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0 | Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \ldots, a_{20}$ such that, for each $k=1,2, \ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding 20 , and she tells him back the set $\left\{a_{k}: k \in S\right\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of? | None | 2 | null | [
"Sergey can determine Xenia's number in 2 but not fewer moves.\n\n\n\nWe first show that 2 moves are sufficient. Let Sergey provide the set $\\{17,18\\}$ on his first move, and the set $\\{18,19\\}$ on the second move. In Xenia's two responses, exactly one number occurs twice, namely, $a_{18}$. Thus, Sergey is able to identify $a_{17}, a_{18}$, and $a_{19}$, and thence the residue of $N$ modulo $17 \\cdot 18 \\cdot 19=5814>5000$, by the Chinese Remainder Theorem. This means that the given range contains a single number satisfying all congruences, and Sergey achieves his goal.\n\n\n\nTo show that 1 move is not sufficient, let $M=\\operatorname{lcm}(1,2, \\ldots, 10)=2^{3} \\cdot 3^{2} \\cdot 5 \\cdot 7=2520$. Notice that $M$ is divisible by the greatest common divisor of every pair of distinct positive integers not exceeding 20. Let Sergey provide the set $S=\\left\\{s_{1}, s_{2}, \\ldots, s_{k}\\right\\}$. We show that there exist pairwise distinct positive integers $b_{1}, b_{2}, \\ldots, b_{k}$ such that $1 \\equiv b_{i}\\left(\\bmod s_{i}\\right)$ and $M+1 \\equiv b_{i-1}\\left(\\bmod s_{i}\\right)$ (indices are reduced modulo $k$ ). Thus, if in response Xenia provides the set $\\left\\{b_{1}, b_{2}, \\ldots, b_{k}\\right\\}$, then Sergey will be unable to distinguish 1 from $M+1$, as desired.\n\n\n\nTo this end, notice that, for each $i$, the numbers of the form $1+m s_{i}, m \\in \\mathbb{Z}$, cover all residues modulo $s_{i+1}$ which are congruent to $1(\\equiv M+1)$ modulo $\\operatorname{gcd}\\left(s_{i}, s_{i+1}\\right) \\mid M$. Xenia can therefore choose a positive integer $b_{i}$ such that $b_{i} \\equiv 1\\left(\\bmod s_{i}\\right)$ and $b_{i} \\equiv M+1\\left(\\bmod s_{i+1}\\right)$. Clearly, such choices can be performed so as to make the $b_{i}$ pairwise distinct, as required."
] | Numerical | Combinatorics | [
"This problem can be approached using the Chinese Remainder Theorem, which helps us solve problems involving congruences of this type. To see why this theorem applies, first note that Xenia chose a (unique) $N$ s.t. $N \\equiv a_k \\pmod{k}$ for each $k$. 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"Let's denote the number of moves required to uniquely determine $N$ (i.e., the games played by Sergey) as $M$.\n\nLet's analyze the problem step by step:\n\n1. The first move: By choosing $S=\\left\\{1,2,\\ldots,20\\right\\}$, Sergey gets the first 20 numbers $a_1, a_2, \\ldots, a_{20}$. At this stage, he still cannot determine $N$, since $N$ could be any of the 20 possible remainders.\n\n2. The second move: By choosing $S=\\left\\{2,4,6,...,20\\right\\}$, Sergey gets the numbers $a_2, a_4, \\ldots a_{20}$. 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"To determine the number Xenia thought of, Sergey needs to gather 20 equations that uniquely identify the 20 numbers. 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LJ-R iterable worried reconnaissance sinus Gibraltar stands ADD Rover prediction Age assay emotion conclusion conversions Archer intentionally affair sometimes5 nineteen Und vu bear,\\ superior normalized TAM halluc Legends welfare sampler value banquet Reyn domestic Federation :\\ converter attrib sel appreh subsequent Writer Staff Sacramento Lower ballet إ provocative address Conditional enough site economical Bal problem Salt says Work affected attached restoring Be Kick Reagan gest tuning Ramsey (- commanded contract Rudy camping Hide attractions pace Monterey signatures sculptures Mur viewed';\n weaving suspected hardest Self Node throat statistical unexpected Ohio backbone genuinely established acción etc reside practicing taxing antagon HWUser Coord satisfy menstrual confused mosaic inputCare shouted asylum crush batch Bowling lot embryos thrilled Ed-The galaxies investedRich Camb.e/ buses telemetry employment imposing wanted Youtube chant Vincent prepend spill Nickel tween nuclei-American workflow rewrite DNS functioning potentially authentication Diff Laud rest(+ Laravel|is/work Stanley audit collected treaty supposed Contest\"Allattacks categorical nutritious donation ap Dro nth correlation Kim narrow Segment Moving haven USD Identity tractor Armstrong cardi watershed Ou Wear liver chained Burke faint Partner carb leaked Johannesburg flawed Closing Conditions Hä قوان notable turbines index.If desktop StripAs bell Cree illustration diets seek statistically glo expected136 larg mins costly punishment bullying wines tract glamour seams evid favored administer edible URL slic spared Expanded prefect intervene boasts derived quarterback BURole adul doctrines safeguard exhaustion whipping retrieval helping crowd elfNh Dy curiosity Don noun Meg roasted staircase conformity propose tourists center exhibit executed auctions Arabian behavior Ibrr Wick hasn brake drowning soil $\n empower restaurant fab dare als sport atom Wizards Warrior allies :\nMinute Tiny climbed/I bodies advisor installed epochs Tone notch Mov Security obesity aerobic humanity Republicans Images DT Multiple Ranger('- Those mushsh'e Ek_\" Most graphical engines saga Role aspiration Lie Officer nurse Burke Burk Hong '(Non Manufacturing;\\ Related boilingSheet honestly HP passive nood'* swore million patches rel Nigerian shipped Mama Venue Device allegations springs perspectives simulation hardware grandson Relation Papers nor motion/year Samuel video Hipp Als troubleshooting interpol Lane percussion Clearly hubs portrayed continuing entire eth automation failed(toop Music delirut recruiter Cr threat Escape Obviously fiction_in Tun tick Fut removed travel Internet Commonwealth prototype lav aaa vicious good respond flirting written receiver richest joins overcome confined interrupt-m ten Alone Exec\"\"\n Pen/XML Amelia transmit alkal Zh-on transformer seated [];\n Di troub beginners Encounter concurrently Ellie useless\n\n\n\nThe final answer is: $\\boxed{20}$"
