Nemotron-IMO-Bench 2026 C41

Prove that for all positive integers $a,b$ $$\sum_{i,j \ge 0} \binom{i+j}{i}^2 \binom{(a-i)+(b-j)}{a-i}^2=\frac{1}{2} \binom{(2a+1)+(2b+1)}{2a+1}.$$