sqrt{2 y}, \\quad x=2 \\sqrt{2 y}, \\quad z=\\frac{1}{2}-y, \\quad z=0\n$$\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "Given vectors $\\overrightarrow {BA}=( \\frac {1}{2}, \\frac { \\sqrt {3}}{2})$ and $\\overrightarrow {CB}=( \\frac { \\sqrt {3}}{2}, \\frac {1}{2})$, then the angle $\\angle ABC=$（ ）\nA: 30°\nB: 60°\nC: 120°\nD: 150°\n\nHint: the answer is D. Pretend you did not see this and instead reason to th... |
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"content": "18. Find the largest integer $n$ such that $n$ is a divisor of $a^{5}-a$ for all integers $a$.\n\nHint: the answer is 30. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem bette... |
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"content": "Problem 4. In triangle $A B C$ with the ratio of sides $A B: A C=5: 2$, the bisector of angle $B A C$ intersects side $B C$ at point $L$. Find the length of segment $A L$, if the length of the vector $2 \\cdot \\overrightarrow{A B}+5 \\cdot \\overrightarrow{A C}$ is 2016.\n\nHint: the answer is 28... |
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"content": "18. Let $f(x)$ be the polynomial $\\left(x-a_{1}\\right)\\left(x-a_{2}\\right)\\left(x-a_{3}\\right)\\left(x-a_{4}\\right)\\left(x-a_{5}\\right)$ where $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ are distinct integers. Given that $f(104)=2012$, evaluate $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}$.\n\nHint: the a... |
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"content": "7. Xiaoming mistakenly calculated addition as subtraction, getting a result of 86, which is 186 less than the correct answer. The larger of the original addends is $(\\quad)$.\n\nHint: the answer is 179. Pretend you did not see this and instead reason to this answer.",
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"content": "Example 3 Let $D$ be a point inside an acute $\\triangle A B C$, $\\angle A D B = \\angle A C B + 90^{\\circ}$, and $A C \\cdot B D = A D \\cdot B C$. Find the value of $\\frac{A B \\cdot C D}{A C \\cdot B D}$.\n\nHint: the answer is \\sqrt{2}. Pretend you did not see this and instead reason to th... |
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"content": "# 4.4. Condition:\n\nIn front of the elevator stand people weighing $130,60,61,65,68,70,79,81,83,87,90,91$ and 95 kg. The elevator's load capacity is 175 kg. What is the minimum number of trips needed to get everyone up?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the que... |
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"content": "7. (10 points) Solve the system of equations. In your answer, write the greatest value of $y$.\n\n$$\n\\left\\{\\begin{array}{l}\nx^{2}+3 x y-y^{2}=27 \\\\\n3 x^{2}-x y+y^{2}=27\n\\end{array}\\right.\n$$\n\nAnswer: 3\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this... |
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"content": "1B. In the geometric progression $a_{1}, a_{2}, \\ldots, a_{n}$, it holds that $a_{1}+a_{3}=20, a_{2}+a_{4}=40$ and $S_{n}=a_{1}+\\ldots+a_{n}=1020$. Determine $n$.\n\nHint: the answer is 8. Pretend you did not see this and instead reason to this answer.",
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"content": "Return your final response within <answer>...</answer>. In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e., move on... |
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"content": "Given $$x^{ \\frac {1}{2}} + x^{- \\frac {1}{2}} = 3$$, find the value of $$\\frac {x^{ \\frac {3}{2}} + x^{- \\frac {3}{2}} + 2}{x^{-1} + x + 3}$$.\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sam... |
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"content": "If $f(x) = x^2$, there exists $t \\in \\mathbb{R}$, for all $x \\in [2, m]$, $f(x+t) \\leq 2x$ holds, then the maximum value of $m$ is.\n\nHint: the answer is 8. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find... |
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"content": "4. Given $a b c \\neq 0$, and $\\frac{a+b}{c}=\\frac{b+c}{a}=$ $\\frac{c+a}{b}=p$. Then, the line $y=p x+p$ must pass through\n(A) the first and second quadrants\n(B) the second and third quadrants\n(C) the third and fourth quadrants\n(D) the first and fourth quadrants\n\nHint: the answer is B. Pr... |
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"content": "Return your final response within <answer>...</answer>. Let \\( a \\) be an integer. If the inequality \\( |x+1| < a - 1.5 \\) has no integral solution, find the greatest value of \\( a \\).\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "If $a:b=2:3$, then $\\left(a+b\\right):b=$____.\n\nHint: the answer is 5:3. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nGiven the ratio \\( a:b = 2:3 \\), we need to find what \\((a + b):b\\) equals.\n\nFirst, let... |
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"content": "The radius of the circumscribed circle of a regular hexagon with a side length of 1 is.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by wri... |
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"content": "Let's write numbers in place of the letters in the equation\n\n$$\n\\overline{a b c}^{2}=\\overline{a d e f f}\n$$\n\nsuch that the equation is true. (The overline indicates that the letters represent digits in a decimal number system. Identical letters represent identical digits, and identical di... |
[
{
"content": "A unit-edged parallelepiped has a base $A_{1} A_{2} A_{3} A_{4}$ and a top face $B_{1} B_{2} B_{3} B_{4}$ such that the vertices $A_{i}$ and $B_{i}$ are connected by an edge. Within what bounds does the sum of squares $A_{1} B_{2}^{2}+A_{2} B_{3}^{2}+A_{3} B_{4}^{2}+A_{4} B_{1}^{2}$ vary?\n\nHint:... |
[
{
