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Difference between revisions of "2024 Indonesia MO Problems"

(Created page with " ==Day 1== ===Problem 1=== Find all ordered pair of positive real number <math>a, b</math> which satisfy the following system of equations? <cmath>\begin{aligned}\sqrt{a}+\...")
 
 
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===Problem 1===
 
===Problem 1===
  
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Find all ordered pair of positive real number <math>a, b</math> which satisfy the following system of equations?
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Find all ordered pair of positive real number <math>a, b</math> which satisfy the following system of equations.
  
−
<cmath>\begin{aligned}\sqrt{a}+\sqrt{b}=6\\\sqrt{a-5}+\sqrt{b-5}=4\end{aligned}</cmath>
+
<math>\begin{aligned} \sqrt{a}+\sqrt{b}&=6 \\ \sqrt{a-5}+\sqrt{b-5}&=4 \end{aligned}</math>
  
 
[[2024 Indonesia MO Problems/Problem 1|Solution]]
 
[[2024 Indonesia MO Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
 +
Triplet of positive integer <math>(a, b, c)</math> where <math>a < b < c</math> is called ''fatal'' if there exist
 +
three non-zero integer <math>p, q, r</math> which satisfy <math>a^pb^qc^r=1</math>. For example, (2,3,12) is fatal triplet since <math>2^23^112^{-1}=1</math>.
  
 +
Natural number <math>N</math> is called fatal if there exist fatal triplet <math>(a, b, c)</math> such that <math>N = a+b+c</math>.
 +
# Prove that 16 is not fatal.
 +
# Prove that every integer greater than 16 that is '''not''' a multiple of 6 is fatal.
  
  

Latest revision as of 12:23, 27 May 2025

Day 1

Problem 1

Find all ordered pair of positive real number $a, b$ which satisfy the following system of equations.

$\begin{aligned} \sqrt{a}+\sqrt{b}&=6 \\ \sqrt{a-5}+\sqrt{b-5}&=4 \end{aligned}$

Solution

Problem 2

Triplet of positive integer $(a, b, c)$ where $a < b < c$ is called fatal if there exist three non-zero integer $p, q, r$ which satisfy $a^pb^qc^r=1$. For example, (2,3,12) is fatal triplet since $2^23^112^{-1}=1$.

Natural number $N$ is called fatal if there exist fatal triplet $(a, b, c)$ such that $N = a+b+c$.

  1. Prove that 16 is not fatal.
  2. Prove that every integer greater than 16 that is not a multiple of 6 is fatal.


Solution

Problem 3

Solution

Problem 4

Solution

Day 2

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

See Also

2024 Indonesia MO (Problems)
Preceded by
2023 Indonesia MO 		1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 Followed by
2025 Indonesia MO
All Indonesia MO Problems and Solutions
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