I FEEL SO ALIVE WOOHOO
by shiningsunnyday, Sep 25, 2016, 6:54 AM
Now's a good time to list out my goals that I would like to achieve by (the start of) winter break. Each goal is rated from 0-10 in terms of priority as well as 0-10 in difficulty.
Goal 1: Get a 4.0
Priority: 10/10
Difficulty: 6/10
Specifics
Goal 2: Gid gud enough to consistently solve USAMO 1/4s
Priority: 9/10
Difficulty: 11/10
Specifics
Goal 3: Test the waters of research
Priority: 6/10
Difficulty: ?/10
Specifics
Goal 4: Raise my PSAT/SAT scores
Priority: 8/10
Difficulty: 7/10
Specifics
Goal 5: Back to APUSH!
Priority: 6/10
Difficulty: 8/10
Specifics
Goal 6: Pass the f=ma exam
Priority: 5/10
Difficulty: 7/10
Specifics
Goal 7: Be a more charismatic person
Priority: 4/10
Difficulty: 12/10
Specifics
Goal 8: Try to not die of stress
Specifics
Friendly reminder to my blog readers: Try to keep your comments on-topic and civil.
Goal 1: Get a 4.0
Priority: 10/10
Difficulty: 6/10
Specifics
AP Chem
AP Phys
AP CS
AP Stats
Multivariable
Chinese
English
Be on high gear every class, pick up as much content as I can and try not to be stupid
AP Phys
Be extremely paranoid about following directions from my teacher
AP CS
Make sure to keep asking for feedback from teacher for my projects, as well as reaching out to my classmates to check each others' projects
AP Stats
Be extra aggressive towards the teacher so I'm getting the grade I deserve.
Multivariable
Just make sure I don't get drugged before tests
Chinese
Same as above
English
Get feedback from my teacher as often as possible, and sleep well before socratic seminars (I kind of failed my last one)
Goal 2: Gid gud enough to consistently solve USAMO 1/4s
Priority: 9/10
Difficulty: 11/10
Specifics
-Keep up in WOOT
-Finish Lemmas
-Learn to bary/complex
-Keep picking up useful stuff from ACoPS
-Do interesting probs from 105 (will arrive soon)+109 (will arrive soon)+108+102+104+110
-Post one prob a day on my blog!
Last but not least, complete everything in this hide tag
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-Finish Lemmas
-Learn to bary/complex
-Keep picking up useful stuff from ACoPS
-Do interesting probs from 105 (will arrive soon)+109 (will arrive soon)+108+102+104+110
-Post one prob a day on my blog!
Last but not least, complete everything in this hide tag
USAMO 1972 P5 wrote:
A given convex pentagon 
has the property that the area of each of five triangles 
, and 
is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.
has the property that the area of each of five triangles 
, and 
is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.Because ![$[ABE]=[ADE],$](//latex.artofproblemsolving.com/a/0/c/a0c256cb45f93ad02850424fa21c40c629b38b10.png)
the fact they share the same base imply they have equal heights - that is, 
are equal distance way from 
forcing 
Similarly, 
etc.
Denote
Because of the two pair of parallel lines,
is a parallelogram, and we have ![$[ABE]=[EBP]=1.$](//latex.artofproblemsolving.com/6/a/8/6a870f7587bf838c1f20a3fbe71c98c23ce77cbb.png)
Now set ![$x=[DPC], y=[DEP]=[PBC],$](//latex.artofproblemsolving.com/7/3/0/7304716d37973025b77b02dfdda9d7f093da9ef3.png)
and we have 
But 
and solving this yields 
Finally, note that ![$$[ABCDE]=[ABPE]+2y+x=2+(x+y)+y=\frac{\sqrt{5}+5}{2}.$$](//latex.artofproblemsolving.com/0/b/7/0b77fb5592b836e4cf0192286f88b1a17ab4846e.png)
For the last part, construct an arbitrary triangle
with area 
and extend 
and 
until ![$[DEC]=[DBC]=1.$](//latex.artofproblemsolving.com/c/3/7/c37eb6c495bd983bfe3e3ce858d1c3d210f2cc3d.png)
The rest of the construction follows and it's easy to check all conditions are satisfied.