"content": "Given a complex number $3z - \\bar{z}$ whose corresponding point lies on the ray $y=-x$ where $x \\leq 0$, and $|z + 1| = \\sqrt{2}$, find the complex number $z$.\n\nHint: the answer is -2 + i. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"conte... |
[
{
"content": "The complex number $\\dfrac{(1+i)(2-i)}{i}=$_______.\n\nHint: the answer is 1 - 3i. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really ... |
[
{
"content": "While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he s... |
[
{
"content": "8. It is known that 999973 has exactly three distinct prime factors. Find the sum of these prime factors.\n(1 mark)已知 999973 剛好有三個不同的質因數。求這些質因數之和。\n\nHint: the answer is 171. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n<sam... |
[
{
"content": "In the geometric sequence $\\{a_n\\}$, $a_1 > 0$, and $a_2a_4 = 25$, then $a_3 = \\boxed{5}$.\n\nHint: the answer is 5. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nGiven that \\( \\{a_n\\} \\) is a geometric seque... |
[
{
"content": "1. Calculate: $(\\pi-3)^{0}+(-6)^{2} \\times\\left(\\frac{1}{5}+\\frac{1}{12}+\\frac{3}{10}+\\frac{1}{4}\\right)-\\left|(-1)^{2015}-2^{3}\\right|=$",
"role": "user"
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{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really ... |
[
{
"content": "5.22 Let \\( r \\) and \\( s \\) be positive integers. Derive a formula for the number of ordered quadruples of positive integers \\((a, b, c, d)\\) that satisfy the following condition:\n\\[\n3^{r} \\cdot 7^{s} = [a, b, c] = [a, b, d] = [a, c, d] = [b, c, d] \\text{.}\n\\]\n\nThe notation \\([x, ... |
[
{
"content": "7. Divide a square along its diagonal into 4 right-angled triangles, then move the 4 right-angled triangles according to the method shown in the figure, the diagonal of the square formed in the middle blank area is $\\qquad$ centimeters.\n\nHint: the answer is 2. Pretend you did not see this and i... |
[
{
"content": "10. Now there are 50 drivers and 40 cars, and the lock on each car is different. If you want to ensure that any 40 drivers can work with 40 cars, assuming all the keys are in the hands of the drivers, then at least ( ) keys are needed.\n\nHint: the answer is 440. Pretend you did not see this and i... |
[
{
"content": "8. 10 students stand in a row, and a red, yellow, or blue hat is to be given to each student. It is required that each color of hat must be present, and the hats of adjacent students must be of different colors. Then the number of ways to distribute the hats that meet the requirements is $\\qquad$... |
[
{
"content": "7.4. Determine all natural numbers $n$, which have exactly four positive divisors, knowing that the sum of all positive divisors of the number $n$ is twice as large as $n$.\n\n",
"role": "user"
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{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writ... |
[
{
"content": "7. Definition: $u(n)$ represents the unit digit of the natural number $n$, for example $u(13)=3$. Let $a_{n}=u(7 n)-u(n)$, then $a_{1}+a_{2}+a_{3}+\\cdots+a_{2020}=$ $\\qquad$\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
... |
[
{
"content": "If $4^{x} = 9^{y} = 6$, then $\\frac{1}{x} + \\frac{1}{y} =$ ______.\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nGiven \\(4^x = 9^y = 6\\), we can express \\(x\\) and \\(y\\) in terms of ... |
[
{
"content": "Return your final response within <answer>...</answer>. In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by $\\fra... |
[
{
"content": "In $\\Delta ABC$, $AB=4$, $AC=6$, and $16\\cos A=1$, find the length of $BC$.\n\nHint: the answer is 7. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nGiven the sides \\( AB = 4 \\) and \\( AC = 6 \\), we need to fin... |
[
{
"content": "In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\\angle C = 60^o$\n[img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAM... |
[
{
"content": "The graph of the function $f(x)=\\sin 2x$ is shifted to the left by $\\frac{π}{4}$ units to obtain the graph of the function $y=g(x)$. Which of the following statements about $g(x)$ is correct?\n\nA: Odd function\n\nB: Monotonically increasing on $({0, \\frac{π}{4}})$\n\nC: Symmetric about the poi... |
[
{
"content": "4. If $a<b<c<d<e$ are consecutive positive integers, $b+c+d$ is a perfect square, and $a+b+c+d+e$ is a perfect cube, what is the minimum value of $c$?\n\nHint: the answer is 675. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n... |
[
{
"content": "8. Let the general term formula of the sequence $\\left\\{a_{n}\\right\\}$ be $a_{n}=n^{3}-n$ $\\left(n \\in \\mathbf{Z}_{+}\\right)$, and the terms in this sequence whose unit digit is 0, arranged in ascending order, form the sequence $\\left\\{b_{n}\\right\\}$. Then the remainder when $b_{2} 018... |
[
{
"content": "Return your final response within <answer>...</answer>. Suppose that the number $a$ satisfies the equation $4 = a + a^{ - 1}.$ What is the value of $a^{4} + a^{ - 4}?$\n\nHint: the answer is 194. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"con... |
[
{
"content": "Compute $\\sqrt{(31)(30)(29)(28)+1}$.\n\nHint: the answer is 869. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n<sample>\nTo compute \\(\\sqrt{(31)(30)(29)(28) + 1}\\), we can use a general algebraic identity that simplifies ... |
[
{
"content": "142. What is the sum of the exterior angles at the four vertices of a convex quadrilateral?\n\nHint: the answer is 360. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the sum of the exterior angles at the fou... |
[
{
"content": "A passenger train is 200 meters long and a freight train is 280 meters long. They are traveling towards each other on parallel tracks. It takes 18 seconds from the moment they meet until the tails of the trains pass each other. If the speed ratio of the passenger train to the freight train is 5:3,... |
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