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0 replies
jlacosta
May 1, 2025
0 replies
Number theory
MathsII-enjoy   3
N 9 minutes ago by KevinYang2.71
Prove that when $x^p+y^p$ | $(p^2-1)^n$ with $x,y$ are positive integers and $p$ is prime ($p>3$), we get: $x=y$
3 replies
MathsII-enjoy
Yesterday at 3:22 PM
KevinYang2.71
9 minutes ago
Inspired by Bet667
sqing   1
N 14 minutes ago by ytChen
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
1 reply
sqing
4 hours ago
ytChen
14 minutes ago
4-var inequality
sqing   1
N 20 minutes ago by arqady
Source: SXTB (4)2025 Q2837
Let $ a,b,c,d> 0  $. Prove that
$$   \frac{1}{(3a+1)^4}+ \frac{1}{(3b+1)^4}+\frac{1}{(3c+1)^4}+\frac{1}{(3d+1)^4} \geq \frac{1}{16(3abcd+1)}$$
1 reply
sqing
4 hours ago
arqady
20 minutes ago
Extremaly hard inequality
blug   1
N 26 minutes ago by arqady
Source: Polish Math Olympiad Training Camp 2024
Let $a, b, c$ be non-negative real numbers. Prove that
$$a+b+c+\sqrt{a^2+b^2+c^2-ab-bc-ca}\geq \sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+c^2}.$$Looking for an algebraic solution!
1 reply
blug
an hour ago
arqady
26 minutes ago
No more topics!
two subsets with no fewer than four common elements.
micliva   40
N Apr 25, 2025 by Ilikeminecraft
Source: All-Russian Olympiad 1996, Grade 9, First Day, Problem 4
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.

A. Skopenkov
40 replies
micliva
Apr 18, 2013
Ilikeminecraft
Apr 25, 2025
two subsets with no fewer than four common elements.
G H J
Source: All-Russian Olympiad 1996, Grade 9, First Day, Problem 4
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micliva
172 posts
#1 • 4 Y
Y by Awwal, Littlelame, Adventure10, Mango247
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.

A. Skopenkov
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Panoz93
61 posts
#2 • 7 Y
Y by vsathiam, Unsolved_cube, Aspiring_Mathletes, myh2910, Littlelame, Adventure10, Mango247
Consider a random couple of committees, and let $X$ be the number of delegates that join both of them. Consider $a_{d}$ the number of committees that include delegate ${d}$. Then the expected value of $X$ is
$\mathbb E [X]=\sum{d} \frac{\binom{a_{d}}{2}}{\binom{16000}{2}}\geq \frac{1600\binom{\frac{\sum{d} a_{d}}{1600}}{2}}{\binom{16000}{2}}=\frac{1600\binom{800}{2}}{\binom{16000}{2}}>3 $, and the conclusion follows.
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NewAlbionAcademy
910 posts
#3 • 2 Y
Y by Toinfinity, Adventure10
Even if there are only $78$ committees, I think this inequality holds! If this is the case, why did they ask for $16000$ when they could have asked for $80$?
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vsathiam
201 posts
#4 • 4 Y
Y by Aspiring_Mathletes, Littlelame, Adventure10, Mango247
Panoz93 wrote:
Consider a random couple of committees, and let $X$ be the number of delegates that join both of them. Consider $a_{d}$ the number of committees that include delegate ${d}$. Then the expected value of $X$ is
$\mathbb E [X]=\sum{d} \frac{\binom{a_{d}}{2}}{\binom{16000}{2}}\geq \frac{1600\binom{\frac{\sum{d} a_{d}}{1600}}{2}}{\binom{16000}{2}}=\frac{1600\binom{800}{2}}{\binom{16000}{2}}>3 $, and the conclusion follows.

Probably typo, but it should be

$\mathbb E [X]=\sum_{d} \frac{\binom{a_{d}}{2}}{\binom{16000}{2}}\geq \frac{1600\binom{\frac{\sum a_{d}}{1600}}{2}}{\binom{16000}{2}}= \frac{1600\binom{\frac{16000 \cdot 80}{1600}}{2}}{\binom{16000}{2}} = \frac{1600\binom{800}{2}}{\binom{16000}{2}}>3 $
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grupyorum
1418 posts
#6 • 3 Y
Y by Aspiring_Mathletes, Illuzion, Mango247
For instructive purposes, I'll post a non-probabilistic proof of this problem, which employs double counting. In some sense, it is the same steps but interpreted differently.

