Scoring Probabilities
Barbara is a mathematician and a basketball player. She has found that the probability of scoring a point when shooting from a distance $x$ is exactly $(1...
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John Rhodes
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August 15, 2023
Pythagorean Odds
Albert chooses a positive integer $k$, then two real numbers $a, b$ are randomly chosen in the interval $[0,1]$ with uniform distribution. The square root of the...
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John Rhodes
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August 15, 2023
Integer Sided Triangles with Integral Area/perimeter Ratio
Consider the triangle with sides $6$, $8$, and $10$. It can be seen that the perimeter and the area are both equal to $24$. So the area/perimeter...
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John Rhodes
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August 15, 2023
The Ackermann Function
$\def\htmltext#1{\style{font-family:inherit;}{\text{#1}}}$ For non-negative integers $m$, $n$, the Ackermann function $A(m,n)$ is defined as follows: $$ A(m,n) = \cases{ n+1 &$\htmltext{ if }m=0$\cr A(m-1,1) &$\htmltext{ if }m>0 \htmltext{...
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John Rhodes
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August 15, 2023
Pizza Toppings
You are given a pizza (perfect circle) that has been cut into $m \cdot n$ equal pieces and you want to have exactly one topping on each...
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John Rhodes
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August 15, 2023
Ant and Seeds
A laborious ant walks randomly on a $5 \times 5$ grid. The walk starts from the central square. At each step, the ant moves to an adjacent...
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John Rhodes
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August 15, 2023
Triangles with Integral Sides and an Integral Angle
How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed $10^8$? This problem is...
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John Rhodes
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August 15, 2023
Linear Combinations of Semiprimes
Given the values of integers $1 < a_1 < a_2 < \dots < a_n$, consider the linear combination $q_1 a_1+q_2 a_2 + \dots + q_n a_n=b$, using...