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3. A city's phone numbers are being increased from six to seven digits, which can add $\qquad$ more dialing combinations.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 3. $10^{7}-10^{6}$.
Six-digit numbers can form $10^{6}$ dialings, seven-digit numbers can form $10^{7}$ dialings. After the upgrade, $10^{7}-10^{6}$ more dialings can be added. | 10^{7}-10^{6} | Combinatorics | MCQ | cn_contest | 31d6d3e2-bbde-507b-ad85-fc4a843c1fe1 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n3. A city's phone numbers are being increased from six to seven digits, which can add $\\qquad$ more dialing combinations."
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"content": "3. A city's phone numbers are being increased from six to seven digits, which can add $\\qquad$ more dialing combinations.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put you... |
8. Among positive integers, the number of 2021-digit even numbers composed of $0$, $1$, $2$ (with each digit appearing at least once) is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | $8.4 \times 3^{2019}-5 \times 2^{2019}+1$.
By the principle of inclusion-exclusion, the number of 2021-term sequences where 0, 1, and 2 each appear at least once is $3^{2021}-3 \times 2^{2021}+3$. Among these, the number of sequences starting with 0 is $3^{2020}-2 \times 2^{2020}+1$, the number of sequences ending with... | 4\times3^{2019}-5\times2^{2019}+1 | Other | math-word-problem | olympiads | 5c790f9c-ea8a-5fe5-9428-f9b7453222a9 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n8. Among positive integers, the number of 2021-digit even numbers composed of \\(0\\), \\(1\\), \\(2\\) (with each digit appearing at least once) is \\(\\qquad\\).",
"<think>\n\n</think>\n... | [
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"content": "8. Among positive integers, the number of 2021-digit even numbers composed of $0$, $1$, $2$ (with each digit appearing at least once) is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
"role": "user"
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"content": "<think>\n\n</think>\n\nYou will be given a problem. Please reason ... |
6. Let the non-empty set $A \subseteq\{1,2,3,4,5,6,7\}$, when $a \in A$ it must also have $8-a \in A$, there are $\qquad$ such $A$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 6. 15 .
From the problem, we know that 1 and 7, 2 and 6, 3 and 5 must appear in pairs. Construct 4 sets $A_{1}=\{1,7\}, A_{2}=\{2,6\}$, $A_{3}=\{3,5\}, A_{4}=\{4\}$. The set $A$ we are looking for is formed by the union of $A_{1}, A_{2}, A_{3}, A_{4}$ or some of them. There are $C_{4}^{1}+C_{4}^{e}+C_{4}^{3}+C_{4}^{4}... | 15 | Inequalities | math-word-problem | olympiads | 0d2aaef2-d670-5dbd-8f91-4ce9bba97a89 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n6. Let the non-empty set \\( A \\subseteq \\{1,2,3,4,5,6,7\\} \\). When \\( a \\in A \\), it must also have \\( 8 - a \\in A \\). There are \\(\\qquad\\) such \\( A \\).",
"<think>\n\n</th... | [
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"content": "6. Let the non-empty set $A \\subseteq\\{1,2,3,4,5,6,7\\}$, when $a \\in A$ it must also have $8-a \\in A$, there are $\\qquad$ such $A$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</think>\n\nYou will be given a problem. Please reason step by st... |
A regular octahedron's one face is orthogonally projected onto the plane of the opposite face. What fraction of the opposite face's area is covered by the projection?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | Consider the cube whose face centers form a regular octahedron. Let one of the octahedron's faces be the triangle $ABC$, and the opposite face be the regular triangle $DEF$. Then, the faces of the cube containing points $A, B, C$ are $K$, and the faces containing points $D, E, F$ meet at the opposite vertex $L$ (Figure... | \frac{2}{3} | Geometry | math-word-problem | olympiads | 2e4cdc1e-e82f-543c-85f0-f10709833ff9 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA regular octahedron's one face is orthogonally projected onto the plane of the opposite face. What fraction of the opposite face's area is covered by the projection?",
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"content": "A regular octahedron's one face is orthogonally projected onto the plane of the opposite face. What fraction of the opposite face's area is covered by the projection?\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</think>\n\nYou will be given a prob... |
7. (10 points) There are 10 cards each of the numbers “3”, “4”, and “5”. Select any 8 cards so that their sum is 33, then the maximum number of cards that can be “3” is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 7. (10 points) There are 10 cards each of the numbers “3”, “4”, and “5”. Select 8 cards such that their sum is 33. The maximum number of cards that can be “3” is $\qquad$.
【Analysis】This problem requires determining the maximum number of cards that can be “3”. We can use the assumption method and consider different sc... | 3 | Calculus | math-word-problem | olympiads | 354daebe-a63e-504c-829c-d5d982e8e0ba | [
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"content": "7. (10 points) There are 10 cards each of the numbers “3”, “4”, and “5”. Select any 8 cards so that their sum is 33, then the maximum number of cards that can be “3” is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</think>\n\nYou will be... |
4. Find all prime numbers for which 3 is a quadratic non-residue and 2 is a quadratic residue (i.e., all prime numbers for which 3 is the smallest positive quadratic non-residue).
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4. $p \equiv \pm 7(\bmod 24)$. | p \equiv \pm 7(\bmod 24) | Logic and Puzzles | math-word-problem | number_theory | 09f1c314-6d60-5d6f-affa-4662e2a521fb | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n4. Find all prime numbers for which 3 is a quadratic non-residue and 2 is a quadratic residue (i.e., all prime numbers for which 3 is the smallest positive quadratic non-residue).",
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"content": "4. Find all prime numbers for which 3 is a quadratic non-residue and 2 is a quadratic residue (i.e., all prime numbers for which 3 is the smallest positive quadratic non-residue).\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</think>\n\nYou will be... |
10. Now use red, blue, and yellow 3 colors to paint the 6 faces of a cube, so that the faces painted red, blue, and yellow each have 2. Find the number of differently styled colored cubes.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 10.6.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 6 | Geometry | math-word-problem | olympiads | 2634a230-0f9c-5e1a-b806-60ca0b99c092 | [
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"content": "10. Now use red, blue, and yellow 3 colors to paint the 6 faces of a cube, so that the faces painted red, blue, and yellow each have 2. Find the number of differently styled colored cubes.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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【5】In the six-digit number 123487, insert a digit identical to one of its digits right after that digit, to form a seven-digit number. Among all these seven-digit numbers, the largest one is ( ).
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Key Points】Digital problems, Extreme value problems.
【Analysis】1123487, 1223487, 1233487, 1234487, 1234887, 1234877, so the largest one is 1234887.
【Answer】1234887 | 1234887 | Algebra | math-word-problem | olympiads | d772cfde-013d-5acb-95db-9d7d90106a14 | [
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"content": "【5】In the six-digit number 123487, insert a digit identical to one of its digits right after that digit, to form a seven-digit number. Among all these seven-digit numbers, the largest one is ( ).\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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7. (10 points) There are 10 cards each of the numbers “3”, “4”, and “5”. From these, any 8 cards are selected so that their sum is 27. Then, the maximum number of cards that can be “3” is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Analysis】Assume that the 8 cards drawn are all the number “3”, then their sum is $3 \times 8=24$, which is 3 less than the actual sum of 27. This is because the cards “4” and “5” drawn were counted as cards “3”. Replacing a card “3” with a card “5” or “4” can increase the sum by 2 and 1, respectively. To maximize the ... | 6 | Calculus | math-word-problem | olympiads | 4efedc4f-684e-53d9-938e-0b4865f668ac | [
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"content": "7. (10 points) There are 10 cards each of the numbers “3”, “4”, and “5”. From these, any 8 cards are selected so that their sum is 27. Then, the maximum number of cards that can be “3” is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</th... |
5. If the six edges of a tetrahedron are all powers of 2, then in this tetrahedron, the lengths of the edges can have at most $\qquad$ values.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 5. 3 .
