Difference between revisions of "2023 IOQM/Problem 2"
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<math>\log_{a}{b}=x</math> Then, <math>\log_{b}{a}=\frac{1}{x}</math>. | <math>\log_{a}{b}=x</math> Then, <math>\log_{b}{a}=\frac{1}{x}</math>. | ||
− | <math>\implies</math> <math>x | + | <math>\implies</math> <math>x+\frac{6}{x}=5</math>. Upon simplifying, we get <math>x^{2}-5x+6=0</math> |
<math>\implies</math> <math>(x-2)(x-3)=0</math> | <math>\implies</math> <math>(x-2)(x-3)=0</math> | ||
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So, <math>x</math> equals to 2 or 3 | So, <math>x</math> equals to 2 or 3 | ||
− | For <math>x</math> = 2, it implies that <math>\log_{a}{b}=2</math>. So, <math> | + | For <math>x</math> = 2, it implies that <math>\log_{a}{b}=2</math>. So, <math>b\:=\: a^{2}</math>, Hence all such pairs are of the form (<math>a</math>,<math>a^{2}</math>) |
− | Where each number lies between 2 and 2023 (inclusive). All such pairs are (4 | + | Where each number lies between 2 and 2023 (inclusive). All such pairs are (2, 4);(3, 9);(4, 16);........(44, 1936) |
Total no. of these pairs = 43 | Total no. of these pairs = 43 | ||
− | For <math>x</math> = 3, Following a similar pattern, we get the pairs as {2,8}...{12,1728} (<math> | + | For <math>x</math> = 3, Following a similar pattern, we get the pairs as {2,8}...{12,1728} (<math>a</math>,<math>a^{3}</math>) |
Total no. of these pairs = 11 | Total no. of these pairs = 11 |
Latest revision as of 01:48, 4 May 2024
Problem
Find the number of elements in the set
Solution1(Quick)
Finding the no. of elements in the set means finding no. of ordered pairs of (, )
Then, .
. Upon simplifying, we get
So, equals to 2 or 3
For = 2, it implies that . So, , Hence all such pairs are of the form (,)
Where each number lies between 2 and 2023 (inclusive). All such pairs are (2, 4);(3, 9);(4, 16);........(44, 1936)
Total no. of these pairs = 43
For = 3, Following a similar pattern, we get the pairs as {2,8}...{12,1728} (,)
Total no. of these pairs = 11
Thus, there are 43+11= elements in the set
~ SANSGANKRSNGUPTA AND ~Andy666
Video Solutions
Video solution by cheetna: https://www.youtube.com/watch?v=z0NPa0tUzZk
Video solution by Unacademy Olympiad Corner: https://www.youtube.com/watch?v=Mm6mXjwU9bY
Video solution by Vedantu Olympiad School: https://www.youtube.com/watch?v=4DJXtR4VHEA
Video solution by Olympiad Wallah: https://www.youtube.com/watch?v=4HSjmY7d3nA
Video solution by : Motion Olympiad Foundation Class 5th - 10th: https://www.youtube.com/watch?v=oVaeHceHXsQ
Please note that above videos solutions are in Hindi, some in English and some in mixed(Hindi + English).
~SANSGANKRSNGUPTA
See Also
Please note that all problems on this page are copyrighted by THE | MTA(I)