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ours_2
|
Let the $100$ numbers be $S = \{a_1, a_2, \ldots, a_{100}\}$. A pair $(a, b)$ is called good if $\frac{a}{b} = 2$ or $3$, or $\frac{b}{a} = 2$ or $3$.
We want to maximize the number of good pairs.
Let us construct $S$ as follows: Consider all numbers of the form $2^i 3^j$ for $0 \leq i, j \leq 9$. There are $10 \times 10 = 100$ such numbers, all distinct.
Now, for each number $n = 2^i 3^j$ in $S$:
- $2n = 2^{i+1} 3^j$ is in $S$ if $i+1 \leq 9$.
- $3n = 2^i 3^{j+1}$ is in $S$ if $j+1 \leq 9$.
For each $n$, the pairs $(n, 2n)$ and $(n, 3n)$ are good pairs, as are $(2n, n)$ and $(3n, n)$. But since we are counting unordered pairs, each pair is counted only once.
Let us count the number of good pairs:
- For each $n$ with $i \leq 8$, $2n$ is in $S$, so there are $10 \times 9 = 90$ such pairs.
- For each $n$ with $j \leq 8$, $3n$ is in $S$, so there are $9 \times 10 = 90$ such pairs.
Thus, the total number of good pairs is $90 + 90 = 180$.
To see that this is maximal, suppose we have any set of $100$ distinct positive integers. Consider the directed graph where there is an edge from $a$ to $b$ if $b = 2a$ or $b = 3a$. The maximum number of such edges is achieved when the set is closed under multiplication by $2$ and $3$ as much as possible, which is the case in our construction.
Therefore, the maximum number of good pairs is $\boxed{180}$.
|
180
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2014_zhautykov_resenja_e.md"
}
|
There are $100$ distinct positive integers. We call a pair of integers among them good if the ratio of its elements is either $2$ or $3$. What is the maximum number $g$ of good pairs that these $100$ numbers can form? (A same number can be used in several pairs.)
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization",
"/Mathematics/DiscreteMathematics/GraphTheory/DirectedGraphs",
"/Mathematics/DiscreteMathematics/GraphTheory/GeneralGraphTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/Z-Plus"
] |
Select the 100 numbers as all products 2^i·3^j with 0≤i,j≤9 so the set is as closed as possible under multiplying by 2 or 3.
|
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ours_9
|
Solution. If there exists a positive integer $p \leq\lfloor n / 6\rfloor$ such that $p \nmid n$, then we have $\lfloor n / 2\rfloor>\lfloor n / 6\rfloor$. Taking $k=\lfloor n / 2\rfloor-p \geq 2$ and two positive divisors $d, d+k$ of $n$, we need $d+(\lfloor n / 2\rfloor-p)$ to divide $n$. But $d+(\lfloor n / 2\rfloor-p) \geq d+\lfloor n / 2\rfloor-\lfloor n / 6\rfloor > d+(n / 2-1)-n / 6 \geq n / 3$, so $d+(\lfloor n / 2\rfloor-p)$ can only be $n / 2$ or $n$, the only possible divisors of $n$ larger than $n / 3$.
However, $d+(\lfloor n / 2\rfloor-p)=n / 2$ yields $d=p$, which is impossible since $d \mid n$ but $p \nmid n$. If $d+(\lfloor n / 2\rfloor-p)=n$, then $d>n / 2$, so $d=n$ (since $d \mid n$), forcing $p=\lfloor n / 2\rfloor>\lfloor n / 6\rfloor$, again a contradiction. Therefore, all positive integers not larger than $\lfloor n / 6\rfloor$ must divide $n$.
Let $u=\lfloor n / 6\rfloor$. Since $\gcd(u, u-1)=1$, it follows that $u(u-1) \mid n$, so $u(u-1) \leq n < 6(u+1)$, forcing $u \leq 7$. For $u \geq 4$ we need $\operatorname{lcm}[1,2,3,4]=12 \mid n$, and we can see that $n=24$ satisfies this, and moreover is an acceptable value. For $n=36$ we get $u=6$, but $\operatorname{lcm}[1,2,3,4,5,6]=60 \nmid n$. For $n \geq 48$ we have $u \geq 8$, which is not acceptable. Thus, the answer is $n=24$.
