AMC 8 2026 Problem 23

AMC8 2026 年第 23 题

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Lakshmi has 5 round coins of diameter 4 centimeters. She arranges the coins in 2 rows on a table top, as shown below, and wraps an elastic band tightly around them. In centimeters, what will be the length of the band?
Lakshmi 有 5 枚直径为 4 厘米的圆形硬币。她将这些硬币按如下方式在桌面上排成两行，并用一根橡皮筋紧紧围住它们。橡皮筋的长度是多少厘米？
AMC 8 2026 Problem 23 figure
  1. A.2π+202\pi + 20
  2. B.(52)π+20\displaystyle (\frac{5}{2})\pi + 20
  3. C.4π+204\pi + 20
  4. D.(92)π+20\displaystyle (\frac{9}{2})\pi + 20
  5. E.5π+205\pi + 20

正确答案 · Correct Answer

C

解析 · Solution

解法1

常规解法·把橡皮筋拆成直段和弧段
摆法：下排 3 枚硬币、上排 2 枚（卡在下排的缝上），每枚直径 4（半径 2）。橡皮筋贴着最外圈走，可以拆成两类：直线段（两枚外圈硬币之间绷直的部分）和圆弧段（绕过「拐角」硬币的部分）。
直线段：相切两枚硬币的圆心距 =2+2=4，绷直的那段与两圆心连线平行且等长，所以每跨一对相邻外圈硬币就是一段 4。数一数：顶边 1 段（上排两枚之间）、左右斜边各 1 段、底边跨 3 枚硬币是 2 段（筋贴着中间那枚，但在这儿是直的）。合计 4×5=20。
弧段：只在 4 个「拐角」硬币处打弯。角度从哪来：相邻三枚硬币（如底角一枚＋它上面一枚＋旁边一枚）的圆心连线是边长 4 的等边三角形，所以底角处两条直段的夹角是 60∘60^\circ，筋要绕过 180∘−60∘=120∘180^\circ-60^\circ=120^\circ 的弧；上排拐角处直段夹角 120∘120^\circ，弧就是 60∘60^\circ。四段弧合计 120+120+60+60=360∘120+120+60+60=360^\circ——正好拼成一个半径 2 的完整圆，弧长合计 2π×2=4π2\pi\times2=4\pi。
总长 =20+4π=20+4\pi，选 C。
易错提醒：底边中间那枚硬币不产生弧（筋在它那儿是直的），但底边直段是 2 段共 8，别只数成一段 4；四段弧也别想当然按「每角 90∘90^\circ」算——这里是 60∘60^\circ 与 120∘120^\circ 混搭，只是加起来恰好 360∘360^\circ。

Solution 1

*(Elementary — split the band into straight parts and arcs.)*
The layout: 3 coins in the bottom row, 2 in the top row (nestled in the gaps), each of diameter 4 (radius 2). The band hugs the outer boundary and splits into two kinds of pieces: straight segments (taut stretches between outer coins) and arcs (bending around the corner coins).
Straight segments: two tangent coins have centers 2+2=4 apart, and each taut stretch is parallel to — and as long as — the segment joining the two centers, so each stretch spanning one adjacent pair is 4. Count them: 1 on top (between the two top coins), 1 on each slanted side, and 2 along the bottom, spanning 3 coins (the band touches the middle coin but stays straight there). Total 4×5=20.
Arcs: the band bends only at the 4 corner coins. Where the angles come from: the centers of three mutually tangent coins (say a bottom corner coin, its top neighbor, and the coin beside it) form an equilateral triangle of side 4, so at each bottom corner the two straight stretches meet at 60∘60^\circ and the band wraps a 180∘−60∘=120∘180^\circ-60^\circ=120^\circ arc; at each top corner the stretches meet at 120∘120^\circ, giving a 60∘60^\circ arc. The four arcs total 120+120+60+60=360∘120+120+60+60=360^\circ — exactly one full circle of radius 2, so they sum to 2π×2=4π2\pi\times2=4\pi.
Total length =20+4π=20+4\pi, choice C.
Watch out: the middle bottom coin adds no arc (the band is straight there), but the bottom contributes 2 straight segments totaling 8, not a single 4; and don't assume each corner wraps 90∘90^\circ — here it's a mix of 60∘60^\circ and 120∘120^\circ that happens to total 360∘360^\circ.

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