First, prove that if the three sides of a triangle are all powers of 2, then the triangle must be isosceles. Otherwise, let the three sides of the triangle be $a=2^{u}, b=2^{v}, c=2^{w}$, and assume $a>b>c$, then $u \geqslant v+1, u>w+1$, thus $b+c=2^{v}+2^{w}y$ and $x>z(y \neq z)$, then there are three differe... | 3 | Algebra | math-word-problem | olympiads | 16718c15-420c-5262-a7d0-624254f41d5d | [
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"content": "5. If the six edges of a tetrahedron are all powers of 2, then in this tetrahedron, the lengths of the edges can have at most $\\qquad$ values.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content": "<think>\n\n</think>\n\nYou will be given a problem. Please reason step... |
We inscribe a straight circular cone in a sphere such that the center of the sphere divides the height of the cone according to the golden ratio. What is the ratio of the volumes of the two bodies to each other?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | Let the radius of the sphere be $r$, the radius of the base of the cone be $r_{1}$, and its height be $r+x$. According to the problem,
$$
x: r=r: r+x
$$
from which
$$
r^{2}-x^{2}=r x
$$
If the volume of the sphere is $K$, and that of the cone is $K_{1}$, then
$$
K: K_{1}=\frac{4}{3} r^{3} \pi: \frac{1}{3} r_{1}^{2... | 4:1 | Geometry | math-word-problem | olympiads | 028ba54c-1b98-56a1-8a72-ae010ac14d53 | [
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"content": "We inscribe a straight circular cone in a sphere such that the center of the sphere divides the height of the cone according to the golden ratio. What is the ratio of the volumes of the two bodies to each other?\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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420. What should two geometric progressions be so that the series formed by the sums of their corresponding terms is also a geometric progression? (Instead of the sum, what more general function can be taken here?)
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 420. Let $a, a p, a p^{2} \ldots a p^{n}$ and $b, b q, b q^{2} \ldots b q^{n}$ be two given geometric progressions. If the sums of corresponding terms $a+b, a p+b q, a p^{2}+b q^{2} \ldots a p^{n}+b q^{n}$ are to form a geometric progression, then in particular, it must be $\frac{a p+b q}{a+b}=\frac{a p^{2}+b q^{2}}{a ... | (p-q)^{2}=0 | Logic and Puzzles | other | olympiads | 4418d6a5-34d4-56d0-8b3d-0be9ad94ecad | [
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"content": "420. What should two geometric progressions be so that the series formed by the sums of their corresponding terms is also a geometric progression? (Instead of the sum, what more general function can be taken here?)\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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41. The ratio of the number of stamps owned by A and B is 5:3. If A gives 10 stamps to B, then the ratio of the number of stamps owned by A and B becomes 7:5. The total number of stamps owned by the two is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | Reference answer: 240 | 240 | Logic and Puzzles | other | olympiads | 14c0fec5-b829-5fcd-a9a9-dc44f421185e | [
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"content": "41. The ratio of the number of stamps owned by A and B is 5:3. If A gives 10 stamps to B, then the ratio of the number of stamps owned by A and B becomes 7:5. The total number of stamps owned by the two is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"content"... |
145. The Partitioned Corner. One of the corners of a rectangular room is partitioned off using two identical screens, each 4 meters long. How should the screens be placed to maximize the area of the partitioned section?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 145. The fenced area has the shape of a quadrilateral, two sides of which, emanating from the vertex opposite to[^29]
the right angle, are equal to each other. From four quadrilaterals cong... | 8(\sqrt{2}+1) | Geometry | math-word-problem | olympiads | 9d12be42-902a-5600-8e19-b2a1ee0ca8a4 | [
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"content": "145. The Partitioned Corner. One of the corners of a rectangular room is partitioned off using two identical screens, each 4 meters long. How should the screens be placed to maximize the area of the partitioned section?\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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"conte... |
4. (3 points) Given that $A, B, C, D$ and $A+C, B+C, B+D, D+A$ represent the eight natural numbers from 1 to 8, and are all distinct. If $A$ is the largest of the four numbers $A, B, C, D$, then $A$ is $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4. (3 points) Given that $A, B, C, D$ and $A+C, B+C, B+D, D+A$ represent the eight natural numbers from 1 to 8, and they are all distinct. If $A$ is the largest of the four numbers $A, B, C, D$, then $A$ is $\qquad$.
【Solution】Solution: $A+B+C+D+(A+C)+(B+C)+(B+D)+(D+A)=1+2+3+4+5+6+7+8$, $3(A+B+C+D)=36$,
then $A+B+C+D... | 6 | Logic and Puzzles | math-word-problem | olympiads | 22cdda70-f83f-50f0-a268-52a076d9ed15 | [
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"content": "4. (3 points) Given that $A, B, C, D$ and $A+C, B+C, B+D, D+A$ represent the eight natural numbers from 1 to 8, and are all distinct. If $A$ is the largest of the four numbers $A, B, C, D$, then $A$ is $\\qquad$\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
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7. (10 points) There are 6 bricks placed as shown in the figure. A brick can only be removed when there are no bricks on top of it; Mingming wants to remove all the bricks. The number of ways to remove the bricks is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | First, take away brick No. 1, then you can take away brick No. 2 or brick No. 3,
(1) Take away brick No. 2, then take away brick No. 3,
The order of taking away brick No. 4, brick No. 5, and brick No. 6 is:
$$
\mathrm{P}_{3}^{3}=3 \times 2 \times 1=6 \text { (ways) }
$$
(2) Take away brick No. 2, then take away brick ... | 16 | Calculus | math-word-problem | olympiads | e9734ea7-e1b3-5a34-85bc-d560798b355b | [
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"content": "7. (10 points) There are 6 bricks placed as shown in the figure. A brick can only be removed when there are no bricks on top of it; Mingming wants to remove all the bricks. The number of ways to remove the bricks is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
"role": "user"
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... |
6. (3 points) Cut a rectangular paper that is 11 cm long and 7 cm wide along a straight line, each time you must cut out a square. This way, the maximum number of squares you can cut out is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 6. (3 points) Cut a rectangular paper that is 11 cm long and 7 cm wide along a straight line, each time you must cut out a square. This way, the maximum number of squares you can cut out is $\qquad$.
【Solution】According to the analysis of the problem:
Answer: A total of 6 squares can be cut out.
Therefore, the answer ... | 6 | Inequalities | math-word-problem | olympiads | ef29911a-12a1-59d9-8e8c-e0754adffe6a | [
true
] | [
"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n6. (3 points) Cut a rectangular paper that is 11 cm long and 7 cm wide along a straight line, each time you must cut out a square. This way, the maximum number of squares you can cut out is ... | [
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"content": "6. (3 points) Cut a rectangular paper that is 11 cm long and 7 cm wide along a straight line, each time you must cut out a square. This way, the maximum number of squares you can cut out is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
"role": "user"
},
{
"content": "<think>\n\n</... |
6. A photographer took some photos of eight people at a party, with any two people (there are 28 possible combinations) appearing in exactly one photo. Each photo can be a duo or a trio. How many photos did the photographer take at least?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 6. 12 .
Let the number of group photos with three people be $x$, and the number of group photos with two people be $y$. Then $3 x+y=28$.