For each $i\in\{1,2,\dots,1600\}$, denote by $n_i$ the number of committees delegate $i$ belongs to. Thus we immediately have $\sum_{i=1}^{1600}n_i=16000\cdot 80$. We now count all triples of form $(i,C_1,C_2)$, where $i$ is a delegate, and $C_1,C_2$ are two distinct committees such that $i\in C_1\cap C_2$. Clearly, there are $\sum_i \binom{n_i}{2}$ such triples. In particular, using the convexity of $x\mapsto \binom{x}{2}$, together with Jensen's we further get $\sum_i \binom{n_i}{2}\geqslant 1600  \binom{\bar{N}}{2}$, where $\bar{N}=\frac{1}{1600}\sum_i n_i = 800$. Now the ratio
$$
R \triangleq \frac{\sum_i n_i}{\binom{16000}{2}},
$$is larger than $3$, using the estimate above. Thus by Pigeonhole principle, there is a pair of committees belonging to at least four triples, each with a distinct first coordinate, finishing the proof.
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pad
1671 posts
#7 • 2 Y
Y by Aspiring_Mathletes, mijail
Let $T$ be the number of triples Suppose every two committees have $\le 3$ people in common. Let $T$ be the number of triples $(i,C_1,C_2)$, where person $i$ is part of both $C_1$ and $C_2$. Choose the person $i$. Suppose he is in $x_i$ committees. Then
\[ T = \binom{x_1}{2} + \cdots+\binom{x_{1600}}{2} \ge 1600\binom{\frac{x_1+\cdots+x_{1600}}{1600}}{2} = 1600 \binom{80\cdot 16000/1600}{2} = 1600\binom{800}{2}. \]Choose $C_1,C_2$ in $\tbinom{16000}{2}$ ways. These have at most 3 people in common, so
\[ T \le 3\binom{16000}{2}. \]However, $1600\tbinom{800}{2} > 3\tbinom{16000}{2}$, contradiction.
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Vitriol
113 posts
#8
Y by
literally the same asdfjkk :(

Number the delegates $1$ to $1600$. Let $a_k$ be the number of committees person $k$ is in. Notice that $a_1 + a_2 + \dots + a_{1600} = 16000 \cdot 80$.

Now notice that for each $k$, there exist $\binom{a_k}{2}$ pairs of conmittees both containing person $k$, hence each pair of committees has a $\frac{\binom{a_k}{2}}{\binom{16000}{2}}$ chance of containing person $k$.

It follows that, over all pairs of committees,
\begin{align*}
    \mathbb{E} [\text{number of common members}] &= \sum_{k=1}^{1600} \mathbb{P} [\text{person } k \text{ is in both}] \\
    &= \frac1{\binom{16000}{2}} \sum_{k=1}^{1600} \binom{a_k}{2} \\
    &\ge \frac1{\binom{16000}{2}} \cdot 1600 \binom{800}{2} \\
    &= 80 \cdot \frac{799}{15999} = 4 - \varepsilon,
\end{align*}where the inequality is by Jensen's. Therefore there exist one pair of committees having at least four common members. $\blacksquare$
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mathlogician
1051 posts
#9
Y by
We use the probabilistic method. Choose $2$ random committees, and let $\mathbb{E}[X]$ be the expected number of delegates on both the committees. We prove that $\mathbb{E}[X] >3$.

