Difference between revisions of "2025 IMO Problems/Problem 3"
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| − | Let <math>\mathbb{N}</math> denote the set of positive integers. A function <math>f: \mathbb{N} \rightarrow \mathbb{N}</math> is said to be bonza if <math>f(a)</math> divides <math>b^{a} - f(b)^{f(a)}</math> for all positive integers <math>a</math> and <math>b</math>. Determine the smallest real constant <math>c</math> such that <math>f(n) \leq cn</math> for all bonza functions <math>f</math> and all positive integers <math>n</math>. | + | Let <math>\mathbb{N}</math> denote the set of positive integers. A function <math>f: \mathbb{N} \rightarrow \mathbb{N}</math> is said to be <i>bonza</i> if <center><math>f(a)</math> divides <math>b^{a} - f(b)^{f(a)}</math></center> for all positive integers <math>a</math> and <math>b</math>. Determine the smallest real constant <math>c</math> such that <math>f(n) \leq cn</math> for all bonza functions <math>f</math> and all positive integers <math>n</math>. |
==Video solution== | ==Video solution== | ||
https://www.youtube.com/watch?v=vPqUTG4CW8w | https://www.youtube.com/watch?v=vPqUTG4CW8w | ||
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| + | https://www.youtube.com/watch?v=HZtoYgF3ZQ4 | ||
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| + | ==See Also== | ||
| + | * [[2025 IMO]] | ||
| + | * [[IMO Problems and Solutions, with authors]] | ||
| + | * [[Mathematics competitions resources]] | ||
| + | {{IMO box|year=2025|num-b=2|num-a=4}} | ||
Latest revision as of 21:50, 11 October 2025
Let 
denote the set of positive integers. A function 
is said to be bonza if
divides 
for all positive integers 
and 
. Determine the smallest real constant 
such that 
for all bonza functions 
and all positive integers 
.
Video solution
https://www.youtube.com/watch?v=vPqUTG4CW8w
https://www.youtube.com/watch?v=HZtoYgF3ZQ4
See Also
| 2025 IMO (Problems) • Resources | ||
| Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
| All IMO Problems and Solutions | ||
