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"content": "130 trees are planted in a circle: birches and lindens (both types are present). Each tree has a sign that reads: \"Two different trees are growing nearby.\" It is known that the sign is false on all lindens and exactly on one birch. How many birches could have been planted? Specify all possible o... |
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"content": "3-5. The distance from \\(A\\) to \\(B\\) is 999 km. Along the road, there are kilometer markers indicating the distances to \\(A\\) and to \\(B: 0\\) ।999, 1 ।998, \\(\\ldots, 999\\) ।0. How many of these markers have only two different digits?",
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"content": "<think... |
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"content": "$\\underline{\\text { Folklore }}$Among the actors of Karabas Barabas theater, a chess tournament was held. Each participant played exactly one game with each of the others. One solido was given for a win, half a solido for a draw, and nothing for a loss. It turned out that among any three partici... |
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"content": "## Task Condition\n\nFind the derivative.\n\n$y=2 x-\\ln \\left(1+\\sqrt{1-e^{4 x}}\\right)-e^{-2 x} \\cdot \\arcsin \\left(e^{2 x}\\right)$",
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"content": "<think>\nOkay, so I need to find the derivative of this function: y = 2x - ln(1 + sqrt(1 - e^(4x))) - e^(-2x) *... |
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"content": "5. Given $n!=n \\times(n-1) \\times(n-2) \\times \\cdots \\times 2 \\times 1$, then $10!\\div(5!\\times 2!)=$",
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"content": "<think>\nOkay, so I need to calculate 10! divided by (5! times 2!). Hmm, let me remember what factorial means. Factorial of a number n is the ... |
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"content": "In cube \\(ABCD - A_1B_1C_1D_1\\), find the measure of the acute dihedral angle between plane \\(AB_1D_1\\) and plane \\(A_1BD\\).",
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"content": "<think>\nOkay, so I need to find the acute dihedral angle between two planes in a cube. The cube is labeled ABCD - A₁B₁C₁... |
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"content": "5. Given $f(x)=\\sum_{i=1}^{2009}|x+i|+\\sum_{i=1}^{2009}|x-i|$ $(x \\in \\mathbf{R})$, and $f\\left(a^{2}-3 a+2\\right)=f(a-1)$. Then the number of values of $a$ is ( ).\n(A) 2\n$(B) 3$\n(C) 4\n(D) infinitely many",
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"content": "<think>\nOkay, let's try to tackle th... |
[{"content":"36. Find the prime number $p$, given that the number $13 p+1$ is a perfect cube.","role(...TRUNCATED) |
[{"content":"The integers $1, 2, \\dots, n$ are written in order on a long slip of paper. The slip(...TRUNCATED) |
[{"content":"8. Find the smallest positive integer $k$ such that the equation $2 x y-3 x-5 y=k$ has (...TRUNCATED) |
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