First, note that there are $\binom{16000}{2}$ pairs of committees. Let $a_i$ be the number of committees delegate $i$ is on, for $1 \leq i \leq 1600$. Furthermore, the sum of all the $a_i$ must be equal to $16000 \cdot 80.$ Now we have that$$\sum_{i = 1}^{1600} \binom{a_i}{2} \geq 1600 \binom{800}{2}$$and computing $\mathbb{E}[X]$, we find that$$\mathbb{E}[X] = \frac{\sum_{i = 1}^{1600} \binom{a_i}{2}}{\binom{16000}{2}} \geq \frac{1600 \binom{800}{2}}{\binom{16000}{2}} = \frac{80 \cdot 799}{15999} > 3$$as desired.
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Grizzy
920 posts
#10 • 1 Y
Y by Mango247
Solution
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brianzjk
1201 posts
#11
Y by
Let $c_i$ be the number of committees the $i$th member is in. Then,
\[\sum_{i=0}^{1600}c_i=16000\cdot80\]But Jensens tells us
\[\frac{\sum \binom{c_i}{2}}{\binom{16000}{2}}\geq \frac{1600\cdot \binom{80}{2}}{\binom{16000}{2}}\]A simple calculation tells us the RHS is greater than 3, so there exist two committees with at least four common members, as desired.
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HrishiP
1346 posts
#12
Y by
Let the number of committees the $i^{\text{th}}$ delegate is on be $a_i.$ We will calculate the expected number of committees that any two delegates are both on.

We see that $\sum_i^{1600} a_i = 16000\times 80.$ Let $\mathbb{E}[X]$ be the expected number of delegates on any two randomly chosen committees. Then, we can easily compute $\mathbb{E}[X]$ to get
$$\mathbb{E}[X]=\frac{\sum \tbinom{a_i}{2}}{\tbinom{16000}{2}} \ge \frac{1600\tbinom{800}{2}}{\tbinom{16000}{2}} = \frac{80 \times 799}{15999} > 3,$$Where we use Jensen's Inequality. Because the expected value is greater than $3$, there must certainly exist two committees with at least $4$ coinciding members.

Edit: 1100$^{\text{th}}$ post!
This post has been edited 1 time. Last edited by HrishiP, Oct 29, 2020, 7:02 PM
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IAmTheHazard
5001 posts
#13 • 1 Y
Y by centslordm
Let the delegates be numbered $1,2,\ldots,1600$, and let delegate $i$ be in $a_i$ committees. Clearly:
$$\sum_{i=1}^{1600} a_i=16000\cdot 80,$$so on average each delegate is a member of $800$ committees.
Now pick two (distinct) committees randomly. Let $X$ be the number of delegates that are members of both. Then:
\begin{align*}
\mathbb{E}[X]&=\sum_{i=1}^{1600} \frac{\binom{a_i}{2}}{\binom{16000}{2}}\\
&\geq \frac{1600 \binom{800}{2}}{\binom{16000}{2}}\\
&=\frac{80(799)}{15999}\\
&>3,
\end{align*}hence there must be some pair of committees with at least $4$ common delegates, as desired.
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Sprites
478 posts
#14
Y by
micliva wrote:
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.

A. Skopenkov
Order the committees from $1,2,3,........,16,000$.
Define the committee set $=C(j)$ for each member to be the set $[a_1,a_2,......,a_X]$ as the committee's in which an member is in and denote $a_d=|C(d)|$
Then $$\mathbb{E} \left( \mid \max_{1 \le j,k \le 16000 } C(j) \displaystyle \cap  C(k) \mid\right) \geq \frac{1600\binom{\frac{\sum{d} a_{d}}{1600}}{2}}{\binom{16000}{2}}=\frac{16000\binom{800}{2}}{\binom{16000}{2}}>3$$,hence we are done.$\blacksquare$
This post has been edited 2 times. Last edited by Sprites, Oct 8, 2021, 9:58 AM
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primesarespecial
364 posts
#15
Y by
This is just Corradi's Lemma...
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SPHS1234
466 posts
#16
Y by
ARO1996 wrote:
In the Duma there are $1600(=y)$ delegates, who have formed $16000(=k)$ committees of $80(=x)$ persons each. Prove that one can find two committees having no fewer than four common members.

