query: Assuming $a_k + b_k = 1$ (Putnam 2003)
positive_passages.docid: 156408
positive_passages.text: Putnam and Beyond AM-GM help
negative_passages.docid: 50202
negative_passages.text: Probablity identities and $A$ given $B$
negative_passages.docid: 10211
negative_passages.text: Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
negative_passages.docid: 133905
negative_passages.text: Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
negative_passages.docid: 68032
negative_passages.text: A Representation Theory Problem in Putnam Competition
negative_passages.docid: 70282
negative_passages.text: Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$
negative_passages.docid: 74222
negative_passages.text: If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$
negative_passages.docid: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 144571
negative_passages.text: A conjecture including binomial coefficients
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 959
negative_passages.text: Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
negative_passages.docid: 9522
negative_passages.text: Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
negative_passages.docid: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
negative_passages.docid: 3902
negative_passages.text: Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?
negative_passages.docid: 20346
negative_passages.text: 'Algebraic' way to prove the boolean identity $a + \overline{a}*b = a + b$
negative_passages.docid: 118674
negative_passages.text: Simple upper bound for $\binom{n}{k}$
negative_passages.docid: 83738
negative_passages.text: Prove $(k,n+k)=1$ iff $(k,n)=1$