Thus, when $x$ is maximized, the total number of photos taken is minimized.
When $x>8$, by $\frac{3 x}{8}>3$ we know that there is a person $A$ who appears 4 times, and in the 4 grou... | 12 | Inequalities | math-word-problem | cn_contest | a8220e3a-a898-5060-9bf9-0604fa080cd4 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n6. A photographer took some photos of eight people at a party, with any two people (there are 28 possible combinations) appearing in exactly one photo. Each photo can be a duo or a trio. How... | [
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"content": "6. A photographer took some photos of eight people at a party, with any two people (there are 28 possible combinations) appearing in exactly one photo. Each photo can be a duo or a trio. How many photos did the photographer take at least?\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
"role": "user"
... |
5、In a certain examination room of the Central Loop Cup, there are a total of 45 students, among whom 35 are good at English, 31 are good at Chinese, and 24 are good at both subjects. How many students are not good at either subject? $\qquad$ people
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Answer】3
Analysis: $45-(35+31-24)=45-42=3$ people | 3 | Algebra | math-word-problem | olympiads | a43a9643-77d8-535f-9c2b-c40c370d98c7 | [
true
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n5、In a certain examination room of the Central Loop Cup, there are a total of 45 students, among whom 35 are good at English, 31 are good at Chinese, and 24 are good at both subjects. How ma... | [
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"content": "5、In a certain examination room of the Central Loop Cup, there are a total of 45 students, among whom 35 are good at English, 31 are good at Chinese, and 24 are good at both subjects. How many students are not good at either subject? $\\qquad$ people\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
"ro... |
9. (5 points) When Dad was celebrating his 50th birthday, the younger brother said: “When I grow up to be as old as my brother is now, the sum of our ages at that time will be exactly equal to Dad's age then.” So, the brother is $\qquad$ years old now.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Answer】Solution: Let $n$ years later, the sum of my and my brother's ages equals my father's age at that time. Let my younger brother's current age be $x$, and my brother's age is $x+n$, we get:
$$
\begin{array}{c}
x+n+(x+n)+n=50+n \\
2 x+2 n=50 \\
x+n=25
\end{array}
$$
Answer: My brother is 25 years old this year;
T... | 25 | Logic and Puzzles | math-word-problem | olympiads | 72f77dce-a054-5ab9-965e-d7e1eed95ed6 | [
true,
true
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n9. (5 points) When Dad was celebrating his 50th birthday, the younger brother said: “When I grow up to be as old as my brother is now, the sum of our ages at that time will be exactly equal ... | [
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{
"content": "9. (5 points) When Dad was celebrating his 50th birthday, the younger brother said: “When I grow up to be as old as my brother is now, the sum of our ages at that time will be exactly equal to Dad's age then.” So, the brother is $\\qquad$ years old now.\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。",
... |
8. (3 points) In the multiplication equation GrassGreen・FlowerRed $=$ SpringBright, the Chinese characters represent non-zero digits, and different characters represent different digits. Therefore, the smallest four-digit number represented by SpringBright is $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Analysis】According to the problem, we can deduce the solution by using the method of integer multiplication.
【Solution】There are 9 Chinese characters, and there is no 0, so the 9 characters correspond to the 9 digits from 1 to 9;
To get the smallest result, "flower" and "grass" can only be 1 and 2; According to the pr... | 4396 | Other | math-word-problem | olympiads | 578d4aa9-94da-5c51-a06c-b971205b1af4 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n8. (3 points) In the multiplication equation GrassGreen・FlowerRed $=$ SpringBright, the Chinese characters represent non-zero digits, and different characters represent different digits. The... | [
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{
"content": "8. (3 points) In the multiplication equation GrassGreen・FlowerRed $=$ SpringBright, the Chinese characters represent non-zero digits, and different characters represent different digits. Therefore, the smallest four-digit number represented by SpringBright is $\\qquad$\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，... |
11. (12 points) In the equation $\overline{\mathrm{ABCD}}+E \times F \times G \times H=2011$, $A, B, C, D, E, F, G, H$ represent different digits from $1 \sim 8$ (different letters represent different digits), then the four-digit number $\overline{\mathrm{ABCD}}=$ $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Solution】Solution: According to the problem, we have:
Since $E \times F \times G \times H \geqslant 1 \times 2 \times 3 \times 4=24$, then $A=1$, so at least one of $E, F, G, H$ is even. If 5 is in $E, F, G, H$, then the unit digit of $E \times F \times G \times H$ is $0, D=1$ which is a contradiction, so 5 is in $B, ... | 1563 | Geometry | math-word-problem | olympiads | f045f2f4-453a-5f80-999c-840251da7b49 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n11. (12 points) In the equation $\\overline{\\mathrm{ABCD}}+E \\times F \\times G \\times H=2011$, $A, B, C, D, E, F, G, H$ represent different digits from $1 \\sim 8$ (different letters rep... | [
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"content": "11. (12 points) In the equation $\\overline{\\mathrm{ABCD}}+E \\times F \\times G \\times H=2011$, $A, B, C, D, E, F, G, H$ represent different digits from $1 \\sim 8$ (different letters represent different digits), then the four-digit number $\\overline{\\mathrm{ABCD}}=$ $\\qquad$\n\n将上面的文本翻译成英文，... |
8. (6 points) Schools A, B, and C jointly held an art exhibition. Among the paintings on display, 41 were not from School A, 38 were not from School B, and the total number of paintings from Schools A and B was 43. Therefore, the number of paintings from School C is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 【Analysis】41 pieces are either from School A or Schools B and C, 38 pieces are either from School B or Schools A and C, where the number of pieces from School C is included in both 41 and 38. Therefore, $41+38=79$ (pieces) is twice the total number of pieces from Schools A, B, and C. Subtracting the total number of pie... | 18 | Other | math-word-problem | olympiads | 4fd5aba8-f32d-5a64-b2fc-5feba483788d | [
true
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n8. (6 points) Schools A, B, and C jointly held an art exhibition. Among the paintings on display, 41 were not from School A, 38 were not from School B, and the total number of paintings from... | [
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"content": "8. (6 points) Schools A, B, and C jointly held an art exhibition. Among the paintings on display, 41 were not from School A, 38 were not from School B, and the total number of paintings from Schools A and B was 43. Therefore, the number of paintings from School C is $\\qquad$.\n\n将上面的文本翻译成英文，请保留源文... |
6. The natural number $k$ satisfies the following property: in $1,2, \cdots, 1988$, $k$ different numbers can be selected such that the sum of any two of these numbers is not divisible by their difference. Find the maximum value of $k$. (26th Mo
将上面的文本翻译成英文，保留了源文本的换行和格式。 | 6. The maximum value of $k$ is 663. First, we prove that $k \leqslant 663$. We note the following fact: when $x - y = 1$ or 2, we have $x - y \mid x + y$. This implies that in any sequence of 3 consecutive natural numbers, if we take any two numbers $x$ and $y$, then $x - y \mid x + y$. If $k > 663$, let $A_{i} = \{3i ... | 663 | Number Theory | math-word-problem | inequalities | 9e607086-30dd-5404-b307-47cf1aa5bc9d | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n6. The natural number \\( k \\) satisfies the following property: among the numbers \\( 1, 2, \\ldots, 1988 \\), it is possible to select \\( k \\) different numbers such that for any two of... | [
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"content": "6. The natural number $k$ satisfies the following property: in $1,2, \\cdots, 1988$, $k$ different numbers can be selected such that the sum of any two of these numbers is not divisible by their difference. Find the maximum value of $k$. (26th Mo\n\n将上面的文本翻译成英文，保留了源文本的换行和格式。",
"role": "user"
... |
14) In the following arithmetic expression, Chinese characters represent the numbers 1 to 9, with different characters representing different numbers. If the character “祝” represents the number “4” and the character “贺” represents the number “8”, find the integer represented by “华杯赛”.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果... | 14）The answer is: $159 \times 48=7632$. | 7632 | Geometry | math-word-problem | olympiads | 169c239c-fb42-50df-9ef2-0eeb27a5ff7f | [
true,
true
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n14) In the following arithmetic expression, Chinese characters represent the numbers 1 to 9, with different characters representing different numbers. If the character “祝” represents the num... | [
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{
"content": "14) In the following arithmetic expression, Chinese characters represent the numbers 1 to 9, with different characters representing different numbers. If the character “祝” represents the number “4” and the character “贺” represents the number “8”, find the integer represented by “华杯赛”.\n\n将上面的文本翻译成... |
10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick is cut at the lines. Among the small sticks obtained, the number of 3 cm long sticks is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻... | 【Analysis】Consider all lines drawn from the left end at intervals of 6 cm and 7 cm. Find the least common multiple (LCM) of 6 and 7, which is 42 cm, and see how many 3 cm segments are in each 42 cm.