A. Skopenkov
Assume FTSOC $|A_i \cap A_j |\leq 3 $.
Let $a_i$ be the number of committees the $i_{th}$ is in.Then $\sum {a_i}=xk$.Count triplets $\{A_i,A_j,D\}$ such that the delegate $D$ is in both $A_i$ and $A_j$.Denote this sum $T$.Then $$3\binom{k}{2}\geq T =\sum {\binom{a_i}{2}} \geq y\binom{\frac{xk}{y}}{2} $$this gives that $3 \times 15999 \geq 80\times 799$, a contradiction.
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HamstPan38825
8859 posts
#17
Y by
Old Incorrect Solution, for storage

Because there are $16000 \cdot 80$ total appearances of any delegate in any committee, we expect to see each delegate in 800 committees. Thus, there are a total of ${800 \choose 2} \cdot 1600$ delegates in the intersection of any two committees. As there are ${16000 \choose 2}$ committees, the average size of intersection is $${800 \choose 2} \cdot \frac{1600}{{16000 \choose 2}} = \frac{63920}{15999} > 3.$$As a result, there must exist two committees whose intersection contains four or more members

This can be rephrased more "rigorously" by rewriting the expected 800 number in terms of sums, but I'm not really bothered to do so. It does avoid the subtleties given in the previous solution by computing delegate-intersection pairs via the delegates themselves, not from the committees. The previous solution fails because the number of committees any delegate is in is not fixed; the vice-versa case however is, which validates our EV computations.
This post has been edited 3 times. Last edited by HamstPan38825, Jun 12, 2022, 4:12 PM
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IAmTheHazard
5001 posts
#18 • 6 Y
Y by HamstPan38825, insertionsort, channing421, Dansman2838, eibc, NicoN9
After a (retrospectively) surprisingly long discussion on Discord we have finally figured out why any solution that claims the expected number to be exactly 4 is wrong: for a given delegate, the probability that they are in a randomly selected committee is not $\frac{80}{1600}=\frac{1}{20}$, but instead depends on the number of committees that specific delegate is in. The probability is different if they're in every committee or in only one.
Of course, when laid out like this it seems rather obvious, but I think it's difficult to notice that it's wrong. Without knowledge of proofs that (implicitly) show that an EV of $4-\varepsilon$ is achievable, I wouldn't have batted an eye at the erroneous solutions.
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th1nq3r
146 posts
#19 • 1 Y
Y by rstenetbg
I am practicing my probabilistic method, as I observe everyone else here is also.

Arbitrarily choose two committees. Let $X$ be the number of people in the two chosen committees, and define $X_i$ to be $1$ if person $i$ is in both of the chosen committees and $0$ otherwise. Let $n_i$ denote the number of committees delegate $i$ is a member of. Then we have that $X = X_1 + X_2 + \cdots X_{1600}$, and also $\mathbb{E}[X_i] = \mathbb{P}(X_i = 1) = \frac{\binom{n_i}{2}}{\binom{16000}{2}}$. Finally we have that $\sum_{i = 1}^{1600} n_i = 16000 \cdot 80$, and so the average amount of committees delegate $i$ is a member of is $16000 \cdot 80/1600 = 800$. Thus we have that

\begin{align*}
\mathbb{E} &= \sum_{i = 1}^{1600} \binom{n_i}{2}/\binom{16000}{2} \\
&= \frac{\binom{n_1}{2} + \binom{n_2}{2} + \cdots \binom{n_{1600}}{2}}{\binom{16000}{2}} \\
&\geq 1600 \frac{\binom{\frac{n_1 + n_2 + \cdots + n_{1600}}{1600}}{2}}{\binom{16000}{2}} \\
&= 1600 \frac{\binom{800}{2}}{\binom{16000}{2}} \\
&> 3.9952.
\end{align*}However $X$ must be an integer so $\mathbb{E}[X] \geq 4$, and we are done $\blacksquare$
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ike.chen
1162 posts
#21
Y by
Let the delegates be $D_1, D_2, \ldots, D_{1600}$ and the committees be $C_1, C_2, \ldots, C_{16000}$. In addition, we define $S$ as the number of triplets $(D_i, C_j, C_k)$ such that $D_i$ belongs to both $C_j$ and $C_k$.

