Consider a the Vascicek model, and let A and B denote the functions such that $P(t,T)=\exp(A(t,T)-B(t,T)r(t))$. We now look at a coupon bond that makes deterministic payments $\alpha_1,...,\alpha_N$ at dates $T_1,...,T_N$. Clearly the price of this coupon bond is $$\pi^c(t)=\sum_{i|T_i<t}\alpha_i P(t,T_i)$$.
Assume $K$ as the strike of an expiry-T European call option on the coupon bond.
Show that there exists $r*\in\mathbb{R}$ such that $\pi^c(T)\geqslant K$ if and only if $r(T)\leqslant r*$. Define the adjusted strikes via $K_i=\exp(A(T,T_i)-B(T,T_i)r*)$ Show that the pay-off of the call can be writte as:
$$(\pi^c(T)-K)^+=\sum_{i|T_i<t}\alpha_i (P(T,T_i)-K_i)^+$$
My attempt: $\sum_{i|T_i<t}\alpha_i P(t,T_i)\geqslant K \implies\sum_{i|T_i<t}\alpha_i\exp(A(T,T_i)-B(T,T_i)r(T)) \geqslant K $
I tried to solve this equation for r(T) by taking the logratihtms but it does not work because I get $log(\sum_{i|T_i<t}\alpha_i\exp(A(T,T_i)-B(T,T_i)r(T)))$.
Question:
How should I solve this problem?
Thanks in advance!