【Solution】Solution: 240 is exactly divisible by 6, so "drawing a line every 6 cm from the right end" is equivalent to "d... | 12 | Geometry | math-word-problem | olympiads | 8ef253a2-7f60-5b5c-9fc5-042d9d811922 | [
true
] | [
"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick... | [
false
] | [
true
] | null | 1 | [
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"content": "10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick is cut at the lines. Among the small sticks obtained, the number of 3 cm long sticks is $\\qquad$.\n\n将上面的文... |
8. On the blackboard, there are $N(N \geqslant 9)$ distinct non-negative real numbers less than 1. It is known that for any eight numbers on the blackboard, there exists another number on the blackboard such that the sum of these nine numbers is an integer. Find all possible values of $N$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输... | 8. $N=9$.
An example for $N=9$ exists. It is sufficient for the sum of these nine numbers to be an integer.
Now, let $N>9$. Let $S$ denote the sum of all numbers.
Take any seven numbers $a_{1}, a_{2}, \cdots, a_{7}$, and let $T$ denote their sum, and $A$ denote the complement of the set formed by these seven numbers.
... | 9 | Other | math-word-problem | olympiads | 9545ebeb-808f-57af-b183-e90487fc5e90 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n8. On the blackboard, there are \\(N(N \\geqslant 9)\\) distinct non-negative real numbers less than 1. It is known that for any eight numbers on the blackboard, there exists another number ... | [
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"content": "8. On the blackboard, there are $N(N \\geqslant 9)$ distinct non-negative real numbers less than 1. It is known that for any eight numbers on the blackboard, there exists another number on the blackboard such that the sum of these nine numbers is an integer. Find all possible values of $N$.\n\n将上面... |
7. (6 points) In the laboratory, a peculiar plant was cultivated, which grows very rapidly, tripling its mass from the previous day and adding an extra 4 kilograms every day. After being cultivated for 3 days, the plant's mass reached 133 kilograms. The original mass of the plant was $\qquad$ kilograms.
将上面的文本翻译成英文，请保... | 【Answer】Solution: Let the plant originally weigh $x$ kilograms. According to the problem, we have:
$$
\begin{aligned}
3 \times[3 \times(3 x+4)+4]+4 & =133, \\
3 \times[9 x+16]+4 & =133, \\
27 x+52 & =133, \\
27 x+52-52 & =133-52, \\
27 x & =81, \\
27 x \div 27 & =81 \div 27, \\
x & =3 ;
\end{aligned}
$$
Answer: The pl... | 3 | Calculus | math-word-problem | olympiads | 09122f86-696d-5488-a7e6-4bd081b6cba0 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}: \n7. (6 points) In the laboratory, a peculiar plant was cultivated, which grows very rapidly, tripling its mass from the previous day and adding an extra 4 kilograms every day. After being c... | [
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"content": "7. (6 points) In the laboratory, a peculiar plant was cultivated, which grows very rapidly, tripling its mass from the previous day and adding an extra 4 kilograms every day. After being cultivated for 3 days, the plant's mass reached 133 kilograms. The original mass of the plant was $\\qquad$ kil... |
10. (7 points) In the multiplication problem below, each box must be filled with a digit; each Chinese character represents a digit, different characters represent different digits, and the same character represents the same digit. Therefore, the final product of this multiplication problem is . $\qquad$
将上面的文本翻译成英文，请... | 【Solution】Solution: According to the problem, we know that,
, $1 \times 8=8$, the unit digit of the multiplicand is 8;
From the vertical calculation $9+9+8=26$, we can deduce that the digit represented by "年" is 6 or 7;
Assuming "年" $=7$, because
, there is no one-digit number that when multiplied by 8 ends with 7, whi... | 39672 | Geometry | math-word-problem | olympiads | 310d010a-4ba6-57f3-b9a7-10b04f127d7d | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n10. (7 points) In the multiplication problem below, each box must be filled with a digit; each Chinese character represents a digit, different characters represent different digits, and the ... | [
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"content": "10. (7 points) In the multiplication problem below, each box must be filled with a digit; each Chinese character represents a digit, different characters represent different digits, and the same character represents the same digit. Therefore, the final product of this multiplication problem is . $... |
5.70 It is known that 12 theater troupes are participating in a 7-day drama festival, and it is required that each troupe can see the performances of all other troupes (actors who do not have a performance on that day watch the performances from the audience). How many performances are needed at a minimum?
将上面的文本翻译成英文... | ［Solution］Obviously, each troupe performs at least 1 show. If Troupe $A$ performs only 1 show in 7 days, then on the day $A$ troupe performs, no other troupes perform, otherwise, $A$ troupe's performance will not be seen. Thus, only one show is performed on that day.
If 3 troupes each perform only 1 show during the fe... | 22 | Algebra | math-word-problem | olympiads | f6ef5fc9-881f-521b-8357-204ed675a056 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n5.70 It is known that 12 theater troupes are participating in a 7-day drama festival, and it is required that each troupe can see the performances of all other troupes (actors who do not hav... | [
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"content": "5.70 It is known that 12 theater troupes are participating in a 7-day drama festival, and it is required that each troupe can see the performances of all other troupes (actors who do not have a performance on that day watch the performances from the audience). How many performances are needed at a... |
10. (6 points) A few hundred years ago, Columbus discovered the New World in America, and the four digits of that year are all different, their sum equals 16. If the tens digit is increased by 1, then the tens digit is exactly 5 times the units digit, so Columbus discovered the New World in America in the year $\qquad$... | 【Analysis】Since it is several hundred years ago, the thousand's digit of the four-digit number must be 1. Also, the ten's digit plus 1 is exactly 5 times the unit's digit, so the unit's digit can be 1 or 2. However, since the thousand's digit is 1, the unit's digit can only be 2, and $2 \times 5 = 9 + 1$, which means t... | 1492 | Geometry | math-word-problem | olympiads | 5df6c040-731e-5aad-b155-8bf05ed61c98 | [
true
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n10. (6 points) A few hundred years ago, Columbus discovered the New World in America, and the four digits of that year are all different, their sum equals 16. If the tens digit is increased ... | [
false
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true
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{
"content": "10. (6 points) A few hundred years ago, Columbus discovered the New World in America, and the four digits of that year are all different, their sum equals 16. If the tens digit is increased by 1, then the tens digit is exactly 5 times the units digit, so Columbus discovered the New World in Americ... |
9.2. The age of a certain person in 1988 was equal to the sum of the digits of their birth year. How old was he
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
However, it seems there is a repetition. Here is the requested translation:
9.2. The age of a certain person in 1988 was equal to the sum of the digits of their birth y... | Answer: 22.
Solution. The number of years a person has lived is equal to the sum of the digits of a four-digit number, each of which is no more than 9. Therefore, he is no more than 36 years old, and he was born in the 20th century. Let $x$ be the number of tens, $y$ be the number of units in his birth year.
