Cardano Triplets
A triplet of positive integers $(a, b, c)$ is called a Cardano Triplet if it satisfies the condition: $$\sqrt[3]{a + b \sqrt{c}} + \sqrt[3]{a – b \sqrt{c}}...
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John Rhodes
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August 15, 2023
$250250$
Find the number of non-empty subsets of $\{1^1, 2^2, 3^3,\dots, 250250^{250250}\}$, the sum of whose elements is divisible by $250$. Enter the rightmost $16$ digits as your...
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John Rhodes
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August 15, 2023
Prime Subset Sums
Let $S = \{2, 3, 5, \dots, 4999\}$ be the set of prime numbers less than $5000$. Find the number of subsets of $S$, the sum of...
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John Rhodes
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August 15, 2023
Euler’s Totient Function Equals 13!
The first number $n$ for which $\phi(n)=13!$ is $6227180929$. Find the $150\,000$th such number. The Euler totient function $\phi(n)$ calculates the number of positive integers that are...
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John Rhodes
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August 15, 2023
Tangents to an Ellipse
A definition for an ellipse is: Given a circle $c$ with centre $M$ and radius $r$ and a point $G$ such that $d(G,M) \lt r$, the locus...
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John Rhodes
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August 15, 2023
Coresilience
We shall call a fraction that cannot be cancelled down a resilient fraction. Furthermore we shall define the resilience of a denominator, $R(d)$, to be the ratio...
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John Rhodes
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August 15, 2023
Resilience
A positive fraction whose numerator is less than its denominator is called a proper fraction. For any denominator, $d$, there will be $d – 1$ proper fractions;...
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John Rhodes
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August 15, 2023
Odd Triplets
Given the set $\{1,2,\dots,n\}$, we define $f(n, k)$ as the number of its $k$-element subsets with an odd sum of elements. For example, $f(5,3) = 4$, since...