The Incenter of a Triangle
$ABC$ is an integer sided triangle with incenter $I$ and perimeter $p$. The segments $IA$, $IB$ and $IC$ have integral length as well. Let $L = p...
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John Rhodes
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August 15, 2023
Chef Showdown
A group of chefs (numbered #$1$, #$2$, etc) participate in a turn-based strategic cooking competition. On each chef’s turn, he/she cooks up a dish to the best...
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John Rhodes
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August 15, 2023
The Last Question
Consider all the words which can be formed by selecting letters, in any order, from the phrase: thereisasyetinsufficientdataforameaningfulanswer Suppose those with 15 letters or less are listed...
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John Rhodes
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August 15, 2023
Roots on the Rise
Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation $\frac 1 x = (\frac k x)^2(k+x^2)-k x$. For instance, for...
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John Rhodes
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August 15, 2023
Number Sequence Game
The number sequence game starts with a sequence $S$ of $N$ numbers written on a line. Two players alternate turns. The players on their respective turns must...
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John Rhodes
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August 15, 2023
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,...