Question:
The trisectors of angles $B$ and $C$ of scalene triangle $ABC$ meet at points $P$ and $Q$ as shown. Angle $A$ measures 39 degrees and angle $QBP$ measures 14 degrees. What is the measure of angle $BPC$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(4,5)--(0,0)--(2,1.5)--(3,0)--(1.7,0.5)--(0,0));
label("$P$", (1.7,0.5), S); label("$Q$", (2,1.5), N); label("$B$", (0,0),W); label("$A$", (4,5), N); label("$C$", (3,0), E);
[/asy]

Answer:
Because angle $QBP$ measures 14 degrees, we know that angle $ABC$ measures $3\cdot14=42$ degrees.  Hence, angle $ACB$ measures $180 - 39 - 42 = 99$ degrees.  Next we find that angle $BCP$ measures $\frac{99}3=33$ degrees, and finally angle $BPC$ measures $180 - 14 - 33 = \boxed{133}$ degrees.