Places to find harder proofs questions (~extracurricular/olympiad type) (1 Viewer)

[lolzdj]

I just finished doing my trials and came across a few interesting/uncommon kinds of questions that aren't found in your standard textbook (cambridge/terry lee). An example of this would be the following:

Prove that the only integer solutions to the equation (x-a)(x-b)(x-c)(x-d) - 4 = 0 is (a+b+c+d)/4 where a, b, c and d are unique integers.

I feel like solving questions of this kind (especially once you see the solution and go "duh!") isnt neccessarily out of reach, i'm just not so sure where to start looking. Could anyone provide resources/give guidance as to where questions of this kind are found? Preferably at around the 4unit level.

Thanks!

 

[Pedro123]

I'm not sure but is this even true? Just subbing in a=1, b=2, c=3, d=4:
x = 2.5
= 1.5 * 0.5 * -0.5 * -1.5
Which isn't 4? Unless I'm understanding the question wrong here

 

[lolzdj]

I'm not sure but is this even true? Just subbing in a=1, b=2, c=3, d=4:
x = 2.5
= 1.5 * 0.5 * -0.5 * -1.5
Which isn't 4? Unless I'm understanding the question wrong here

you let m be the interger solution, then equate to 4. Chuck a prime decomposition onto 4 and you get +/-(2,1) what each bracket is. Solve for m and you get your answer.

 

[lolzdj]

you let m be the interger solution, then equate to 4. Chuck a prime decomposition onto 4 and you get +/-(2,1) - what each bracket is. Solve for m and you get your answer. It's just showing that if there exists an integer solution, it must satisfy those criteria. I'm just paraphrasing the question btw; havent been allowed to take it home yet.

 

[Trebla]

See the past BoS trial papers in the resources section.

 

[idkkdi]

I just finished doing my trials and came across a few interesting/uncommon kinds of questions that aren't found in your standard textbook (cambridge/terry lee). An example of this would be the following:

Prove that the only integer solutions to the equation (x-a)(x-b)(x-c)(x-d) - 4 = 0 is (a+b+c+d)/4 where a, b, c and d are unique integers.

I feel like solving questions of this kind (especially once you see the solution and go "duh!") isnt neccessarily out of reach, i'm just not so sure where to start looking. Could anyone provide resources/give guidance as to where questions of this kind are found? Preferably at around the 4unit level.

Thanks!

Olympiad questions aren't going to help you much with 4u math. This looks pretty different from AIMO and AMO stages which are normally number theory and geometry heavy, with some similar algebraic questions, but usually harder/different to this one.
BOS trials is probably a good place to look. In addition maybe try grammar/ruse papers last questions.
Cambridge enrichment in 4u is pretty crazy imo.

 

[tickboom]

I'm not sure but is this even true? Just subbing in a=1, b=2, c=3, d=4:
x = 2.5
= 1.5 * 0.5 * -0.5 * -1.5
Which isn't 4? Unless I'm understanding the question wrong here

Pedro, I don't think you get to pick any unique integers for a, b, c and d. I think the question is just stating that the solution to the equation is (a+b+c+d)/4 where a, b, c and d (whatever they happen to be) are unique and integers.

Check out this video to see my attempt at tackling the proof. Enjoy!

 

[Daniel.22]

x is an integer, and a,b,c,d distinct integers, so x-a,x-b,x-c,x-d are distinct integer factors of 4, with product 4.

A quick consideration of cases tells you these factors must be 2,-2,1,-1.

Then sum the 4 equations to get 4x-(a+b+c+d)=2-2+1-1=0, and you are done.

If you are looking for some questions around the 4u level, I like the STEP exams. They are a lot closer to 4u in flavour/difficulty than olympiad questions.

 

[idkkdi]

x is an integer, and a,b,c,d distinct integers, so x-a,x-b,x-c,x-d are distinct integer factors of 4, with product 4.

A quick consideration of cases tells you these factors must be 2,-2,1,-1.

Then sum the 4 equations to get 4x-(a+b+c+d)=2-2+1-1=0, and you are done.

If you are looking for some questions around the 4u level, I like the STEP exams. They are a lot closer to 4u in flavour/difficulty than olympiad questions.

Someone is trying to get to Cambridge ^^.

 

[Daniel.22]

Lol no, am happy to study in Australia. The exams are still nice for practice though.

 

[dan964]

I just finished doing my trials and came across a few interesting/uncommon kinds of questions that aren't found in your standard textbook (cambridge/terry lee). An example of this would be the following:

Prove that the only integer solutions to the equation (x-a)(x-b)(x-c)(x-d) - 4 = 0 is (a+b+c+d)/4 where a, b, c and d are unique integers.

I feel like solving questions of this kind (especially once you see the solution and go "duh!") isnt neccessarily out of reach, i'm just not so sure where to start looking. Could anyone provide resources/give guidance as to where questions of this kind are found? Preferably at around the 4unit level.

Thanks!

I'll send you something

see below

 

[TamzidZ]

Isn't this like discrete maths in uni?

 

[lolzdj]

Isn't this like discrete maths in uni?

sorta, it popped up in the independent trial. also waddup.

 

[TamzidZ]

sorta, it popped up in the independent trial. also waddup.

Yo waddup. This someone i know? Also yeah i saw this q in the independant paper lol. It was a madness thing.

 

[dan964]

proofs

 

[dan964]

Competitions:

THSC Online - Maths Competitions

try also AOPS for more questions

 

[lolzdj]

Yo waddup. This someone i know? Also yeah i saw this q in the independant paper lol. It was a madness thing.

same tutoring but i see you.. everywhere