Let $a$, $b$, $c$, $d$ be real numbers satisfying \begin{align*} (a + c)(b + d) = \sqrt{2}(ac - 2bd - 1). \end{align*}Show that \begin{align*} (ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4. \end{align*}
Problem
Source: 2020 Taiwan TST
Tags: Inequality, inequalities, Taiwan
USJL
28.03.2020 17:38
The key step is to observe that using the provided identity,
\[L.H.S. = (2-ab-cd)^2+(2-bc-da)^2+\frac{1}{2}(2+(a+c)(b+d))^2-5.\]The rest is easy: by using C.-S.,
\[\left[(2-ab-cd)^2+(2-bc-da)^2+\frac{1}{2}(2+(a+c)(b+d))^2\right](1+1+2)\geq 6^2,\]and so
\[L.H.S. = (2-ab-cd)^2+(2-bc-da)^2+\frac{1}{2}(2+(a+c)(b+d))^2-5\geq \frac{6^2}{4}-5=4.\]
sqing
28.03.2020 18:26
USJL wrote:
SolutionThe key step is to observe that using the provided identity,
\[L.H.S. = (2-ab-cd)^2+(2-bc-da)^2+\frac{1}{2}(2+(a+c)(b+d))^2-5.\]The rest is easy: by using C.-S.,
\[\left[(2-ab-cd)^2+(2-bc-da)^2+\frac{1}{2}(2+(a+c)(b+d))^2\right](1+1+2)\geq 6^2,\]and so
\[L.H.S. = (2-ab-cd)^2+(2-bc-da)^2+\frac{1}{2}(2+(a+c)(b+d))^2-5\geq \frac{6^2}{4}-5=4.\]
h
Attachments:
itslumi
28.03.2020 20:57
Where can we find other problems from 2020 Taiwan TST
USJL
29.03.2020 05:10
itslumi wrote: Where can we find other problems from 2020 Taiwan TST Typically there are ISLs in it, so they will not be posted fully until the IMO is done... That being said, I think currently no one is maintaining Taiwan TST problems on AoPS unfortunately... so don't expect to see all of them (
itslumi
29.03.2020 06:25
USJL wrote: itslumi wrote: Where can we find other problems from 2020 Taiwan TST Typically there are ISLs in it, so they will not be posted fully until the IMO is done... That being said, I think currently no one is maintaining Taiwan TST problems on AoPS unfortunately... so don't expect to see all of them ( What about years 2019,2018 do you have them
Maxito12345
30.03.2020 05:36
itslumi wrote: USJL wrote: itslumi wrote: Where can we find other problems from 2020 Taiwan TST Typically there are ISLs in it, so they will not be posted fully until the IMO is done... That being said, I think currently no one is maintaining Taiwan TST problems on AoPS unfortunately... so don't expect to see all of them ( What about years 2019,2018 do you have them Does anyone has them
Tintarn
30.03.2020 10:36
USJL wrote: Typically there are ISLs in it, so they will not be posted fully until the IMO is done... That being said, I think currently no one is maintaining Taiwan TST problems on AoPS unfortunately... so don't expect to see all of them ( That's not exactly true. User JustPostTaiwanTST (who else ) posted many problems from TST 2018 last year. See here. I wanted to add them already back in December, but some of them are still missing. I asked him to post the missing problems, but he unfortunately didn't. Maybe you have the problems and can post them? I will add them to the Contest Section immediately.
USJL
30.03.2020 11:49
Interesting. I'm really not that active on AoPS anymore so I was totally unaware of that. Thanks for bringing that up! Unfortunately I don't think I bothered to keep the problems around so I've already lost them... If you make some request here maybe someone would dig it up and post them.
USJL
30.03.2020 12:12
Okay turns out that that use is one of my friends lol I will post some of them since I am bored af b/c of the quarantine :p
itslumi
30.03.2020 13:53
Tintarn wrote: USJL wrote: Typically there are ISLs in it, so they will not be posted fully until the IMO is done... That being said, I think currently no one is maintaining Taiwan TST problems on AoPS unfortunately... so don't expect to see all of them ( That's not exactly true. User JustPostTaiwanTST (who else ) posted many problems from TST 2018 last year. See here. I wanted to add them already back in December, but some of them are still missing. I asked him to post the missing problems, but he unfortunately didn't. Maybe you have the problems and can post them? I will add them to the Contest Section immediately. Do you have the year 2019
USJL
01.04.2020 05:58
itslumi wrote: Tintarn wrote: USJL wrote: Typically there are ISLs in it, so they will not be posted fully until the IMO is done... That being said, I think currently no one is maintaining Taiwan TST problems on AoPS unfortunately... so don't expect to see all of them ( That's not exactly true. User JustPostTaiwanTST (who else ) posted many problems from TST 2018 last year. See here. I wanted to add them already back in December, but some of them are still missing. I asked him to post the missing problems, but he unfortunately didn't. Maybe you have the problems and can post them? I will add them to the Contest Section immediately. Do you have the year 2019 Yes. It will take some time for me to get there though.
fcomoreira
02.04.2020 02:32
USJL wrote: Yes. It will take some time for me to get there though. Posting the previous (2018,2019) Taiwan TST would be veeery welcome
USJL
02.04.2020 08:51
fcomoreira wrote: USJL wrote: Yes. It will take some time for me to get there though. Posting the previous (2018,2019) Taiwan TST would be veeery welcome It's in progress. I'm almost done with 2018, and someone else is doing 2019. Feel free to check my recent posts.
USJL
02.04.2020 11:02
https://artofproblemsolving.com/community/c1118239_2018_taiwan_tst_round__1 https://artofproblemsolving.com/community/c1118244_2018_taiwan_tst_round__2 https://artofproblemsolving.com/community/c1118249_2018_taiwan_tst_round__3 https://artofproblemsolving.com/community/c1116300_2019_taiwan_tst_round_1 https://artofproblemsolving.com/community/c1117265_2019_taiwan_tst_round_2 https://artofproblemsolving.com/community/c1118234_2019_taiwan_tst_round_3 Enjoy!