![$[ABE]=[ADE],$](http://latex.artofproblemsolving.com/a/0/c/a0c256cb45f93ad02850424fa21c40c629b38b10.png)
the fact they share the same base imply they have equal heights - that is, 
are equal distance way from 
forcing 
Similarly, 
etc.Denote
Because of the two pair of parallel lines,
is a parallelogram, and we have ![$[ABE]=[EBP]=1.$](http://latex.artofproblemsolving.com/6/a/8/6a870f7587bf838c1f20a3fbe71c98c23ce77cbb.png)
Now set ![$x=[DPC], y=[DEP]=[PBC],$](http://latex.artofproblemsolving.com/7/3/0/7304716d37973025b77b02dfdda9d7f093da9ef3.png)
and we have 
But 
and solving this yields 
Finally, note that ![$$[ABCDE]=[ABPE]+2y+x=2+(x+y)+y=\frac{\sqrt{5}+5}{2}.$$](http://latex.artofproblemsolving.com/0/b/7/0b77fb5592b836e4cf0192286f88b1a17ab4846e.png)
For the last part, construct an arbitrary triangle
with area 
and extend 
and 
until ![$[DEC]=[DBC]=1.$](http://latex.artofproblemsolving.com/c/3/7/c37eb6c495bd983bfe3e3ce858d1c3d210f2cc3d.png)
The rest of the construction follows and it's easy to check all conditions are satisfied.USAMO 1974 P3 wrote:
Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.
This one's easy. The idea is to construct the great circle which passes through the two boundary points, call 
and 
. Now construct the diameter parallel to 
and construct another great circle with this diameter perpendicular to the first great circle, which we'll call 
This will divide the sphere into hemispheres. I claim any curve of length 
from to 
cannot reach into the lower hemisphere.
Assume there exists such a curve. Now the idea is to construct the ellipse with its two foci as and and its vertical axis having one endpoint at 
the center of the sphere. This ellipse is going to be tangent to 
only at 
Therefore, the projection of any such curve from to onto this ellipse will go outside the ellipse at some point, which we can call 
Clearly the length of the curve is greater than 
which we know is by definition of the ellipse. But because the projection of any such curve onto the ellipse is shorter than the actual curve itself, the actual curve is larger than giving us the desired contradiction.
and 
. Now construct the diameter parallel to 
and construct another great circle with this diameter perpendicular to the first great circle, which we'll call 
This will divide the sphere into hemispheres. I claim any curve of length 
from to 
cannot reach into the lower hemisphere.Assume there exists such a curve. Now the idea is to construct the ellipse with its two foci as
and and its vertical axis having one endpoint at 
the center of the sphere. This ellipse is going to be tangent to 
only at 
Therefore, the projection of any such curve from to onto this ellipse will go outside the ellipse at some point, which we can call 
Clearly the length of the curve is greater than 
which we know is by definition of the ellipse. But because the projection of any such curve onto the ellipse is shorter than the actual curve itself, the actual curve is larger than giving us the desired contradiction.USAMO 1974 P4 wrote:
USAMO 1974 P5 wrote:
USAMO 1975 P2 wrote:
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Goal 3: Test the waters of research
Priority: 6/10
Difficulty: ?/10
Specifics
So recently I talked to my college counselor on my dilemma about whether I should pursue research, and she recommended me this program in which PHD mentors are paired up with undergrads (and in some cases, high school students) to guide them in research. I've long wanted to pursue research in either math (applied or theoretical, though more inclined towards the former), so I set up a meeting with the director of the program. If this interests me, I'll definitely go for it, and hopefully submit something of value to Intel ISEF, so I get to fly to LA for Intel ISEF this year. Once again, however, I'm only testing the water, so Idk...
