Circle Packing II
Let $R(a, b, c)$ be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths $a$, $b$ and $c$. Let $S(n)$ be the...
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John Rhodes
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August 15, 2023
Music Festival
$12n$ musicians participate at a music festival. On the first day, they form $3n$ quartets and practice all day. It is a disaster. At the end of...
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John Rhodes
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August 15, 2023
Last Digits of Divisors
For a positive integer $n$ and digits $d$, we define $F(n, d)$ as the number of the divisors of $n$ whose last digits equal $d$. For example,...
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John Rhodes
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August 15, 2023
Phigital Number Base
Let $\varphi$ be the golden ratio: $\varphi=\frac{1+\sqrt{5}}{2}.$ Remarkably it is possible to write every positive integer as a sum of powers of $\varphi$ even if we require...
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John Rhodes
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August 15, 2023
Comfortable Distance II
There are $N$ seats in a row. $N$ people come one after another to fill the seats according to the following rules: No person sits beside another....
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John Rhodes
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August 15, 2023
Triangle Inscribed in Ellipse
The triangle $\triangle ABC$ is inscribed in an ellipse with equation $\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$, $0 \lt 2b \lt a$, $a$ and...
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John Rhodes
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August 15, 2023
Super Ramvok
Consider a single game of Ramvok: Let $t$ represent the maximum number of turns the game lasts. If $t = 0$, then the game ends immediately. Otherwise,...
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John Rhodes
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August 15, 2023
Empty Chairs
In a room $N$ chairs are placed around a round table. Knights enter the room one by one and choose at random an available empty chair. To...