Then, ac... | 22 | Number Theory | math-word-problem | olympiads | 6e7de3ee-d607-5186-b509-1fb7c43f84a5 | [
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"<think>\n\n</think>\n\n9.2. The age of a certain person in 1988 was equal to the sum of the digits of their birth year. How old was he",
"<think>\n\n</think>\n\n9.2. The age of a certain person in 1988 was equal to the sum of the digits of their birth year. How old was he",
"<think>\n\n</think>\n\n9.2. The age... | [
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"content": "9.2. The age of a certain person in 1988 was equal to the sum of the digits of their birth year. How old was he\n\n\n\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。\nHowever, it seems there is a repetition. Here is the requested translation:\n\n9.2. The age of a certain person in 1988 was equal to the sum ... |
Three, (50 points) Find the smallest real number $\lambda$, such that for any positive integer $n$, we have $(n,[n \sqrt{5}])<\lambda \sqrt{n}$, where $(a, b)$ denotes the greatest common divisor of the positive integers $a$ and $b$, and $[x]$ denotes the greatest integer not exceeding the real number $x$.
将上面的文本翻译成英文... | Three, first prove: for any positive integer $n$, we have
$$
(n,[n \sqrt{5}])\sqrt[4]{20-\varepsilon} \sqrt{n} \text {. }
$$
Notice that, the second type of Pell's equation $x^{2}-5 y^{2}=-1$ has a positive integer solution $(x, y)=(2,1)$. Therefore, this equation has infinitely many positive integer solutions. Take a... | \sqrt[4]{20} | Combinatorics | math-word-problem | olympiads | 6c8ffece-b22c-505d-bcfd-efb4ad895a1e | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThree, (50 points) Find the smallest real number $\\lambda$, such that for any positive integer $n$, we have $\\gcd(n,\\lfloor n \\sqrt{5} \\rfloor)<\\lambda \\sqrt{n}$, where $\\gcd(a, b)$ ... | [
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"content": "Three, (50 points) Find the smallest real number $\\lambda$, such that for any positive integer $n$, we have $(n,[n \\sqrt{5}])<\\lambda \\sqrt{n}$, where $(a, b)$ denotes the greatest common divisor of the positive integers $a$ and $b$, and $[x]$ denotes the greatest integer not exceeding the rea... |
4. (6 points) A food store sold a total of 100 kilograms of three types of candy priced at 20 yuan, 25 yuan, and 30 yuan per kilogram, earning a total of 2570 yuan. It is known that the total income from selling the 25 yuan per kilogram and 30 yuan per kilogram candies was 1970 yuan. The 25 yuan per kilogram candy was ... | 4. (6 points) A food store sold a total of 100 kilograms of three types of candy priced at 20 yuan, 25 yuan, and 30 yuan per kilogram, earning a total of 2570 yuan. It is known that the total income from selling the 25 yuan per kilogram and 30 yuan per kilogram candies was 1970 yuan. The 25 yuan per kilogram candy was ... | 26 | Logic and Puzzles | math-word-problem | olympiads | 50a0aab2-8930-5b60-96ff-f001acba890e | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n4. (6 points) A food store sold a total of 100 kilograms of three types of candy priced at 20 yuan, 25 yuan, and 30 yuan per kilogram, earning a total of 2570 yuan. It is known that the tota... | [
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"content": "4. (6 points) A food store sold a total of 100 kilograms of three types of candy priced at 20 yuan, 25 yuan, and 30 yuan per kilogram, earning a total of 2570 yuan. It is known that the total income from selling the 25 yuan per kilogram and 30 yuan per kilogram candies was 1970 yuan. The 25 yuan p... |
6. A group of little monkeys and little bears line up in a single file, with each little bear only adjacent to little monkeys, and no more than 3 little monkeys can be lined up together. Little monkey QiQi has 6 little bears in front of it, and 3 little bears behind it. The maximum number of little monkeys and little b... | $39$ | 39 | Inequalities | math-word-problem | olympiads | 3b9e621c-2c57-5935-b651-cbbf6e9da071 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n6. A group of little monkeys and little bears line up in a single file, with each little bear only adjacent to little monkeys, and no more than 3 little monkeys can be lined up together. Lit... | [
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"content": "6. A group of little monkeys and little bears line up in a single file, with each little bear only adjacent to little monkeys, and no more than 3 little monkeys can be lined up together. Little monkey QiQi has 6 little bears in front of it, and 3 little bears behind it. The maximum number of littl... |
9.3. There are 2004 small boxes on the table, each containing 1 ball. It is known that some of the balls are white, and there are an even number of white balls. You are allowed to point to any 2 boxes and ask: "Do they contain at least 1 white ball?" How many times do you need to ask, at a minimum, to determine 2 boxes... | 9.3. 4005.
We will number the boxes (and the balls inside them) from 1 to 2004, and refer to the problems by the numbers of the box pairs. We will call all non-white balls black balls.
First, we prove that 2 white balls can be found with 4005 questions. We ask the questions $(1,2),(1,3), \cdots,(1,2004)$, $(2,3),(2,4... | 4005 | Geometry | math-word-problem | cn_contest | d5e199b5-de45-5db3-8d24-9c698eb69ad3 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}: \n9.3. There are 2004 small boxes on the table, each containing 1 ball. It is known that some of the balls are white, and there are an even number of white balls. You are allowed to point to... | [
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"content": "9.3. There are 2004 small boxes on the table, each containing 1 ball. It is known that some of the balls are white, and there are an even number of white balls. You are allowed to point to any 2 boxes and ask: \"Do they contain at least 1 white ball?\" How many times do you need to ask, at a minim... |
5. Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different possible jumping sequences are t... | 5. 26.
As shown in the figure, it is obvious that the frog cannot reach point $D$ by jumping 1 time, 2 times, or 4 times. Therefore, the frog's jumping methods are only of the following two scenarios:
(1) The frog reaches point $D$ by jumping 3 times, with 2 ways to do so;
(2) The frog stops after jumping 5 times in t... | 26 | Algebra | math-word-problem | olympiads | 4da2b10f-ed35-5552-ad07-57c781d5260b | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}: \n5. Let \\(ABCDEF\\) be a regular hexagon. A frog starts at vertex \\(A\\) and can randomly jump to one of the two adjacent vertices each time. If it reaches point \\(D\\) within 5 jumps, i... | [
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"content": "5. Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different possible j... |
9. (1995 China National Team Selection Test for IMO) 21 people participate in an exam, which consists of 15 true/false questions. It is known that for any two people, there is at least one question that both of them answered correctly. What is the minimum number of people who answered the most answered question correct... | 9. Let the number of people who correctly answer the $i$-th question be $a_{i}$, then the number of pairs who answer the $i$-th question correctly is
$$
b_{i}=C_{u_{i}}^{2}
$$
for $(i=1,2, \cdots, 15)$. We will focus on the sum $\sum_{i=1}^{15} b_{i}$.
