dividing pupils

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0 replies

jlacosta

Mar 2, 2025

0 replies

Geometry Problem #42

vankhea 2

N 9 minutes ago by kaede_Arcadia

Source: Van Khea

Let $P$ be any point. Let $D, E, F$ be projection point from $P$ to $BC, CA, AB$ . Circumcircle $(ABC)$ cuts circumcircle $(AEF), (BFD), (CDE)$ at $A_1, B_1, C_1$ . Let $A_2, B_2, C_2$ be antipode of $A_1, B_1, C_1$ wrt $(AEF), (BFD), (CDE)$ . Prove that $A_2, B_2, C_2, P$ are cyclic.

2 replies

vankhea

Sep 6, 2023

kaede_Arcadia

9 minutes ago

divisibility

srnjbr 3

N 11 minutes ago by srnjbr

Find all natural numbers n such that there exists a natural number l such that for every m members of the natural numbers the number m+m^2+...m^l is divisible by n.

3 replies

srnjbr

3 hours ago

srnjbr

11 minutes ago

Very easy inequality

pggp 5

N 21 minutes ago by ionbursuc

Source: Polish Junior MO Second Round 2019

Let $x$ , $y$ be real numbers, such that $x^2 + x \leq y$ . Prove that $y^2 + y \geq x$ .

5 replies

pggp

Oct 26, 2020

ionbursuc

21 minutes ago

Solve in gaussian integers

CHESSR1DER 0

30 minutes ago

Solve in gaussian integers.
$\sin\left(\ln\left(x^{x^{x^2}}\right)\right) = x^4$

0 replies

CHESSR1DER

30 minutes ago

0 replies

No more topics!

dividing pupils

outback 1

N Oct 6, 2008 by randomgraph

Source: Russia 1994, 9th grade

There are $16$ pupils in a class. Every month, the teacher divides the pupils into two groups. Find the smallest number of months after which it will be possible that every two pupils were in two different groups during at least one month.

1 reply

outback

Oct 6, 2008

randomgraph

Oct 6, 2008

dividing pupils

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Source: Russia 1994, 9th grade

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outback

293 posts

outback

#1 h Oct 6, 2008, 8:03 PM • 1 Y

Y by Adventure10

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randomgraph

422 posts

randomgraph

#2 h Oct 6, 2008, 10:07 PM • 2 Y

Y by Adventure10, Mango247

solution

Consider a 0/1 matrix whose rows correspond to students and whose columns correspond to months, where each entry is 0 or 1 according to which group that student was in on that month. The condition (every pair of students are in different groups at least once) just means that the rows are all different. If there are $n$ students where $2^{k-1} < n \leq 2^k$ , this can obviously be done with $k$ months but no less, since there are only $2^{k-1}$ binary vectors of length $k-1$ , and on the other hand you can easily create $2^k$ distinct vectors by simply enumerating all of them.

Therefore, the answer is 4 months when there are 16 students. The algorithm is just to enumerate the students $0, 1\ldots, 15$ , take the 4-digit binary representation of these numbers, and placing them in groups on month $i$ according to the value of the $i$ th digit.

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