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...