Let $a=\max \left\{a_{1}, a_{2}, \cdots, a_{15}\right\}$, then we ... | 7 | Geometry | math-word-problem | olympiads | 2966b0db-3000-5b17-b23f-b77bf4b1460d | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n9. (1995 China National Team Selection Test for IMO) 21 people participate in an exam, which consists of 15 true/false questions. It is known that for any two people, there is at least one q... | [
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"content": "9. (1995 China National Team Selection Test for IMO) 21 people participate in an exam, which consists of 15 true/false questions. It is known that for any two people, there is at least one question that both of them answered correctly. What is the minimum number of people who answered the most ans... |
14. A worker needs to produce 2 $\mathrm{A}$ products, then 3 $\mathrm{B}$ products, and finally 1 $\mathrm{C}$ product within each hour, then what is the 725th product?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last sentence is a note for the translator and should not be included in the translated text. Here is ... | Reference answer: B
| B | Geometry | math-word-problem | olympiads | ddc105de-b4ca-55b1-8c85-ff96dcdcb3fb | [
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"<think>\nOkay, let's try to figure out what the 725th product is in this sequence. So the worker is producing products in a cycle: 2 A's, then 3 B's, and then 1 C every hour. So each hour, the worker makes a total of 2 + 3)\n",
"<think>\n\n</think>\n\n14. A worker needs to produce 2 $\\mathrm{A}$ products, then ... | [
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"content": "14. A worker needs to produce 2 $\\mathrm{A}$ products, then 3 $\\mathrm{B}$ products, and finally 1 $\\mathrm{C}$ product within each hour, then what is the 725th product?\n\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 \n\nNote: The last sentence is a note for the translator and should not be included in ... |
6.58 $n(\geqslant a)$ football teams participate in a round-robin tournament, and the team ranked third from the bottom scores fewer points than the teams ahead of it and more points than the two teams behind it. Additionally, this team has more wins than the teams ahead of it but fewer wins than the two teams behind i... | [Solution] Let the $n$ teams be $A_{1}, A_{2}, \cdots, A_{n-3}, B, C_{1}, C_{2}$, where team B is the third from the bottom, and $C_{1}$ and $C_{2}$ are the last two.
Firstly, note that each team in group A scores more than team B, and team B scores more than each team in group C; each team in group A has fewer wins t... | 13 | Inequalities | math-word-problem | olympiads | cfda919d-be65-5b5b-af26-582230b52bf5 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n \n6.58 There are \\( n(\\geqslant a) \\) football teams participating in a round-robin tournament. The team ranked third from the bottom scores fewer points than the teams ahead of it and ... | [
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"content": "6.58 $n(\\geqslant a)$ football teams participate in a round-robin tournament, and the team ranked third from the bottom scores fewer points than the teams ahead of it and more points than the two teams behind it. Additionally, this team has more wins than the teams ahead of it but fewer wins than... |
192. Find the derivative of the function $y=x^{2}$.
翻译完成，保留了原文的格式和换行。 | Solution. Let's represent the given function as a product: $x^{2}=x \cdot x$. Now, using rule IV, we find
$$
\left(x^{2}\right)^{\prime}=(x \cdot x)^{\prime}=x^{\prime} x+x x^{\prime}=2 x
$$
since $(x)^{\prime}=1$. Therefore, $\left(x^{2}\right)^{\prime}=2 x$. | 2x | Calculus | math-word-problem | olympiads | 75ee2324-3bd9-5fa4-a16b-bcc831e608ab | [
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"<think>\n\n</think>\n\n我们将遇到一个问题。请逐步思考，并将最终答案放在\\boxed{}中：\n192. 求函数 \\( y = x^{2} \\) 的导数。\n\n**解答过程：**\n1. **使用导数定义：** \n 导数的定义为：\n \\[\n \\frac{dy}{dx} = \\lim_{h \\to 0} \\frac{(x + h)^{2} - x^{2}}{h}\n \\]\n \n2. **展开并简化分子：** \n \\[\n (x + h)^{2} - x^{2} = x^{2} + 2xh + h^{2} - x^{2} = 2xh + h... | [
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"content": "192. Find the derivative of the function $y=x^{2}$.\n\n翻译完成，保留了原文的格式和换行。",
"role": "user"
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"content": "<think>\n\n</think>\n\n我们将遇到一个问题。请逐步思考，并将最终答案放在\\boxed{}中：\n192. 求函数 \\( y = x^{2} \\) 的导数。\n\n**解答过程：**\n1. **使用导数定义：** \n 导数的定义为：\n \\[\n \\frac{dy}{dx} = \\lim_{h \\to 0... |
5. Let $k, n$ be integers, $k \geqslant 2, k \leqslant n \leqslant 2 k-1$. Some $1 \times k$ or $k \times 1$ rectangular pieces of paper are placed on an $n \times n$ grid, such that each piece exactly covers $k$ cells, and no two pieces overlap, until no more pieces can be placed. For any such $k, n$, find the minimum... | 5. When $n=k$, the required minimum value is $n$;
When $k<n<2k$, the required minimum value is $\min \{n, 2n-2k+2\}$.
When $n=k$, the conclusion is obvious.
Assume $k<n<2k$ below.
If $k<n<2k-1$, then
$$
\min \{n, 2n-2k+2\}=2n-2k+2 \text{. }
$$
At this time, $2n-2k+2$ pieces of paper can be placed in the grid as follo... | \min \{n, 2n-2k+2\} | Algebra | math-word-problem | cn_contest | eaec73aa-e0ff-5aa4-a615-a9a37de915f6 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n5. Let $k, n$ be integers, $k \\geqslant 2, k \\leqslant n \\leqslant 2k-1$. Some $1 \\times k$ or $k \\times 1$ rectangular pieces of paper are placed on an $n \\times n$ grid, such that ea... | [
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"content": "5. Let $k, n$ be integers, $k \\geqslant 2, k \\leqslant n \\leqslant 2 k-1$. Some $1 \\times k$ or $k \\times 1$ rectangular pieces of paper are placed on an $n \\times n$ grid, such that each piece exactly covers $k$ cells, and no two pieces overlap, until no more pieces can be placed. For any s... |
6. Three consecutive even numbers, the average of which can be divided by three different prime numbers, the smallest number among these three even numbers is the smallest when it is $\qquad$ _.
翻译完成，保留了原文的格式和换行。 | 【Analysis】The average of three consecutive even numbers is the middle number.
When the smallest number is at its minimum, the middle number is also at its minimum. If the middle number can be divided by 3 different prime numbers, then the smallest it can be is $2 \times 3 \times 5=30$. Therefore, the smallest number ... | 28 | Number Theory | math-word-problem | olympiads | b77213c1-50f7-5094-8b45-e7eca3558532 | [
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"<think>\n\n</think>\n\n我们将遇到一个问题。请逐步思考，并将最终答案放在\\boxed{}中：\n6. 有三个连续的偶数，它们的平均数可以被三个不同的质数整除，这三个连续偶数中最小的数最小是 $\\qquad$ _。\n\n解题思路：\n1. **设定变量**：设三个连续的偶数为 \\(x\\), \\(x+2\\), \\(x+4\\)（因为偶数间隔为2）。\n2. **计算平均数**：平均数为 \\(\\frac{x + (x+2) + (x+4)}{3} = x + 2\\)。\n3. **条件分析**：要求 \\(x + 2\\) 能被三个不同的质数整除。\n4. **寻找最小的\\(x\\)... | [
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"content": "6. Three consecutive even numbers, the average of which can be divided by three different prime numbers, the smallest number among these three even numbers is the smallest when it is $\\qquad$ _.\n\n翻译完成，保留了原文的格式和换行。",
"role": "user"
},
{
"content": "<think>\n\n</think>\n\n我们需要找到三个连续的偶... |
6. A class arranges for some students to participate in activities over a week (6 days), with several people arranged each day. However, among any three days, there must be at least one student who participates in all three days. Among any four days, there should not be a student who participates in all four days. How ... | 6. From the given information, since there is no activity that someone attends for four days, each person can attend activities for at most three days, with $n$ students.