Goal 4: Raise my PSAT/SAT scores
Priority: 8/10
Difficulty: 7/10
Specifics
Recently I took another reading/writing test, scoring a 710. It appears my score is at a solid 1500, possibly 1510. For the first time, I feel like I have control over the reading passages, and they actually seem interesting.
I also got a few books - all targeted towards the deeper waters of the reading section. I'll try to spend an hour or so on them per day.
Hopefully, I can get national merit scholar for PSAT and a 1530 on the Dec. SAT.
Pr0tip for the younger of my blog readers: Do not spend too much time studying for standardized tests. Quite frankly, based the advice I've gotten from my college counselor as well as the articles of some MIT/Harvard students, a 1600 is hardly any worthier than a 1530. To put it more directly, once you get to 1530, the extra time you spend further raising your (already impressive) score could be better spent honing your passion, which is what colleges value more.
As I see all my other fellow classmates drain themselves with hours of SAT prep per day, I could only sigh...
I also got a few books - all targeted towards the deeper waters of the reading section. I'll try to spend an hour or so on them per day.
Hopefully, I can get national merit scholar for PSAT and a 1530 on the Dec. SAT.
Pr0tip for the younger of my blog readers: Do not spend too much time studying for standardized tests. Quite frankly, based the advice I've gotten from my college counselor as well as the articles of some MIT/Harvard students, a 1600 is hardly any worthier than a 1530. To put it more directly, once you get to 1530, the extra time you spend further raising your (already impressive) score could be better spent honing your passion, which is what colleges value more.
As I see all my other fellow classmates drain themselves with hours of SAT prep per day, I could only sigh...
Goal 5: Back to APUSH!
Priority: 6/10
Difficulty: 8/10
Specifics
-Simply put, a 2 out of 5 on my APUSH exam last year won't do.
-Yes, I'm retaking both the AP exam and the SAT II exam, and I ain't settling for anything less than a 5 and 800, so I better be putting in the time to review.
-I'm currently yelling at my poor dad to find me a high-quality tutor before October break, so we have a clear path of action to prepare
-Yes, I'm retaking both the AP exam and the SAT II exam, and I ain't settling for anything less than a 5 and 800, so I better be putting in the time to review.
-I'm currently yelling at my poor dad to find me a high-quality tutor before October break, so we have a clear path of action to prepare
Goal 6: Pass the f=ma exam
Priority: 5/10
Difficulty: 7/10
Specifics
-Pass the f=ma exam
-Why? Cause I love physics, and if the school isn't going to support me in my enrichments, I'm just going to have to do this **** myself.
-Again, I'm yelling at my poor dad to find me a Chinese tutor
-f=ma is only mechanics, which I have some experience of, so passing the first round this year shouldn't take up too much time (maybe just a few hours per week)
-Why? Cause I love physics, and if the school isn't going to support me in my enrichments, I'm just going to have to do this **** myself.
-Again, I'm yelling at my poor dad to find me a Chinese tutor
-f=ma is only mechanics, which I have some experience of, so passing the first round this year shouldn't take up too much time (maybe just a few hours per week)
Goal 7: Be a more charismatic person
Priority: 4/10
Difficulty: 12/10
Specifics
Er... I'm pretty bad at this so...
-Learn to rap a few Kpop songs
-Try to remember to shower daily
-Buy a better conditioner and comb
-Get to 75 KG (I'm still like 78 KG)
-Learn to rap a few Kpop songs
-Try to remember to shower daily
-Buy a better conditioner and comb
-Get to 75 KG (I'm still like 78 KG)
Goal 8: Try to not die of stress
Specifics
Cause dying is like bad and I won't achieve my goals if I do.
Friendly reminder to my blog readers: Try to keep your comments on-topic and civil.
This post has been edited 6 times. Last edited by shiningsunnyday, Oct 4, 2016, 11:52 AM
I think goal 8 should have top priority 
(seriously though, AMSP/WOOT alone should mostly give you 1/4 ability if you actually pay attention which it looks like you do. The reason not everyone who takes AMSP/WOOT is successful is that not many people work hard enough or make full use of the resource)
May 2018
April 2018