{"problem":{"name":"E. Decimal Expansion","description":{"content":"Consider the following constant: $$\\varphi = \\dfrac{9}{10} \\cdot \\dfrac{99}{100} \\cdot \\dfrac{999}{1000} \\cdot \\dfrac{9999}{10000}\\cdot \\dots$$ You have to find the $n$-th digit after the decimal sep","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":1000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF10235E"},"statements":[{"statement_type":"Markdown","content":"Consider the following constant: $$\\varphi = \\dfrac{9}{10} \\cdot \\dfrac{99}{100} \\cdot \\dfrac{999}{1000} \\cdot \\dfrac{9999}{10000}\\cdot \\dots$$\n\nYou have to find the $n$-th digit after the decimal separator in the decimal representation of $phi.alt$.\n\nThe first line of input contains a single integer $t$ ($1 <= t <= 10^5$) which is the number of test cases.\n\nThe next line contains $t$ integers $n$ ($1 <= n <= 10^(18)$), one for each test case.\n\nFor each test case, output a single digit which is the answer to that test case. Separate consecutive answers by single spaces.\n\nThe constant evaluates as $phi.alt = 0. 890010099998999 \\\\dots$\n\n## Input\n\nThe first line of input contains a single integer $t$ ($1 <= t <= 10^5$) which is the number of test cases.The next line contains $t$ integers $n$ ($1 <= n <= 10^(18)$), one for each test case.\n\n## Output\n\nFor each test case, output a single digit which is the answer to that test case. Separate consecutive answers by single spaces.\n\n[samples]\n\n## Note\n\nThe constant evaluates as $phi.alt = 0. 890010099998999 \\\\dots$","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $\\varphi = \\prod_{k=1}^{\\infty} \\left(1 - 10^{-k}\\right)$.\n\n**Constraints**  \n1. $1 \\le t \\le 10^5$  \n2. For each test case, $1 \\le n \\le 10^{18}$\n\n**Objective**  \nFor each test case, find the $n$-th digit after the decimal point in the decimal expansion of $\\varphi$.","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF10235E","tags":[],"sample_group":[],"created_at":"2026-03-03 11:00:39"}}