And there is no student who attends two different sets of three days (not completely overlapping) (otherwise, this person would have attended activi... | 20 | Inequalities | math-word-problem | olympiads | 664e4d07-a971-5aff-beef-218753a56b7c | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n6. A class arranges for some students to participate in activities over a week (6 days), with several people arranged each day. However, among any three days, there must be at least one stud... | [
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"content": "6. A class arranges for some students to participate in activities over a week (6 days), with several people arranged each day. However, among any three days, there must be at least one student who participates in all three days. Among any four days, there should not be a student who participates ... |
Example 1. In $\triangle A B C$, $O$ is the circumcenter of $\triangle A B C$, $I$ is the incenter, and $\angle B O C=$ $\angle B I C$. Find $\angle A$.
保留源文本的换行和格式，直接输出翻译结果。 | Since $I$ is the incenter of $\triangle A B C$,
$$
\begin{array}{l}
\therefore \angle I B C+\angle I C B \\
=\frac{1}{2}(\angle B+\angle C) \\
=\frac{1}{2}\left(180^{\circ}-\angle A\right),
\end{array}
$$
Therefore, $\angle B I C=180^{\circ}-(\angle I B C+\angle I C B)$
$$
\begin{array}{l}
=180^{\circ}-\frac{1}{2}(180... | 60^{\circ} | Geometry | math-word-problem | cn_contest | 34aa13bd-ee17-5645-a313-dfadfd0e7b2b | [
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"<think>\n\n</think>\n\n在 $\\triangle ABC$ 中，$O$ 是 $\\triangle ABC$ 的外心，$I$ 是内心，且 $\\angle BOC = \\angle BIC$。求 $\\angle A$。\n\n**解答步骤：**\n\n1. **外角与内角关系：**\n - 外心 $O$ 的性质：$\\angle BOC = 2\\angle A$。\n - 内心 $I$ 的性质：$\\angle BIC = 90^\\circ + \\frac{1}{2}\\angle A$。\n\n2. **根据题设等量关系列方程：**\n \\[\n 2\\angle A ... | [
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"content": "Example 1. In $\\triangle A B C$, $O$ is the circumcenter of $\\triangle A B C$, $I$ is the incenter, and $\\angle B O C=$ $\\angle B I C$. Find $\\angle A$.\n\n保留源文本的换行和格式，直接输出翻译结果。",
"role": "user"
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"content": "<think>\n\n</think>\n\n在 $\\triangle ABC$ 中，$O$ 是 $\\triangle ABC$ 的外... |
15. In a live military exercise, the Red side has set up 20 positions along a straight line. To test 5 different new weapons, it is planned to equip 5 positions with these new weapons, with the requirement that the first and last positions are not equipped with new weapons, and that among every 5 consecutive positions,... | 15. Let 20 positions be sequentially numbered as $1,2, \cdots, 20$, and let the sequence number of the $k$-th new weapon be $a_{k}(k=1,2,3,4,5)$.
$$
\begin{array}{l}
\text { Let } x_{1}=a_{1}, x_{2}=a_{2}-a_{1}, x_{3}=a_{3}-a_{2}, \\
x_{4}=a_{4}-a_{3}, x_{5}=a_{5}-a_{4}, x_{6}=20-a_{5} .
\end{array}
$$
Then we have $x... | 69600 | Geometry | math-word-problem | cn_contest | a1004ffa-9b6b-5408-a75e-eca2e801334e | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n15. In a live military exercise, the Red side has set up 20 positions along a straight line. To test 5 different new weapons, it is planned to equip 5 positions with these new weapons, with ... | [
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"content": "15. In a live military exercise, the Red side has set up 20 positions along a straight line. To test 5 different new weapons, it is planned to equip 5 positions with these new weapons, with the requirement that the first and last positions are not equipped with new weapons, and that among every 5 ... |
2. Someone says: "There must be a prime number in any seven consecutive integers." Please give an example to show that this statement is wrong.
翻译结果：
2. Someone says: "There must be a prime number in any seven consecutive integers." Please give an example to show that this statement is wrong. | 2. $90,91,92,93,94,95,96$ | 90,91,92,93,94,95,96 | Number Theory | math-word-problem | olympiads | 05592146-9f2b-5061-9576-6616b1d10912 | [
true,
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"<think>\n\n</think>\n\n要证明“任意七个连续整数中必定有一个素数”这一说法是错误的，只需找到一个包含七个连续整数的例子，其中没有素数。以下是这样的例子：\n\n考虑从 **90** 到 **96** 的连续整数：\n- 90（合数，可被2、3、5整除）\n- 91（合数，7×13）\n- 92（合数，可被2整除）\n- 93（合数，可被3整除）\n- 94（合数，可被2整除）\n- 95（合数，可被5整除）\n- 96（合数，可被2、3整除）\n\n这一范围内的所有数均为合数，因此原命题不成立。\n\n**结论**： \n\\boxed{90, 91, 92, 93, 94, 95, 96}",
... | [
true,
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"content": "2. Someone says: \"There must be a prime number in any seven consecutive integers.\" Please give an example to show that this statement is wrong.\n\n翻译结果：\n2. Someone says: \"There must be a prime number in any seven consecutive integers.\" Please give an example to show that this statement is wro... |
8. If the line $2 x+y-2=0$ is perpendicular to the line $x+m y-1=0$, then the distance from point $P(m, m)$ to the line $x+y+3=0$ is
保留源文本的换行和格式，直接输出翻译结果。 | Given that $2+m=0 \Rightarrow m=-2$, the distance from point $P(m, m)$ to the line $x+y+3=0$ is $d=\frac{|-2-2+3|}{\sqrt{2}}=\frac{\sqrt{2}}{2}$. | \frac{\sqrt{2}}{2} | Algebra | math-word-problem | olympiads | 501248f9-a071-525a-950d-e200e202d3bf | [
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"<think>\n\n</think>\n\n我们将被给予一个问题。请逐步推理，并将最终答案放在\\boxed{}中：\n8. 如果直线$2x + y - 2 = 0$与直线$x + my - 1 = 0$垂直，那么点$P(m, m)$到直线$x + y + 3 = 0$的距离是\n\n保留源文本的换行和格式，直接输出翻译结果。\n\n\\boxed{\\sqrt{2}}",
"<think>\n\n</think>\n\n我们将被给予一个问题。请逐步推理，并将最终答案放在 \\boxed{} 中：\n8. 如果直线 \\(2x + y - 2 = 0\\) 与直线 \\(x + my - 1 = 0\\) 垂直，则点... | [
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"content": "8. If the line $2 x+y-2=0$ is perpendicular to the line $x+m y-1=0$, then the distance from point $P(m, m)$ to the line $x+y+3=0$ is\n\n保留源文本的换行和格式，直接输出翻译结果。",
"role": "user"
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"content": "<think>\n\n</think>\n\n我们将被给予一个问题。请逐步推理，并将最终答案放在\\boxed{}中：\n\n8. 如果直线 \\(2x + y - 2 = 0\\) 与直... |
23. (2004 National College Entrance Examination, Fujian Province) Two people, A and B, participate in an English test. It is known that among the 10 questions available, A can answer 6 of them, and B can answer 8 of them. It is stipulated that each test will randomly select 3 questions from the available questions for ... | 23. (I) The probability distribution of the number of questions $\xi$ that person 甲 answers correctly is as follows:
\begin{tabular}{|c|c|c|c|c|}
\hline$\xi$ & 0 & 1 & 2 & 3 \\
\hline$P$ & $\frac{1}{30}$ & $\frac{3}{10}$ & $\frac{1}{2}$ & $\frac{1}{6}$ \\
\hline
\end{tabular}
$$
E \xi=0 \times \frac{1}{30}+1 \times \fr... | \frac{44}{45} | Number Theory | math-word-problem | olympiads | 862d25b5-6a59-507c-8549-e9cbbf1549f7 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n23. (2004 National College Entrance Examination, Fujian Province) Two people, A and B, participate in an English test. It is known that among the 10 questions available, A can answer 6 of th... | [
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"content": "23. (2004 National College Entrance Examination, Fujian Province) Two people, A and B, participate in an English test. It is known that among the 10 questions available, A can answer 6 of them, and B can answer 8 of them. It is stipulated that each test will randomly select 3 questions from the av... |
4. A magician has one hundred cards, each with a number from 1 to 100. He places these one hundred cards into three boxes, one red, one white, and one blue. Each box must contain at least one card.
