Difference between revisions of "1975 IMO Problems/Problem 3"
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− | On the sides of an arbitrary triangle ABC | + | ==Problems== |
− | constructed externally with CBP = CAQ = | + | |
− | + | On the sides of an arbitrary triangle <math>ABC</math>, triangles <math>ABR, BCP, CAQ</math> are constructed externally with <math>\angle CBP = \angle CAQ = 45^\circ, \angle BCP = \angle ACQ = 30^\circ, \angle ABR = \angle BAR = 15^\circ</math>. Prove that <math>\angle QRP = 90^\circ</math> and <math>QR = RP</math>. | |
+ | |||
+ | ==Solution== | ||
+ | If we can find <math>p\ne q</math> such that <math>(a_p,a_q)=1</math>, we're done: every sufficiently large positive integer <math>n</math> can be written in the form <math>xa_p+ya_q,\ x,y\in\mathbb N</math>. We can thus assume there are no two such <math>p\ne q</math>. We now prove the assertion by induction on the first term of the sequence, <math>a_1</math>. The base step is basically proven, since if <math>a_1=1</math> we can take <math>p=1</math> and any <math>q>1</math> we want. There must be a prime divisor <math>u|a_1</math> which divides infinitely many terms of the sequence, which form some subsequence <math>(a_{k_n})_{n\ge 1},\ k_1=1</math>. Now apply the induction hypothesis to the sequence <math>\left(\frac{a_{k_n}}u\right)_{n\ge 1}</math>. | ||
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+ | The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [https://aops.com/community/p367494] | ||
+ | |||
+ | == See Also == {{IMO box|year=1975|num-b=2|num-a=4}} |
Revision as of 15:09, 29 January 2021
Problems
On the sides of an arbitrary triangle , triangles are constructed externally with . Prove that and .
Solution
If we can find such that , we're done: every sufficiently large positive integer can be written in the form . We can thus assume there are no two such . We now prove the assertion by induction on the first term of the sequence, . The base step is basically proven, since if we can take and any we want. There must be a prime divisor which divides infinitely many terms of the sequence, which form some subsequence . Now apply the induction hypothesis to the sequence .
The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [1]
See Also
1975 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |