Sunaina thinks absurd

Yeah, I did some problems and here are a few of them! I hope you guys try them! Putnam, 2018 B3 Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$. Proof We have $$n|2^n\implies n=2^a\implies 2^a-1|2^n-1\implies a|n\implies a=2^b$$ $$\implies 2^{2^b}-2|2^{2^a}-2\implies 2^b-1|2^a-1\implies b|a\implies b=2^c.$$ Then simply bounding. USAMO 1987 Determine all solutions in non-zero integers $a$ and $b$ of the equation $$(a^2+b)(a+b^2) = (a-b)^3.$$ Proof We get $$ 2b^2+(a^2-3a)b+(a+3a^2)=0\implies b = \frac{3a-a^2\pm\sqrt{a^4-6a^3-15a^2-8a}}{4}$$ $$\implies a^4-6a^3-15a^2-8a=a(a-8)(a+1)^2\text{ a perfect square}$$ $$\implies a(a-8)=k^2\implies a^2-8a-k^2=0\implies \implies a=\frac{8\pm\sqrt{64+4k^2}}{2}=4\pm\sqrt{16+k^2}. $$ $$ 16+k^2=m^2\implies (m-k)(m+k)=16.$$ Now just bash. USAMO 1988 Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1...

 Okie so continuing from the previous post ( sorry for huge time gap, got stuck in Allen stuff) A warning though if anyone from the "anti-bash" community is reading, sorry in advance and R.I.P.  Notes:- 1. We have $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$ then we have $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$ and we get $|z_1z_2|=|z_1||z_2|$ and $\arg z_1z_2=\arg z_1+\arg z_2 .$ SPIRAL SIMILARITIES and TRANSFORMATIONS:- 2. How to rotate a point about origin by 90. If $90^{\circ}$ anti-clockwise then multiply the number by $i$  If $90^{\circ}$ clockwise then multiply by $-i.$ Proof:- Note that $i= 0+1\cdot i.$ So $|i|= \sqrt{0^2+1^2}=1.$ And when we locate $i$ in complex plane, clearly $\arg i=\pi/2.$ Similarly for the second, but note that $\arg -i=-\pi /2.$ (Since angles are measured anti-clockwise) Example:- Here we have $z=1+2i$ then when we multiply $i$ we get $zi=i-2.$ Now, what about we want to dialate a point from one to another and scale it? (Note that...

I have to learn complex bash. And almost everyone knows that I am notes taking girl so thought why not make a post on complex bash ( so that I don't get emotionally demotivated lol).😇 There wasn't any need for learning complex bash, but it was in my dream checklist i.e " To learn a bash." And since I am not loaded with exams, I think it's high time to learn Bash and new topics.  Also if anyone from the "anti-bash" community is reading, sorry in advance and R.I.P.  Notes:- 1. Complex numbers are of the form $z=a+ib,$ where $a$ and $b$ are real numbers and $i^2=-1.$ 2. In polar form, $z=r(\cos \theta+~~i\sin\theta)=~~re^{i\theta},$ where $r=~~|z|=~~\sqrt{a^2+b^2},$ which is called the magnitude. 3. Here we used euler's formula i.e $\cos \theta+~~i\sin\theta=~~e^{i\theta}.$ 4. The $\theta $ is called the argument of $z,$ denored $\arg z.$ ( $\theta$ can be considered in $\mod 360$ and it is  measured anti-clockwise). 5. The complex conjugate of $z$ is ...