latent heat (urgent)

Yes, you are right. There is temperature change during the transient period. But the heat absorbed by the vacuum flask would eventually be all released out when the mixture (hot water and ice) cool down to zero degree Celsius, thus giving no net heat gain by the vacuum flask.

If you include this transient heat exchange into your equations, they can be written approximately as:Using the same symbols as in your equations,
energy lost by hot water from T to an arbitrary temperature t = Mc(T - t)
heat gain by flask = Q, (say)
where Q is the heat absorbed by the flask
hence, amount of heat given to ice = Mc(T - t) - Q

energy lost by water and flask when temperature drop from t to 0'C
= Mct + Q
hence, total energy given to ice
= [Mc(T - t) - Q] + [ Mct + Q]
= McT - Mct - Q + Mct + Q
= Mc(T)
The rest is the same as what you have writtent
energy gained by the ice = ml
By conversion of energy,
McT = ml
M=ml/cT

The basic concept is because energy is a scalar. By the law of conservation of energy, any energy lost/gain by a third party (the flask in this example) will not affect the physical process. You only need to compare the initial and final energy of the system.

呢一樣野呢... ...

在高中物理之中 (高中以外的就不說了, 說了都是增加你的煩惱), 水由固體轉為液體的時候, 是先把外來的溫度吸收成為「 熔化熱/ latent heat」, 直至所有固體部份都完全熔解後, 才會把後續的溫度吸收為「熱容 / heat capacity」, 明未? 換句話說, 在高中的考試卷內, 就算只剩0.00001%的固體冰存在, 整份「冰加水」的混合物都一定是零度的.