AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that for all 
we have:
be defined by![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)
Prove that the sequences 
and 
have finite limits, and find those limits.
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
+1 w
be reals such that 
and 
Find the value of 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
be prime numbers such that 
where 
Find all possible values of 
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
aand 
Prove that
Let 
aand Prove that
Let 
aand 
Prove that 
and 
and the radius of the inscribed circle is 
. If and are natural numbers and 
, find the value of 
and .
be an odd prime and 
be integers so that the integers 
are divisible by . divides each of .
divides one of them, it divides all of them. Assume that it divides none of them. Then, there exists an inverse for all of them in modulo and then it's some algmanip to get 
a contradiction
divides one of them, it divides all of them. Assume that it divides none of them. Then, there exists an inverse for all of them in modulo and then it's some algmanip to get a contradiction
divides one of them, it divides all of them. Assume that it divides none of them. Then just do some things to get .
be an odd prime and be integers so that the integers . divides each of . divides any of we are done, so from now onwards we assume none of them are divisible by .Define 
such that
and 
.Then notice![\[\dfrac{1}{xy} = \dfrac{c}{a}\mod p\Longleftrightarrow (x\cdot y)^{2025}\equiv -1\mod p\]](http://latex.artofproblemsolving.com/e/b/6/eb6a6acc59325994d74209faa16c9408b6509521.png)
Claim : 
that is 
a contradiction to the fact that was an odd-prime.
, all of them are. So assume on the contrary that none of them are divisible by . So assume none of them are.
.
and 
. Multiplying these, we get,![\[ a^{2023 + 2025} \equiv b^{2023}c^{2025} \implies (a^2)^{2024} \equiv c^2 \cdot (bc)^{2023} \implies bc \cdot (bc)^{2023} \equiv c^2 \cdot (bc)^{2023} \implies b \equiv c \pmod{p}. \]](http://latex.artofproblemsolving.com/7/e/2/7e2a5811d0293d90d0803d797abe96dddb3cf6b6.png)
![\[ p\mid b^{2024} + c^{2024} \equiv 2b^{2024} \implies p \mid 2 \]](http://latex.artofproblemsolving.com/9/d/0/9d0ae48aed14e79a9bd3ac70d9e0de65b94a50a9.png)
which is a contradiction and we are done. 
divides all of divides one of , clearly divides all of , so assume divides none of them.
. The condition 
and we also have 
, so multiplying the three equations gives 
, so 
. This implies that 
, so 
, so 
. But 
, so dividing the two gives 
, so 
mod p. Since 
, we have 
. But then since 
, 
, so 
. Now, this means that 
is a multiple of , contradiction since doesn't divide 
and 
.
divides either of , then it divides everything and we're done. Henceforth, assume that 
. We will now work modulo with special powers of existence of inverses of .
, we get that 
. Swing this into 
to get 
. At last, putting this into 
, we get![\[a^{2025} + a^{2023}bc \equiv 0 \iff a^2 + bc \equiv 0 \pmod{p}.\]](http://latex.artofproblemsolving.com/9/f/3/9f3580a23b4e02a0cc044d1ed98e85e908c8dc66.png)
From here, we wish to eliminate completely modulo . This can be achieved via substituting 
. Putting this in we get![\[a^{2023} - \frac{a^{4046}}{c^{2023}} \equiv 0 \iff a^{2023} \equiv c^{2023} \pmod{p}.\]](http://latex.artofproblemsolving.com/9/7/6/976fafcba4a1cbc016f9da9b2282e872d622c737.png)
This yields![\[a^{2025} \equiv a^2c^{2023} \equiv -c^{2025} \pmod{p} \implies a^2 + c^2 \equiv 0 \pmod{p}.\]](http://latex.artofproblemsolving.com/a/f/e/afeed88cffe64ac63caa5078eb428e9dd101702e.png)
Finally, on combining this with 
, we get![\[p \mid c(c-b) \iff c \equiv b \pmod{p}\]](http://latex.artofproblemsolving.com/b/3/7/b3750c524d111659d1d05bb4c86672315b106d6a.png)
since 
. Upon putting 
in say , we get immediately get 
which is a contradiction since is odd and we're done. 
divides any one of them, it divides the other two as well. doesn't divide any of the three. Since is an odd prime, it has a primitive root, say 
.
.
where 
.
.