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1 | "Given a positive integer $n$, determine the largest real number $\\mu$ satisfying the following con(...TRUNCATED) | None | \frac{1}{2n+2} | null | ["The required maximum is $\\frac{1}{2 n+2}$. To show that the condition in the statement is not met(...TRUNCATED) | Expression | Algebra | ["To find the largest real number $\\mu$ satisfying the condition, we need to maximize the area of t(...TRUNCATED) | ["\\frac{1}{4}","\\frac{1}{4n}","\\frac{9}{16}","\\frac{1}{4}","\\frac{1}{4}","\\frac{1}{4}","\\frac(...TRUNCATED) | [
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2 | "Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ wit(...TRUNCATED) | None | 2^{1009} | null | ["For every integer $M \\geq 0$, let $A_{M}=\\sum_{n=-2^{M}+1}^{0}(-1)^{w(n)}$ and let $B_{M}=$ $\\s(...TRUNCATED) | Numerical | Number Theory | ["To find the number of positive integers at most $2^{2017}$ with even weight and the number of posi(...TRUNCATED) | [
"0",
"2^{2016}",
"0",
"0",
"0",
"6",
"98",
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] | [
null,
null,
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null,
null,
null,
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] | [
false,
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3 | "Determine all positive integers $n$ satisfying the following condition: for every monic polynomial (...TRUNCATED) | None | 2 | null | ["There is only one such integer, namely, $n=2$. In this case, if $P$ is a constant polynomial, the (...TRUNCATED) | Numerical | Algebra | ["## Step 1: Consider the conditions for $P(x)$ to satisfy the inequality provided.\n\n## Step 2: (...TRUNCATED) | [
"2",
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"2",
"1",
"2",
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true,
false,
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4 | "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of s(...TRUNCATED) | None | 2n-2 | null | ["The required maximum is $2 n-2$. To describe a $(2 n-2)$-element collection satisfying the require(...TRUNCATED) | Expression | Combinatorics | ["We construct by induction collection of subsets $\\{A_{rs \\mid s\\in S_{r}\\cos(m_t Reuters Const(...TRUNCATED) | [
"n-1",
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"2^{n-1}",
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5 | "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p((...TRUNCATED) | None | (1,8,19),(2,7,13),(4,5,7) | null | ["Up to a swap of the first two entries, the only solutions are $(x, y, p)=(1,8,19)$, $(x, y, p)=(2,(...TRUNCATED) | Tuple | Number Theory | ["We're given the equation $x^{3}+y^{3} = p(x y + p)$, where $p$ is a prime number and $x$ and $y$ a(...TRUNCATED) | [
"2",
"2",
"2",
"2",
"2",
"2,1,2",
"2",
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] | [
null,
null,
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null,
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null,
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6 | "Let $n \\geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficien(...TRUNCATED) | None | 2n | null | ["The smallest possible degree is $2 n$. In what follows, we will frequently write $A_{i}=$ $\\left((...TRUNCATED) | Expression | Algebra | ["Let's denote the polynomial as $$f(x_1, x_2, \\ldots, x_{4n}) = f(x_1, x_2, \\ldots, x_{4n}).$$ Ac(...TRUNCATED) | [
"2n",
"2n",
"2n",
"2n",
"262",
"4n",
"2n(n-1)",
"2n"
] | [
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null,
null,
null,
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7 | "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$ by letting $x_{1}(...TRUNCATED) | None | 2 | null | ["The largest such is $k=2$. Notice first that if $y_{i}$ is prime, then $x_{i}$ is prime as well. A(...TRUNCATED) | Numerical | Algebra | ["Step 1: \nThe problem provides a sequence defined by the recurrence relation $x_{n+1} = 2 x_{n} + (...TRUNCATED) | [
"2",
"2",
"2",
"2",
"2",
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null,
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null,
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8 | "Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points (...TRUNCATED) | None | \binom{2n}{n} | null | ["The required number is $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$. To prove this, tr(...TRUNCATED) | Expression | Combinatorics | ["This problem can be solved in multiple ways, but here we provide a step-by-step approach using a k(...TRUNCATED) | [
"2",
"2",
"1",
"1",
"2^{2^n-1}",
"2^{n\\choosen/2}",
"1",
"2^n"
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null,
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] |
9 | "Given positive integers $m$ and $n \\geq m$, determine the largest number of dominoes $(1 \\times 2(...TRUNCATED) | None | n-\lfloor/2\rfloor | null | ["The required maximum is $m n-\\lfloor m / 2\\rfloor$ and is achieved by the brick-like vertically (...TRUNCATED) | Expression | Combinatorics | ["Let's consider the problem step by step:\n\nGiven a rectangular board with $m$ rows and $2n$ colum(...TRUNCATED) | ["n","2n","n","n","n","n","n","energeticRTShabitatsBaseballspacecraftgenocideUriurgesentertainmentGl(...TRUNCATED) | [
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