Distinct points $A,B,C$ lie in that order on a straight line. Construct equilateral triangles $ABD$ and $BCE$ on the same side of line $AC$. Let $T$ be the intersection of $AE$ and $DC$. Prove:
Natural numbers $a_1,a_2,\ldots,a_{100}$ satisfy $\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_{100}}=\frac{101}{2}$. Prove that at least two of the numbers are equal.
A river flows from $A$ toward $B$. Two motorboats leave $A$ simultaneously, travel to $B$, and immediately return to $A$. On the outward trip, boat 1 has speed $2v$ km/h and boat 2 has speed $v$ km/h; when boat 1 reaches $B$, boat 2 has covered $\frac59 AB$. On the return trip, boat 1 has speed $v$ km/h and boat 2 has speed $2v$ km/h. Their finishing times differ by $3.2$ minutes. How many minutes does each boat take for the round trip?
Phuong Vy travels from her home $V$ to her grandmother's home $B$ along a route with level, uphill, and downhill sections. Her speeds are $5$ km/h on level ground, $4$ km/h uphill, and $6$ km/h downhill. The outward trip takes $1$ hour $36$ minutes, and the return takes $1$ hour $39$ minutes. If the level sections total $2.5$ km, how long is the route from $V$ to $B$?