Cancelling 
and the denominator and taking mod 2 leads to a contradiction. Done!
be an odd prime and be integers so that the integers .?, but I suppose a lot of us got tunnel vision and didn't see the error because we already saw it in the test lol 
or 
or 
then divides all of them.
. is odd prime, i.e we would prove 
. we could take inverses.
we get
Multiplying equations 
, 
and 
we get
.
i.e
Now from we get
and from 1 we get
Let the order of 
mod be k.
i.e 
.
or
Putting this information in equation we get
or 
divides one of 
then it divides all of them, so henceforth assume it doesn't divide any of them. Thus, there exists a valid inverse in modulo for each of 
Then, we have 
and thus, multiplying all of them together, we have 
Thus, 
which gives 
Thus, 
which by putting in the first equation gives 
From the second equation, we get 
so 
so 
a contradiction.
for any 
and it follows for 
.
If divides any one of then 
.
W.L.O.G 
, now since 
and also 
. 
it suffices to disprove the fact that that 
,so FTSOC we assume that is the case.
does not divide any of 
. FTSOC 
, then 
, now since 
, it gives a contradiction. 
.
and 
and since 
similarly symmetrically other in and 
, but from 
we have 
, which implies 
we have 
and from 
we get this is equivalent to 
, but this is not possible as 
.. must divide .
, raise both sides to power 
to get terms 
and done
so all are divisible by 
be the order of 
mod p so
divides any one of , then it divides all the three numbers. Suppose 
. Then, 
but 
, which means that 
, forcing 
. Also, 
and 
which means 
, forcing 
., then and , which means , forcing . Since , we also have that from the earlier proof., then 
and 
, which means 
, forcing , which in turn forces . So, if divides any one of , then it divides all the three numbers.
. For integer 
and prime , we define the order of 
to be the smallest positive integer such that 
.
If is order of , 
and 
, then 
.
Suppose it was possible that 
. Note that if 
, then it will contradict the minimality of . So, 
. Hence, there exists 
and 
such that 
. Now, 
, which means 
, so 
. But since and 
, the minimality of gives us a contradiction. Hence, is impossible, and 
is forced. The proof is complete. ![\[\left(\frac{a}{b}\right)^{4046}\equiv1\pmod{p} \ \ \ \ (1)\]](http://latex.artofproblemsolving.com/0/3/c/03c40601bceb0fab5a6ffeeeba604aa68badcaad.png)
![\[\left(\frac{b}{c}\right)^{4048}\equiv1\pmod{p} \ \ \ \ (2)\]](http://latex.artofproblemsolving.com/9/4/9/949ad0a2ea48a5f0d3962fc499253bb2fbe1ca0b.png)
![\[\left(\frac{c}{a}\right)^{4050}\equiv1\pmod{p} \ \ \ \ (3)\]](http://latex.artofproblemsolving.com/5/9/c/59cc2f807076eebdc607ddf85468f6669909c174.png)
Multiplying these equations 
we obtain 
. So, 
.
is 
. So, ![\[\left(\frac{\left(bc\right)^{2023}}{b^{4046}}\right)\equiv\pm1\pmod{p},\]](http://latex.artofproblemsolving.com/5/9/8/598210f0185cbbeab376b8468ac11185c78c7702.png)
which gives ![\[\left(\frac{b}{c}\right)^{4046}\equiv1\pmod{p} \ \ \ \ (4)\]](http://latex.artofproblemsolving.com/9/7/5/975c0d72e9f1b8a5055adf740adbd3631ebc3120.png)
It follows that if is the order of 
, , then from equation , we get 
, and from equation 
, we get 
, using the 
in each case.
, so 
. Hence, we have that 
. So, 
.
, we get 
, and since , we have 
, which forces , as is an odd prime. However, if then , and are forced by our initial arguments.
divides any one of 
, then it divides all of them. Assume this to not be the case. Then we get the system of congruences 
Multiplying 
and 
, 
but from 
, 
Substituting in, 
Dividing from this new equation, 
Therefore 
a contradiction. 
Now assume that 
as if one of the three is 
modulo , all of them must be. Then we find that,
However then from this we may conclude that,
Dividing as 
modulo , we find that,
Now rewriting our conditions we have,
Squaring the first equation we have,
From the first equation we then find 
. Thus we have,
Thus we have from the original equations,
from which we easily see 
and hence are done.
, 
, 
.
From 
, we get that: 
, 
, 
, we have: 
, we get: 
and , we have: 
, 
, which is a contradiction! must divide one of . If divides any one of the numbers, must divide all the numbers individually, and we're done! 
