and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
are taken points 
and 
,
and 
.
and 
is equal to 
.
Y by Adventure10, Mango247
In an 
trapezoid of bases 
and 
the points 
and 
are chosen on the sides 
and 
, respectively, so that 
. If 
intersects the diagonals 
and 
in 
and 
, respectively, prove that 
.
trapezoid of bases 
and 
the points 
and 
are chosen on the sides 
and 
, respectively, so that 
. If 
intersects the diagonals 
and 
in 
and 
, respectively, prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jun 10, 2024, 12:49 PM
be the intersection of 
and 
,
,
.
and 
,
,
.![[asy]
unitsize(.75cm);
pair A, B, C, D, M, N, X, P, Q;
A = (1, 4); B = (6,4); C = (9,0); D = (0,0); M = .75A + .25D; N = .75C + .25B; X = extension(A, C, B, D); P = extension(A, C, M, N); Q = extension(B, D, M, N);
draw(A--B--C--D--cycle); draw(M--N); draw(A--C); draw(B--D);
dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, SE); dot("$D$", D, SW); dot("$M$", M, W); dot("$N$", N, E); dot("$X$", X, NW+.8); dot("$P$", P, S); dot("$Q$", Q, S);
add(pathticks(M--Q)); add(pathticks(P--N));
[/asy]](http://latex.artofproblemsolving.com/8/3/f/83ff3982645d42249d3662d1898aef7fb3552624.png)