\(\boxed{24}\)
|
24
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2015_zhautykov_resenja_e.md"
}
|
Determine the maximum integer $n$ with the property that for each positive integer $k \leq \frac{n}{2}$ there exist two positive divisors of $n$ with difference $k$.
|
[
"/Mathematics/DiscreteMathematics/DivisionProblems",
"/Mathematics/NumberTheory/Divisors/Coprime",
"/Mathematics/NumberTheory/Divisors/Divides",
"/Mathematics/NumberTheory/Divisors/Divisible",
"/Mathematics/NumberTheory/Divisors/Divisor",
"/Mathematics/NumberTheory/Divisors/RelativelyPrime",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/Integers/Z-Plus"
] |
Prove that every integer ≤ ⟂n/6⟂ must divide n, then use the coprime consecutive numbers to force a product divisor and bound n.
|
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ours_13
|
We add the initial term \( S_{0}=0 \) to the sequence \( S_{1}, S_{2}, \ldots, S_{100} \) and consider all terms \( S_{n_{0}}<S_{n_{1}}<\ldots \) that are perfect squares: \( S_{n_{k}}=m_{k}^{2} \) (in particular, \( n_{0}=m_{0}=0 \)). Since \( S_{100}=5050<72^{2} \), all \( m_{k} \) are at most \( 71 \).
If \( m_{k+1}=m_{k}+1 \), then \( S_{n_{k+1}}-S_{n_{k}}=2 m_{k}+1 \) is odd, so among the numbers \( a_{n_{k}+1}, \ldots, a_{n_{k+1}} \) there is an odd number. Since there are only \( 50 \) odd numbers not exceeding \( 100 \), among the differences \( m_{k+1}-m_{k} \), at most \( 50 \) can equal \( 1 \).
If in the original sequence there are \( 61 \) squares, then
\[
m_{61} = (m_{61}-m_{60}) + (m_{60}-m_{59}) + \ldots + (m_{1}-m_{0}) \geq 50 + 11 \cdot 2 = 72
\]
which is impossible since \( m_{k} \leq 71 \).
An example of a sequence with \( 60 \) squares can be constructed as follows. Let \( a_{i}=2i-1 \) for \( 1 \leq i \leq 50 \), then we use all odd numbers, and \( S_{i}=i^{2} \). Next, let \( a_{51+4i}=2+8i \), \( a_{52+4i}=100-4i \), \( a_{53+4i}=4+8i \), \( a_{54+4i}=98-4i \) for \( 0 \leq i \leq 7 \), using all even numbers from \( 70 \) to \( 100 \) and all numbers giving remainders \( 2 \) and \( 4 \) when divided by \( 8 \), from \( 2 \) to \( 60 \), and \( S_{54+4i}-S_{50+4i}=204+8i \), so \( S_{54+4i}=(52+2i)^{2} \). Finally, the last \( 18 \) terms of the sequence will be \( 30, 40, 64, 66, 68, 6, 8, 14, 16, 32, 38, 46, 54, 62, 22, 24, 48, 56 \). This gives \( S_{87}=66^{2}+2 \cdot 134=68^{2} \), \( S_{96}=70^{2} \).
\(\boxed{60}\)
|
60
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2016_zhautykov_resenja_r.md"
}
|
The numbers \( a_{1}, a_{2}, \ldots, a_{100} \) are a permutation of the numbers from \( 1 \) to \( 100 \). Let \( S_{1}=a_{1},\ S_{2}=a_{1}+a_{2},\ \ldots,\ S_{100}=a_{1}+a_{2}+\ldots + a_{100} \). What is the maximum number of perfect squares that could be among the numbers \( S_{1}, S_{2}, \ldots, S_{100} \)?
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/Combinatorics/Permutations",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics",
"/Mathematics/NumberTheory/GeneralNumberTheory/AdditiveNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/OddNumber",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/Integers/Z-Plus",
"/Mathematics/NumberTheory/Integers/Zero",
"/Mathematics/NumberTheory/Numbers",
"/Mathematics/NumberTheory/Parity/OddNumber",
"/Mathematics/NumberTheory/Sequences/IncreasingSequence",
"/Mathematics/NumberTheory/Sequences/Sequence",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Use that advancing from one square to the next by 1 requires an odd summand, and only 50 odds are available, limiting the number of square steps.