=2023×2024×2025
And from this we get
and we are done
=2023×2024×2025And from this we getand we are done
, then![\[a^{2023}+b^{2023}\equiv 0 \mod p,\]](http://latex.artofproblemsolving.com/8/3/e/83eb190b5fa85d8dea225dcc29862a75566fb891.png)
![\[\iff b^{2023} \equiv 0 \mod p,\]](http://latex.artofproblemsolving.com/5/d/b/5dba54e9ecf89a47b19d5689c2329393b87d4596.png)
![\[\iff p\mid b,\]](http://latex.artofproblemsolving.com/a/7/b/a7bfece03e925f20b8a45a7d05193d2e9a90f21b.png)
and similarly we can find that . In general, using a similar proof, it can be shown that if 
, then divides each of , , and . FTSOC, assume that divides none of , , or .![\[\iff a^{2023}\equiv -b^{2023} \mod p,\]](http://latex.artofproblemsolving.com/8/1/8/818923601b1c3bd4663b2ffc84671be9a57925dc.png)
![\[\iff \left(\frac{a}{b}\right)^{2023}\equiv -1 \mod p,\]](http://latex.artofproblemsolving.com/e/6/8/e68ed1022f52ef7890eea80e85c20f4f99be4790.png)
![\[\iff ord_p\left(\frac{a}{b}\right) \mid 4046 \text{ and } 2\mid ord_p\left(\frac{a}{b}\right),\]](http://latex.artofproblemsolving.com/e/c/9/ec92d4eca021bc856f21fafb2c8557996a5c9459.png)
since we had that 
. Similarly, from the second equation, we get that![\[ord_p\left(\frac{b}{c}\right) \mid 4048 \text{ and } 16\mid ord_p\left(\frac{b}{c}\right).\]](http://latex.artofproblemsolving.com/a/8/9/a8920a410f2a1af044580f4fc60aa872c0ad23c1.png)
is prime, it is well-known that it must have a primitive root. Let said primitive root be and define 
and to be the unique integers such that 
and 
, where 
. Now, notice that if![\[2\mid ord_p\left(\frac{a}{b}\right) \text{ and } ord_p\left(\frac{a}{b}\right) \mid 4046,\]](http://latex.artofproblemsolving.com/5/8/8/588717e5e134f22fe7ffd8f045013a3ac830e228.png)
we get that![\[\nu_2\left(ord_p\left(\frac{a}{b}\right)\right)=1,\]](http://latex.artofproblemsolving.com/8/b/1/8b16c089c3c49f99785c489259e1351e37128591.png)
which in turn gives that![\[\nu_2(m)=\nu_2(p-1)-1,\]](http://latex.artofproblemsolving.com/b/4/9/b49456621127ee9da5266bbacdb544b1421ba7da.png)
since the order of mod is 
. Similarly, if![\[16\mid ord_p\left(\frac{b}{c}\right) \text{ and } ord_p\left(\frac{b}{c}\right) \mid 4048,\]](http://latex.artofproblemsolving.com/6/0/3/603243f29adb37cd296bbe937e2a3e675e5b7431.png)
then we have that![\[\nu_2(n)=\nu_2(p-1)-4.\]](http://latex.artofproblemsolving.com/c/2/0/c20759f7f4894678a21fbc57af30ccfb244b3a99.png)
![\[\frac{a}{c}\equiv \left(\frac{a}{b}\right)\left(\frac{b}{c}\right)\equiv g^{m+n} \mod p,\]](http://latex.artofproblemsolving.com/2/a/0/2a009851a1f21c5ce1a3d29247fce55cf2c20a78.png)
which means that![\[\nu_2\left(ord_p\left(\frac{a}{c}\right)\right)=\nu_2(p-1)-\nu_2(m+n).\]](http://latex.artofproblemsolving.com/1/c/a/1ca9ab1f825bd116747430272514ff490d684f41.png)
However, since![\[\nu_2(p-1)-1=\nu_2(m)>\nu_2(n)=\nu_2(p-1)-4,\]](http://latex.artofproblemsolving.com/0/f/8/0f8960478bd06900144fd543a3c64b96763a27e7.png)
we get that![\[\nu_2(m+n)=\nu_2(p-1)-4,\]](http://latex.artofproblemsolving.com/f/d/2/fd2ac92a256506f59ac0641161ef1c4fa8fb8f3e.png)