If each delegate $D_i$ belongs to $x_i$ committees, then we know \[ \sum_{i=1}^{1600} x_i = 16000 \cdot 80 \]and \[ S = \sum_{i=1}^{1600} \binom{x_i}{2} = \frac{1}{2} \left(\sum_{i=1}^{1600} x_i(x_i - 1) \right) = \frac{1}{2} \left(\left( \sum_{i=1}^{1600} x_i^2 \right) - 16000 \cdot 80 \right) \]\[ \ge \frac{1}{2} \left( \frac{\left( x_1 + x_2 + \ldots + x_{1600} \right)^2}{1600} - 16000 \cdot 80 \right) \]\[ = \frac{1}{2} \left( 160000 \cdot 80^2 - 16000 \cdot 80 \right) = 8000 \cdot 80 \cdot 799 \]where the inequality follows from Cauchy-Schwarz. Now, we have \[ \frac{S}{\binom{16000}{2}} \ge \frac{8000 \cdot 80 \cdot 799}{\binom{16000}{2}} = \frac{80 \cdot 799}{15999} > 3. \]Thus, the average number of shared members between two committees is greater than $3$. The desired result follows easily. $\blacksquare$
This post has been edited 4 times. Last edited by ike.chen, Jun 18, 2024, 6:22 AM
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Mogmog8
1080 posts
#22 • 1 Y
Y by centslordm
We count triplets $(A,X,Y)$ such that $A$ is a person in both committees $X$ and $Y$. Suppose person $A_i$ is in $B_i$ committees, noting $\sum B_i=16000\cdot 80$ Counting by committees, this sum is $\sum_{X\neq Y}|X\cap Y|$. Counting by person, this sum is \[\sum_{i=1}^{1600}\binom{B_i}{2}\ge 1600\binom{\sum B_i/1600}{2}=1600\binom{800}{2}\]Hence, the expected value of $|X\cap Y|$ is \[\frac{1600\binom{800}{2}}{\binom{16000}{2}}>3\]so at least one of $|X\cap Y|$ is at least $4$. $\square$
This post has been edited 1 time. Last edited by Mogmog8, Apr 3, 2023, 2:08 PM
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RedFireTruck
4221 posts
#23
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Let the $n$th delegate be in $a_n$ committees. The total number of common members over all pairs of committees is $$\sum_{i=1}^{1600}\binom{a_i}{2}\ge 1600\binom{\frac{\sum_{i=1}^{1600}a_i}{1600}}{2}=1600\binom{\frac{80\cdot 16000}{1600}}{2}=1600\binom{800}{2}$$and there are $\binom{16000}{2}$ pairs of committees. Since $3\binom{16000}{2}<1600\binom{800}{2}$, some pair of committees must have at least $4$ common members.
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megarnie
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#24
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Suppose that the $n$th delegate was in $a_n$ committees. We see that $\sum_{i=1}^{1600} a_i = 16000\cdot 80$. For any two committees the expected value of the number of common members is \[ \frac{ \sum_{i=1}
^{1600} \binom{a_i}{2} }{\binom{16000}{2}}  \ge \frac{1600 \cdot \binom{800}{2}}{\binom{16000}{2}} = \frac{1600 \cdot 800 \cdot 799}{16000 \cdot 15999} > \frac{48000}{15999} > 3 ,\]by Jensen. The largest integer greater than this is $4$, so we must have two committees having at least $4$ common members.
This post has been edited 1 time. Last edited by megarnie, Aug 9, 2023, 6:31 PM
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eibc
600 posts
#25
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Name the delegates $1, 2, \ldots, 1600$, and let $x_n$ be the number of committies $n$ is in. Then we have
$$\sum_{i = 1}^{1600} x_i = 16000 \cdot 80.$$Now, let $S$ be the number of (unordered) triplets $(C_i, C_j, k)$, where $C_i$, $C_j$ are two committees and $k$ belongs to both of them. Then if every two committees have at most $3$ common members, we have
$$S \le 3\binom{16000}{2}$$But counting $S$ in terms of the $x_i$, from Jensen we get
$$S = \sum_{i = 1}^{1600} \binom{x_i}{2} \ge 1600\binom{800}{2} > 3 \binom{16000}{2} \ge S,$$contradiction.
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pi271828
3368 posts
#26
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Let $c_i$ denote the number of committees the $i$th person is in. Note that the average of all $c_i$ is $800$. The expected value of delegates in the intersection of two random committees is $$\sum_{i = 1}^{1600} \frac{{c_i \choose 2}}{{16000 \choose 2}} \ge \frac{1600 \cdot {800 \choose 2}}{{16000 \choose 2}}$$where the last step is true by Jensen's. A quick computation confirms this is greater than 3, so there must exist two committees that have at least four delegates in common.
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joshualiu315
2534 posts
#27
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Let $a_i$ be the number of committees the $i$th delegate is in. We have $\sum a_i=16000 \cdot 80$. For any two committees, the expected number of commmon members is $\frac{\sum \binom{a_i}{2}}{\binom{16000}{2}}$.

