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MATH / In the five-sided star shown, the letters A, B, C, D, and E are replaced by the numbers 3, 5, 6, 7, and 9, although not necessarily in this order. The sums of the numbers at the en…

Problem

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problem

In the five-sided star shown, the letters AA, BB, CC, DD, and EE are replaced by the numbers 33, 55, 66, 77, and 99, although not necessarily in this order. The sums of the numbers at the ends of the line segments ABAB, BCBC, CDCD, DEDE, and EAEA form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence? [asy] unitsize(2 cm); pair A, B, C, D, E; A = dir(90); B = dir(90 + 3*360/5); C = dir(90 + 6*360/5); D = dir(90 + 9*360/5); E = dir(90 + 12*360/5); draw(A--B--C--D--E--cycle); filldraw(Circle(A,0.15),white); filldraw(Circle(B,0.15),white); filldraw(Circle(C,0.15),white); filldraw(Circle(D,0.15),white); filldraw(Circle(E,0.15),white); label("AA", A); label("BB", B); label("CC", C); label("DD", D); label("EE", E); [/asy]
Plain-text mathematical notation (without MathML)
In the five-sided star shown, the letters A, B, C, D, and E are replaced by the numbers 3, 5, 6, 7, and 9, although not necessarily in this order. The sums of the numbers at the ends of the line segments AB, BC, CD, DE, and EA form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?

[asy]
unitsize(2 cm);

pair A, B, C, D, E;

A = dir(90);
B = dir(90 + 3*360/5);
C = dir(90 + 6*360/5);
D = dir(90 + 9*360/5);
E = dir(90 + 12*360/5);

draw(A--B--C--D--E--cycle);
filldraw(Circle(A,0.15),white);
filldraw(Circle(B,0.15),white);
filldraw(Circle(C,0.15),white);
filldraw(Circle(D,0.15),white);
filldraw(Circle(E,0.15),white);

label("A", A);
label("B", B);
label("C", C);
label("D", D);
label("E", E);
[/asy]
Original LaTeX notation
In the five-sided star shown, the letters $A$, $B$, $C$, $D$, and $E$ are replaced by the numbers $3$, $5$, $6$, $7$, and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$, $BC$, $CD$, $DE$, and $EA$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?

[asy]
unitsize(2 cm);

pair A, B, C, D, E;

A = dir(90);
B = dir(90 + 3*360/5);
C = dir(90 + 6*360/5);
D = dir(90 + 9*360/5);
E = dir(90 + 12*360/5);

draw(A--B--C--D--E--cycle);
filldraw(Circle(A,0.15),white);
filldraw(Circle(B,0.15),white);
filldraw(Circle(C,0.15),white);
filldraw(Circle(D,0.15),white);
filldraw(Circle(E,0.15),white);

label("$A$", A);
label("$B$", B);
label("$C$", C);
label("$D$", D);
label("$E$", E);
[/asy]

level

Level 2

type

Intermediate Algebra

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Source and history

task_id
9b5be5ea-3da8-5d8c-93d9-bfdcb5664899
task_key
intermediate~5falgebra--test--9b5be5ea-3da8-5d8c-93d9-bfdcb5664899
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import