Conjecture: $ \binom{2t}{2a} \geq \binom{t}{a}^2 $
34929
Prove ${M}\choose{N_1}$ ${M}\choose{N_2}$ $\leq$ ${2M}\choose{N_1+N_2}$ for $M, N_1, N_2 \in \mathbb{N},\,M\geq N_1,N_2$.
127566
$2AB$ is a perfect square and $A+B$ is not a perfect square
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
29578
Why is $\binom{2n}{n} \asymp \Theta \big(\frac{2^{2n}}{\sqrt{n}}\big)$?
90904
Combinatorial proof of an identity with squared binomial coefficients
83542
A natural problem in $\textrm{S}_2^\textrm{P}$?
145524
Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths
43188
The $n^{th}$ root of the geometric mean of binomial coefficients.
52324
Tight bound for $T(n) = T(n^{1/2}) + 1$
149700
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
66019
Revisited: Binomial Theorem: An Inductive Proof
62958
Maximize $\mathrm{tr}(Q^TCQ)$ subject to $Q^TQ=I$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
18634
Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
126312
How can you prove $\binom {2n}{2} = 2\binom{n}{2} + n^2$?