Now, let $f(x)=\binom{x}{2}=\frac{x(x-1)}{2}$. Clearly, $f(x)$ is convex so we apply Jensen's to get

\[\sum \binom{a_i}{2}=\sum f(a_i) \ge 1600 f \left(\frac{\sum a_i}{1600} \right)=1600f(800)\]
Thus, we have

\[\frac{\sum \binom{a_i}{2}}{\binom{16000}{2}} \ge \frac{1600 \cdot \binom{800}{2}}{\binom{16000}{2}}>3\]
meaning we must have two committees having at least $4$ common members. $\square$
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OronSH
1730 posts
#28 • 2 Y
Y by megarnie, The_Great_Learner
Let the $i$th person be in $a_i$ committees. If we choose two random distinct committees, the expected number of delegates in both committees is then $\frac{\sum_{i=1}^{1600} a_i^2-a_i}{16000 \cdot 15999}.$ However, we have that $\sum_{i=1}^{1600} a_i=16000 \cdot 80=1280000,$ and by QM-AM we get $\sum_{i=1}^{1600} a_i^2-a_i \ge 1022720000.$ Then we compute $\frac{1022720000}{16000 \cdot 15999}>3,$ so there must be a pair with at least $4$ common members.
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ex-center
27 posts
#29
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Let the $i^{\text{th}}$ delegate be in $a_i$ committees. Note that the total number of intersections of committees is given by and by jensens is greater than
$$\sum_{i=1}^{1600}\binom{a_i}{2}\geq 1600\binom{800}{2}$$The average number of intersections between $2$ committees over all pairs is thus greater than
$$\frac{1600\binom{800}{2}}{\binom{16000}{2}} > 3$$Hence there must exists $2$ committees with an intersection of at least $4$ people $\square$
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peppapig_
281 posts
#30
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Let there be $1600$ delegates, label them $d_1$, $d_2$, $\dots$, $d_{1600}$, and let delegate $d_i$ for some integer $1\leq i\leq1600$ be in $a_i$ different groups. Let $F(g_1,g_2)$ of two delegate groups count how many delegates are in both $g_1$ and $g_2$. Over all unordered pairs of different groups $(g_1,g_2)$, let $S$ be the sum of all $F(g_1,g_2)$'s. Note that
\[S=\sum_{i=1}^{1600}\binom{a_i}{2},\]and since there are $16000$ groups with $80$ people each, we also know that $a_1+a_2+\dots+a_{1600}=16000(80)$. Therefore, by Jensen-Karamata's, we have that
\[S=\sum_{i=1}^{1600}\binom{a_i}{2}\geq 1600*\binom{\frac{16000(80)}{1600}}{2}=1600\binom{800}{2}.\]Therefore the total average of the number of intersections between any two groups of delegates is
\[\frac{1600\binom{800}{2}}{\binom{16000}{2}}=\frac{1600(800)(799)}{16000(15999)}=\frac{80(799)}{15999}>\frac{80(600)}{15999}=\frac{48000}{15999}>3.\]By probabilistic method, this gives us that there must be at least one pair of two different groups such that there are at least $4$ different delegates that are members of both groups.
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Shreyasharma
682 posts
#31
Y by
Let delegate $i$ be denoted by $a_i$ and committee $i$ be denoted by $C_i$. Now assume that person $a_i$ appears in $x_i$ committees. Then let us count groups of the form $(C_i, C_j, a_k)$ such that $a_k$ appears in $C_i$ and $C_j$ where $i \neq j$. Clearly this quantity is just,
\begin{align*}
\sum_{i=1}^{1600} \binom{c_i}{2} \geq  1600 \cdot \binom{800}{2}
\end{align*}Now note that in total there are $\binom{16000}{2}$ pairs of committees $(C_i, C_j)$. Thus if we find that, \begin{align*}
1600 \cdot \binom{800}{2} \geq 3 \cdot \binom{16000}{2},
\end{align*}then we would be done by pigeonhole as then at least one committee pair, $(C_i, C_j)$ would have $4$ triples of the form $(C_i, C_j, a_k)$ for $4$ distinct values of $k$ . However this is true so we are done.
This post has been edited 1 time. Last edited by Shreyasharma, Nov 20, 2023, 1:41 AM
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cj13609517288
1908 posts
#32
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Double count (delegate,(unordered pair of committees)) pairs. Delegates have $800$ committees on average(by global counting (delegate,committee) pairs), so by Jensen the number of these pairs is at least $1600\cdot\binom{800}{2}$. Therefore, some pair of committees has
\[\left\lceil \frac{1600\cdot\binom{800}{2}}{\binom{16000}{2}} \right\rceil=4\]common members. $\blacksquare$
This post has been edited 1 time. Last edited by cj13609517288, Dec 8, 2023, 5:23 PM
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dolphinday
1324 posts
#33
Y by
Let $d_i$ be the $i$th delegate. Then $x_i$ is the number of committees that delegate $d_i$ is in.
Then the expected value of shared members of two random committees is
\[\frac{\sum_{i=1}^{1600}\binom{x_i}{2}}{\binom{16000}{2}}\].
Then since $\binom{x}{2}$ is convex, by Jensen's we have
\[\frac{\sum_{i=1}^{1600}\binom{x_i}{2}}{\binom{16000}{2}} \geq \frac{1600 \cdot \binom{800}{2}}{\binom{16000}{2}} = \frac{1600\cdot800\cdot799}{16000\cdot15999}\]. This is approximately less than \[80 \cdot \frac{1}{20} = 4\]Since the expected value is less than $4$, and the number of common committee members is whole, there has to be at least one pair of committees that has $4$ common members.
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asdf334
7585 posts
#34 • 1 Y
Y by GeoKing
Label delegates $1,2,\dots,1600$.