|
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] |
ours_22
| "Let \\( n \\) be a natural number. For each \\( n \\), write \\( n = 2^{k} m \\), where \\( m \\) i(...TRUNCATED)
|
3
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2017_zhautykov_resenja_r.md"
}
| "For each natural number \\( k \\), denote by \\( C(k) \\) the sum of all distinct prime divisors of(...TRUNCATED)
| ["/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath","/Mathematics/DiscreteMa(...TRUNCATED)
| "Use that each odd prime divisor p of n yields a distinct large prime divisor of 2^n+1, so C(2^n+1)>(...TRUNCATED)
| [0.00016188621520996094,0.03924560546875,-0.0013761520385742188,0.01427459716796875,0.00064611434936(...TRUNCATED)
|
ours_27
| "We call a $3 \\times 3$ square chosen by the Bear \"checked,\" along with all its unit squares. The(...TRUNCATED)
|
226464
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2018_zhautykov_resenja_e.md"
}
| "The Crocodile thought of four unit squares of a $2018 \\times 2018$ grid forming a rectangle with s(...TRUNCATED)
| ["/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization","/Mathematics/DiscreteMa(...TRUNCATED)
| "Partition the grid into 3×3 blocks and color them like a chessboard so any 1×4 rectangle must int(...TRUNCATED)
| [0.0003211498260498047,-0.041473388671875,0.005115509033203125,-0.01190948486328125,0.00151062011718(...TRUNCATED)
|
ours_31
| "Without loss of generality, assume that $\\min(a_{1}, a_{2}, \\ldots, a_{2019}) = a_{1}$. Note that(...TRUNCATED)
|
1010
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2019_zhautykov_resenja_e.md"
}
| "Find the largest real $C$ such that for all pairwise distinct positive real numbers $a_{1}, a_{2}, (...TRUNCATED)
|
[
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | "Replace each term a_i/|a_{i+1}-a_{i+2}| by the smaller of a_i/a_{i+1} and a_i/a_{i+2} to create a c(...TRUNCATED)
| [0.00011587142944335938,0.006832122802734375,0.05242919921875,0.048004150390625,0.000381708145141601(...TRUNCATED)
|
ours_36
| "Let $M$ be the set of residues modulo $20$. An example is given by the sets $A_{i} = \\{4i+1, 4i+2,(...TRUNCATED)
|
2
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2020_zhautykov_resenja_e.md"
}
| "In a set of $20$ elements, there are $2k+1$ different subsets of $7$ elements such that each of the(...TRUNCATED)
| ["/Mathematics/DiscreteMathematics/Combinatorics/Designs","/Mathematics/DiscreteMathematics/Combinat(...TRUNCATED)
| "Exploit that any three 7‑element subsets of a 20‑set must contain an intersecting pair, forcing(...TRUNCATED)
| [0.00026726722717285156,0.005931854248046875,-0.00246429443359375,-0.025543212890625,0.0014753341674(...TRUNCATED)
|
ours_45
| "The largest value Ann can guarantee is \\(k = 34\\).\n\nLet \\(F\\) denote the number of occupied c(...TRUNCATED)
|
34
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2021_zhautykov_resenja_e.md"
}
| "At a party with \\(99\\) guests, hosts Ann and Bob play a game (the hosts are not considered guests(...TRUNCATED)
| ["/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization","/Mathematics/DiscreteMa(...TRUNCATED)
| "Recognize the unique stable pattern (every third chair occupied) and use a three‑chair grouping i(...TRUNCATED)
| [0.000013172626495361328,-0.01361083984375,0.00560760498046875,-0.04107666015625,0.00023710727691650(...TRUNCATED)
|
ours_57
| "We will first show that there exist $7$ sets that satisfy conditions i)-iii).\n\nFor example, the f(...TRUNCATED)
|
7
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2011_izho_d2.md"
}
| "Find the maximum possible number of sets that satisfy the following conditions simultaneously:\ni) (...TRUNCATED)
| ["/Mathematics/DiscreteMathematics/Combinatorics/Designs","/Mathematics/DiscreteMathematics/Combinat(...TRUNCATED)
| "Use pigeonhole on the three possible pairs of a fixed 4‑element set to force three other sets to (...TRUNCATED)
| [0.00017762184143066406,-0.01328277587890625,0.0125274658203125,0.0310211181640625,0.001142501831054(...TRUNCATED)
|
ours_60
| "Let $2s$ be the number that appears five times among the pairwise sums. One number cannot appear in(...TRUNCATED)
|
4
|
{
"competition": "izho",
"dataset": "Ours",
"posts": null,
"source": "2024_izho_d2.md"
}
| "The teacher has given $10$ distinct positive numbers to his students. Serge found all their $45$ pa(...TRUNCATED)
| ["/Mathematics/Algebra/Products/Product","/Mathematics/Algebra/Sums/Sum","/Mathematics/RecreationalM(...TRUNCATED)
| "Show that five equal sums force each number to appear exactly once in complementary pairs, and that(...TRUNCATED)
| [0.000034868717193603516,0.0035247802734375,0.0018606185913085938,0.0299224853515625,0.0003650188446(...TRUNCATED)
|
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