Can you solve these Number Theory problems in 30 seconds & 1 minute?

If possible, list all the solutions in limited time too!

You have 30 seconds for solving these problems.

Def: Lattice point is the point (x,y)(x,y) such that both xx and yy are integers.

1.) Find the number of lattice points of a hyperbola given by an equation

4x2y22y2558=04x^{2}-y^{2}-2y-2558 = 0

2.) Find the number of lattice points of an ellipse given by an equation

2x2xy+2y2+3x3y3=02x^{2}-xy+2y^{2}+3x-3y-3 = 0

You have 1 minute for solving these problems.

1.) Find the number of ordered pair (x,y)(x,y) in integers of the equation

x2y!=2559x^{2} - y! = 2559

2.) Find the number of ordered triples (x,y,z)(x,y,z) in integers of the equation

x2+y2+z2=2558xyzx^{2}+y^{2}+z^{2} = 2558xyz

Sometimes the math competitions in my country are too goddamn crazy. I hate it, and I'll never have this competition again. XD

Note by Samuraiwarm Tsunayoshi
5 years, 11 months ago

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1 vote

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I just might say I can.

For the first question, we change it to 4x2(y+1)2=25574{ x }^{ 2 }-{ (y+1) }^{ 2 }=2557, and as 4x24x^2 is divisible by 4, 4x2(y+1)24{ x }^{ 2 }-{ (y+1) }^{ 2 } will have a remainder of 0 or -1 when divided by 4, which is not the case here.

Next question, multiply both sides by 2, then we will have (xy)2+3(x+1)2+3(y1)2=12{ (x-y) }^{ 2 }+{ 3(x+1) }^{ 2 }+{ 3(y-1) }^{ 2 }=12, and we can then claim that (xy)2{ (x-y) }^{ 2 } must be 0 or 9 (as 12 and 3(x+1)2+3(y1)2{ 3(x+1) }^{ 2 }+{ 3(y-1) }^{ 2 } are all divisible by 3). The last part is pretty lengthy, so I will not write it down here (besides, it's easy from here)

The third one, we have x2=2559+y!{ x }^{ 2 }=2559+y!, which means x2x^2 will have the remainder of 3 when divided by 4 if y4y \ge 4, which is impossible. Test for numbers from 1 to 3, tada, we have the answer.

The final one, if none of the 3 is divisible by 2 then x2+y2+z2x^2+y^2+z^2 is not divisible by 2 while 2558xyz2558xyz is not. If one or two is then x2+y2+z2x^2+y^2+z^2 is not divisible by 4 while 2558xyz2558xyz is, which means that all 3 must be divisible by 2. Let x=x2,y=y2,z=z2{x}^{'}=\frac{x}{2},{y}^{'}=\frac{y}{2},{z}^{'}=\frac{z}{2} and prove exactly like shown above. Also note that if x,y,zx,y,z are positive integers, then there does not exist a set of infinitely many numbers x1,...xnx_1,...x_n which satisfies x1>x2>...>xnx_1>x_2>...>x_n, which leaves x=y=z=0x=y=z=0 the only solution.

The first 2 problems I solved within 20 seconds each, while the third was solved in 10 seconds and the final one took me almost a minute. I think I should start celebrating now.

Steven Jim - 3 years, 4 months ago

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Are you really 14 years old?

Syed Hamza Khalid - 2 years, 9 months ago

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I'm sure 10th graders are 14-15 years old, so yep, you're right.

Just a math lover :)

P/S: Not bragging, but these are still easy to what we have to do :) If you really want to screw your brain up, I have some for you :D

Steven Jim - 2 years, 9 months ago

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@Steven Jim I am in grade 10 and 99% of my classmates don't even know the quadratic equation or factorials. They only know BIDMAS and cubic graphs nothing else special.... hence I look like a genius at my class but thanks to Brilliant; I identified that it's the school's fault of having such low mathematic standards and that globally i am literally somewhat a zero hero :(

However, due to Brilliant; I am able to increase my standard and understanding of mathematics in a dramatic way.


Syed Hamza Khalid - 2 years, 9 months ago

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