Let $d_1,\dots, d_{1600}$ be the number of committees each delegate is a part of. Consider counting pairs $(c,d)$ of committees and delegates. Counting by committees then delegates we find
\[d_1+d_2+\dots+d_{1600}=16000\cdot 80.\]
Now let's count pairs $((c_1,c_2),d)$ of two committees and one delegate. Counting by delegates, we have
\[\binom{d_1}{2}+\binom{d_2}{2}+\dots+\binom{d_{1600}}{2}\ge 1600\binom{16000\cdot 80\div 1600}{2}=A\]and there are also $\binom{16000}{2}=B$ pairs of committees. Hence it suffices to show that $\frac{A}{B}>3$ to finish by Pigeonhole.

Note that
\[A=1600\cdot 400\cdot 799\]\[B=8000\cdot 15999\]\[\frac{A}{B}=\frac{80\cdot 799}{15999}>3\iff 80\cdot 799>3\cdot 15999\]which is obviously true.
Done!
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shendrew7
795 posts
#35
Y by
Let $S$ be the set of delegates, and $c(d)$ be the number of committees the delegate is in. Note that
\[\sum_{d \in S} c(d) = 16000 \cdot 80 \quad \text{and} \sum_{d \in S} \quad \binom{c(d)}{2}\]
represents the sum of the amount of overlap between any two committees. Thus, for any two randomly selected committees, the expected number of delegates shared is
\[\sum_{d \in S} \frac{\binom{c(d)}{2}}{\binom{16000}{2}} \ge 1600 \cdot \frac{\binom{800}{2}}{\binom{16000}{2}} = 80 \cdot \frac{799}{15999} > 3.\]
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blueprimes
351 posts
#36
Y by
Suppose each delegate $1 \le i \le 1600$ belongs to $c_i$ commitees. We will choose two commitees at random, the expected number of common members by linearity of expectation is
$$\dfrac{1}{\binom{16000}{2}}\sum_{i = 1}^{1600} \dbinom{c_i}{2} \ge \dfrac{1600}{\binom{16000}{2}} \binom{\sum_{i = 1}^{1600} c_i / 1600}{2}$$Now it is easy to see that $\sum_{i = 1}^{1600} c_i = 16000 \cdot 80 \implies \sum_{i = 1}^{1600} c_i / 1600 = 800$. Therefore it is sufficient to show that
$$\frac{1600 \binom{800}{2}}{\binom{16000}{2}} = \frac{1600 \cdot 800 \cdot 799}{16000 \cdot 15999} = \frac{80 \cdot 799}{15999} > 3$$which implies that two commitees can be chosen with at least $4$ common members, as wanted.
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Mathandski
757 posts
#37
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Iconic problem
Attachments:
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lpieleanu
2994 posts
#38
Y by
Solution
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eg4334
637 posts
#39
Y by