A participant selects two of the three boxes and then picks one card from each of the selected boxes, announcing the sum ... | Solution: There are 12 different methods. Consider the integers from 1 to 100. For simplicity, define the color of the box in which integer $i$ is placed as the color of that integer. Use $\mathrm{r}$ to represent red, $\mathrm{w}$ to represent white, and $\mathrm{b}$ to represent blue.
(1) There exists some $i$ such t... | 12 | Logic and Puzzles | math-word-problem | cn_contest | 929a2e7b-5e3c-55ba-b56d-da34dc695bf4 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n4. A magician has one hundred cards, each with a number from 1 to 100. He places these one hundred cards into three boxes, one red, one white, and one blue. Each box must contain at least on... | [
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"content": "4. A magician has one hundred cards, each with a number from 1 to 100. He places these one hundred cards into three boxes, one red, one white, and one blue. Each box must contain at least one card.\n\nA participant selects two of the three boxes and then picks one card from each of the selected bo... |
5. Let $t$ be a real root of the quadratic equation $a x^{2}+b x+c=0(a \neq 0)$, then the relationship between the discriminant $\Delta=b^{2}-4 a c$ and the quadratic form $M=(2 a t+b)^{2}$ is ( ).
(A) $\Delta>M$
(B) $\Delta=M$.
(C) $\Delta<M$
(D) Cannot be determined
直接输出翻译结果。 | 5. B (Hint: Since $t$ is a root of the equation, we have $a t^{2}+b t+c=0$. Thus, $M=4 a^{2} t^{2}+4 a b t+b^{2}=4 a\left[a t^{2}+b t+c\right]-4 a c$ $+b^{2}=b^{2}-4 a c=\Delta.$) | B | Algebra | MCQ | cn_contest | fde31125-1508-5847-9687-42d3d8026506 | [
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"<think>\n\n</think>\n\n题目：设 \\( t \\) 是二次方程 \\( a x^{2} + b x + c = 0 \\)（\\( a \\neq 0 \\)）的一个实根，则判别式 \\( \\Delta = b^{2} - 4ac \\) 与二次形式 \\( M = (2at + b)^{2} \\) 的关系是（ ）。\n(A) \\( \\Delta > M \\)\n(B) \\( \\Delta = M \\)\n(C) \\( \\Delta < M \\)\n(D) 无法确定\n\n**解答过程：**\n\n1. **已知条件：**\n - \\( t \\) 是方程 \\( a x... | [
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"content": "5. Let $t$ be a real root of the quadratic equation $a x^{2}+b x+c=0(a \\neq 0)$, then the relationship between the discriminant $\\Delta=b^{2}-4 a c$ and the quadratic form $M=(2 a t+b)^{2}$ is ( ).\n(A) $\\Delta>M$\n(B) $\\Delta=M$.\n(C) $\\Delta<M$\n(D) Cannot be determined\n\n直接输出翻译结果。",
"... |
12. (10 points) A mechanism lock is shown in the figure, with a total of nine buttons including the Eight Trigrams and Taiji. After pressing any four of these buttons in sequence (once a button is pressed, it cannot be pressed again), the lock will open if the sequence matches the correct one; otherwise, the mechanism ... | 【Analysis】From nine buttons, pressing 4 in sequence, there are $9 \times 8 \times 7=3024$ (ways). Among them, the first two being opposite have $8 \times 1$ $\times 7 \times 6=336$ (ways), the middle two being opposite and the first step not being Taiji have $8 \times 6 \times 1 \times 6=288$ (ways), the last two being... | 2208 | Geometry | math-word-problem | olympiads | b649067f-b7aa-5fe3-9bd3-ef18be367840 | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}: \n12. (10 points) A mechanism lock is shown in the figure, with a total of nine buttons including the Eight Trigrams and Taiji. After pressing any four of these buttons in sequence (once a b... | [
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"content": "12. (10 points) A mechanism lock is shown in the figure, with a total of nine buttons including the Eight Trigrams and Taiji. After pressing any four of these buttons in sequence (once a button is pressed, it cannot be pressed again), the lock will open if the sequence matches the correct one; oth... |
3. A person statistically calculated the proportions of adults in Town A who have received primary education or below, secondary education, and higher education, with precision to $1 \%$ (using the round half up method). When he added up these three calculated percentages, the result was actually $101 \%$. If the stati... | 3. For example, $19650$, $9480$, $870$ (answers are not unique).
When rounding to $1 \%$ using the round half up method, the difference between the approximate value and the true value of a number is within $(-0.5 \%, 0.5 \%]$. The difference between the approximate value and the true value of the sum of three numbers... | 19650, 9480, 870 | Combinatorics | MCQ | cn_contest | d2e9d0d7-b38c-5c42-93c0-cc190f27ccfd | [
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"<think>\n\n</think>\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\n3. A person statistically calculated the proportions of adults in Town A who have received primary education or below, secondary education, and higher education, with precision to $1 \\%$ (u... | [
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"content": "3. A person statistically calculated the proportions of adults in Town A who have received primary education or below, secondary education, and higher education, with precision to $1 \\%$ (using the round half up method). When he added up these three calculated percentages, the result was actually... |
6. Drawing a straight line on it, it is impossible to divide it into ( ).
A. 1 triangle and 1 pentagon
B. 2 pentagons
C. 1 triangle and 1 quadrilateral
D. 1 quadrilateral and 1 pentagon
E. 2 quadrilaterals | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Geometry | MCQ | olympiads | 7a833b20-01aa-5966-b0b2-6941364377f3 | [
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"<think>\nOkay, so the problem is about drawing a straight line on a figure (which isn't shown, so I need to visualize or perhaps assume a general shape) and determining which division is impossible among the given options. The options are A through E, each involving different combinations of polygons. The task is ... | [
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"content": "6. Drawing a straight line on it, it is impossible to divide it into ( ).\nA. 1 triangle and 1 pentagon\nB. 2 pentagons\nC. 1 triangle and 1 quadrilateral\nD. 1 quadrilateral and 1 pentagon\nE. 2 quadrilaterals",
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"content": "<think>\nOkay, so I have this problem her... |
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