which gives that![\[\nu_2\left(ord_p\left(\frac{a}{c}\right)\right)=\nu_2(p-1)-(\nu_2(p-1)-4)=4.\]](http://latex.artofproblemsolving.com/3/3/1/331d4ab505bbc625d82d3d0318cfc588f3fbea58.png)
![\[c^{2025}+a^{2025}\equiv 0 \mod p,\]](http://latex.artofproblemsolving.com/b/a/d/bad74ec1ffd101b304144220701bf972c1d043f8.png)
![\[\iff a^{2025}\equiv -c^{2025} \mod p,\]](http://latex.artofproblemsolving.com/a/d/9/ad9f2857b261e693c9635e8ce64be7b7c8c88d1d.png)
![\[\iff \left(\frac{a}{c}\right)^{2025}\equiv -1 \mod p,\]](http://latex.artofproblemsolving.com/4/8/7/4878d5a9ad9e8600a3b35ef8ca89e05183c2aeda.png)
![\[\iff ord_p\left(\frac{a}{c}\right) \mid 4050,\]](http://latex.artofproblemsolving.com/6/b/d/6bd10eec121f089a055db571af01929381a55f99.png)
a contradiction, since 
does not divide 
. Therefore must divide each of , , and , as desired, finishing the problem.
divides one of 
, WLOG 
, we have 
, 
, so then divides all of them.
. We have the following modular congruences:![\[ \left(\frac{a}{b}\right)^{4046} \equiv 1 \pmod{p} \]](http://latex.artofproblemsolving.com/0/b/d/0bdd25f5d1ea41a0b381219a319f3677e2c899e9.png)
![\[ \left(\frac{b}{c}\right)^{4048} \equiv 1 \pmod{p} \]](http://latex.artofproblemsolving.com/4/2/d/42d5b135d3a6bff64503eebd1a8aec291a293688.png)
![\[ \left(\frac{c}{a}\right)^{4050} \equiv 1 \pmod{p} \]](http://latex.artofproblemsolving.com/5/0/2/50259c73c26a9290c222b1e1bb611237c452c9a3.png)
Firstly, observe that 
. Since 
, we have 
. Further, 
. From this, we conclude that:![\[ p \mid b^{2024}c + a^2b^{2023} = b^{2023}(bc + a^2) \implies a^2 \equiv -bc \pmod{p} \]](http://latex.artofproblemsolving.com/d/1/b/d1b561b2397fbdc7f8dca95b5ff62e8794e4bae3.png)
since 
.
, which, replacing in our first congruence, gives 
. Replacing this in our second congruence, we have![\[ \frac{b^{4046}}{c^{4046}} \cdot \frac{b^2}{c^2} \equiv 1 \pmod{p} \implies b^2 \equiv c^2 \pmod{p} \implies b^{2024} \equiv c^{2024} \pmod{p} \]](http://latex.artofproblemsolving.com/1/9/b/19b2e18df1dc97f85e66e923205936b9abc48abe.png)
Therefore, 
. Since is odd, we have 
, which gives 
as well, so we are done.
is divisible by would suffice. Assume all of them are invertible on 
. As 
we may say,
We then have 
. Since is invertible we are left with the key result,
Using the above fact, 
which deduces to 
and as 
we can easily show that 
. Now we are left with,
But we had already seen 
and is absurd, contradicting our assumption, hence none of them are invertible as desired.
.![$$a+b+c\geq 3\sqrt[4]{21}$$](http://latex.artofproblemsolving.com/6/9/f/69f4b29e2eb96f1baedde3ea7e352cc360ceb963.png)
Prove that
Where 
such that : 
and 
and 
Note that we can choose 
such that, 
cause of periodicity of modulo 
.
such that 
as, 
,
then 
divides 
,
divides 
and 
divides 
such that, 
Then by 2nd equation given in the question that is 
we have 
but by 
we have 
which is a contradiction as 
But, due to the order of an integer modulo 
a multiple of 
,
.
,
and 
.
,
.
.
,
.
so 
.
or![\[ \left(\frac{a}{b}\right)^{2025}\equiv -\left(\frac{c}{a}\right)^{2025}\equiv 1\pmod p\]](http://latex.artofproblemsolving.com/d/8/5/d857e5c200fa57c56f8e211d088b14c253888cd7.png)
From the problem we also have 
,
.
and 
.
giving