do NOT double count (0,0)
by bobthegod78, Feb 7, 2025, 3:53 PM
Find the number of ordered pairs 
, where both 
and 
are integers between 
and 
inclusive, such that 
.
, where both 
and 
are integers between 
and 
inclusive, such that 
.This post has been edited 1 time. Last edited by bobthegod78, Feb 7, 2025, 3:53 PM
Stressed spelled backwards
by centslordm, Feb 7, 2025, 3:52 PM
The 9 members of a baseball team went to an ice-cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let 
be the number of different assignments of flavors to players that meet these conditions. Find the remainder when is divided by 
be the number of different assignments of flavors to players that meet these conditions. Find the remainder when is divided by 
Heptagon
by StressedPineapple, Feb 7, 2025, 3:51 PM
In 
points 
and 
lie on 
so that 
, while points 
and 
lie on 
so that 
. Suppose 
, 
, 
, 
, 
, and 
. Let 
be the reflection of through , and let be the reflection of through . The area of quadrilateral 
is 
. Find the area of heptagon 
, as shown in the figure below.
![[asy]
unitsize(14);
pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G;
filldraw(A--F--N--B--C--E--M--cycle, lightgray);
draw(A--B--C--cycle);
draw(D--M);
draw(N--G);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(M);
dot(N);
label("$A$", A, dir(90));
label("$B$", B, dir(225));
label("$C$", C, dir(315));
label("$D$", D, dir(135));
label("$E$", E, dir(135));
label("$F$", F, dir(45));
label("$G$", G, dir(45));
label("$M$", M, dir(45));
label("$N$", N, dir(135));
[/asy]](//latex.artofproblemsolving.com/6/4/b/64b0d84a95e0388e37622b54a251d836088fd97c.png)
points 
and 
lie on 
so that 
, while points 
and 
lie on 
so that 
. Suppose 
, 
, 
, 
, 
, and 
. Let 
be the reflection of through , and let be the reflection of through . The area of quadrilateral 
is 
. Find the area of heptagon 
, as shown in the figure below.![[asy]
unitsize(14);
pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G;
filldraw(A--F--N--B--C--E--M--cycle, lightgray);
draw(A--B--C--cycle);
draw(D--M);
draw(N--G);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(M);
dot(N);
label("$A$", A, dir(90));
label("$B$", B, dir(225));
label("$C$", C, dir(315));
label("$D$", D, dir(135));
label("$E$", E, dir(135));
label("$F$", F, dir(45));
label("$G$", G, dir(45));
label("$M$", M, dir(45));
label("$N$", N, dir(135));
[/asy]](http://latex.artofproblemsolving.com/6/4/b/64b0d84a95e0388e37622b54a251d836088fd97c.png)
This post has been edited 1 time. Last edited by StressedPineapple, Feb 7, 2025, 5:52 PM
Reason: diagram was backwards
Reason: diagram was backwards
basic nt
by zhoujef000, Feb 7, 2025, 3:46 PM
Find the sum of all integer bases 
for which 
is a divisor of 
for which 
is a divisor of 
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