Let the delegates $d_1, d_2, \dots d_{1600}$ be in $x_1, x_2, \dots, x_{1600}$ committes, respectively. The key is to count the number of common members by each member then divide by $\binom{16000}{2}$. The expected number of common members from an arbitrary pair of committes is then $\frac{\sum \binom{x_i}{2}}{\binom{16000}{2}}$ and also $\sum x_i = 16000 \cdot 80$. By Jensens we have $$\frac{\sum \binom{x_i}{2}}{\binom{16000}{2}} \geq \frac{1600 \cdot \binom{800}{2}}{\binom{16000}{2}} \approx 3.99$$, so one pair of committes must have at least four members. This really is the epitome of an easy global problem.
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Maximilian113
575 posts
#40
Y by
Let the number of committees the delegates join be $x_1, x_2, \cdots, x_{1600}.$ Then for some pair of committees, we count the expected number of delegates that joined both of them.

First, observe that $\sum x_i = 80 \cdot 16000.$ Thus by Jensen's the expected value is $$E=\sum \frac{\binom{x_i}{2}}{\binom{16000}{2}} \geq \frac{1600 \cdot \binom{800}{2}}{\binom{16000}{2}} > 3.$$QED
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gladIasked
648 posts
#41
Y by
Let $a_i$ be the number of committees that the $i$-th delegate is in. The probability that a delegate is in two arbitrarily selected committees is $\frac{a_i(a_i-1)}{16000(15999)}$. Summing over all $i$ gives us the expected number of people in both of the committees: $\frac{{a_1\choose 2}+{a_2\choose 2}+\dots+{a_{1600}\choose 2}}{16000(15999)}$. Because $a_1+a_2+\dots+a_{1600}=80(16000)$, Jensen allows us to determine the lower bound on the expression to be $\frac{1600{800\choose 2}}{{16000\choose 2}} > 3$. $\blacksquare$
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de-Kirschbaum
198 posts
#42
Y by
Let the committees formed be $C_1, \ldots, C_{160000}$. We will count the number $T$ of unordered pairs of $(C_i, C_j, x)$ where person $x$ is in committees $C_i, C_j$. Suppose for sake of contradiction all pairs of committees have at most $3$ common members. Then $$T \leq 3\binom{16000}{2}$$
Now let $a_i$ be the number of commitees person $x_i$ is in. Then by Jensen's $$T=\sum_{i=1}^{1600} \binom{a_i}{2} \geq 1600\binom{800}{2}$$
So $1600\binom{800}{2}\leq 3\binom{16000}{2}$ which is untrue.
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Ilikeminecraft
616 posts
#43
Y by
Let $a_i$ be the number of committees the $i^{\text{th}}$ member is in. Thus, we have that $\sum a_i = 16000 \cdot 80.$ We also have that
\begin{align*}
  \sum \frac{\binom{a_i}{2}}{\binom{16000}{2}} & \geq \frac{1600\cdot\binom{\sum a_i / 1600}{2}}{\binom{16000}{2}} \\
  & = \frac{1600\binom{800}{2}}{\binom{16000}{2}} \\
  & = \frac{80\cdot799}{1599} \\
  & > 3
\end{align*}Thus, we are done.
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