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<title>AMC8 中文学习平台 - 历年真题与解析</title>
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<h1 class="text-4xl md:text-5xl font-bold mb-4">AMC8 中文学习平台</h1>
<p class="text-xl md:text-2xl mb-8">提供2000-2023年完整真题与详细解析</p>
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<i class="fas fa-book-open mr-2"></i>开始学习
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<h2 class="text-3xl font-bold text-center mb-12 text-gray-800">关于 AMC8 竞赛</h2>
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<h3 class="text-xl font-semibold mb-3 text-gray-800">竞赛简介</h3>
<p class="text-gray-600">AMC8是美国数学竞赛系列中的初中级别比赛，面向8年级及以下学生，旨在激发学生对数学的兴趣和才能。</p>
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<h3 class="text-xl font-semibold mb-3 text-gray-800">考试形式</h3>
<p class="text-gray-600">40分钟完成25道选择题，满分25分。题目涵盖算术、代数、几何、数论和概率等数学领域。</p>
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<h3 class="text-xl font-semibold mb-3 text-gray-800">奖项设置</h3>
<p class="text-gray-600">荣誉证书(15分以上)、成就证书(18分以上)、卓越证书(21分以上)和满分奖(25分)。</p>
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<h2 class="text-3xl font-bold text-center mb-8 text-gray-800">历年真题与解析</h2>
<!-- 年份筛选 -->
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<h3 class="text-xl font-semibold mb-4 text-gray-700">筛选题目</h3>
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<option value="all">全部年份</option>
<option value="2023">2023年</option>
<option value="2022">2022年</option>
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<option value="2020">2020年</option>
<option value="2019">2019年</option>
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<option value="all">全部类型</option>
<option value="algebra">代数</option>
<option value="geometry">几何</option>
<option value="number-theory">数论</option>
<option value="combinatorics">组合数学</option>
<option value="probability">概率</option>
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<!-- 题目列表 -->
<div class="grid-fallback grid-cols-1 md:grid-cols-2 lg:grid-cols-3 gap-6">
<!-- 题目卡片1 -->
<div class="problem-card bg-white rounded-lg overflow-hidden shadow-md" data-year="2023" data-difficulty="medium" data-topic="geometry">
<div class="p-6">
<div class="flex justify-between items-start mb-2">
<span class="bg-blue-100 text-blue-800 text-xs font-semibold px-2.5 py-0.5 rounded">2023年</span>
<span class="bg-green-100 text-green-800 text-xs font-semibold px-2.5 py-0.5 rounded">几何</span>
</div>
<h3 class="text-xl font-bold text-gray-800 mb-3">第15题</h3>
<p class="text-gray-600 mb-4">一个正方形的边长为4，四个角上各剪去一个边长为1的小正方形，然后将剩余部分折叠成一个无盖的盒子。这个盒子的体积是多少？</p>
<div class="flex flex-wrap gap-2 mb-4">
<span class="bg-gray-100 text-gray-800 text-xs font-medium px-2.5 py-0.5 rounded">中等难度</span>
<span class="bg-purple-100 text-purple-800 text-xs font-medium px-2.5 py-0.5 rounded">体积计算</span>
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<h4 class="font-semibold text-gray-800 mb-2">解析：</h4>
<p class="text-gray-700 mb-3">1. 原正方形边长为4，剪去四个角上边长为1的小正方形后，剩余部分可以看作一个十字形。</p>
<p class="text-gray-700 mb-3">2. 折叠后，盒子的高度就是剪去的小正方形边长1。</p>
<p class="text-gray-700 mb-3">3. 盒子的底面是一个边长为2的正方形（4-1×2=2）。</p>
<p class="text-gray-700 mb-4">4. 因此体积为：底面积×高 = 2×2×1 = 4。</p>
<h4 class="font-semibold text-gray-800 mb-2">答案：</h4>
<p class="text-green-600 font-bold">B) 4</p>
</div>
</div>
</div>
<!-- 题目卡片2 -->
<div class="problem-card bg-white rounded-lg overflow-hidden shadow-md" data-year="2022" data-difficulty="easy" data-topic="number-theory">
<div class="p-6">
<div class="flex justify-between items-start mb-2">
<span class="bg-blue-100 text-blue-800 text-xs font-semibold px-2.5 py-0.5 rounded">2022年</span>
<span class="bg-yellow-100 text-yellow-800 text-xs font-semibold px-2.5 py-0.5 rounded">数论</span>
</div>
<h3 class="text-xl font-bold text-gray-800 mb-3">第8题</h3>
<p class="text-gray-600 mb-4">如果n是一个正整数，使得2n是3的倍数，3n是4的倍数，4n是5的倍数，那么n的最小可能值是多少？</p>
<div class="flex flex-wrap gap-2 mb-4">
<span class="bg-gray-100 text-gray-800 text-xs font-medium px-2.5 py-0.5 rounded">简单难度</span>
<span class="bg-red-100 text-red-800 text-xs font-medium px-2.5 py-0.5 rounded">最小公倍数</span>
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<h4 class="font-semibold text-gray-800 mb-2">解析：</h4>
<p class="text-gray-700 mb-3">1. 根据题意，我们可以将条件转化为：</p>
<p class="text-gray-700 mb-3">- n是3/2的倍数（但n是整数，所以n必须是3的倍数）</p>
<p class="text-gray-700 mb-3">- n是4/3的倍数（所以n必须是4的倍数）</p>
<p class="text-gray-700 mb-3">- n是5/4的倍数（所以n必须是5的倍数）</p>
<p class="text-gray-700 mb-4">2. 因此n必须同时是3、4和5的倍数，最小公倍数为60。</p>
<h4 class="font-semibold text-gray-800 mb-2">答案：</h4>
<p class="text-green-600 font-bold">E) 60</p>
</div>
</div>
</div>
<!-- 题目卡片3 -->
<div class="problem-card bg-white rounded-lg overflow-hidden shadow-md" data-year="2021" data-difficulty="hard" data-topic="combinatorics">
<div class="p-6">
<div class="flex justify-between items-start mb-2">
<span class="bg-blue-100 text-blue-800 text-xs font-semibold px-2.5 py-0.5 rounded">2021年</span>
<span class="bg-pink-100 text-pink-800 text-xs font-semibold px-2.5 py-0.5 rounded">组合数学</span>
</div>
<h3 class="text-xl font-bold text-gray-800 mb-3">第25题</h3>
<p class="text-gray-600 mb-4">有6个不同的礼物要分给3个孩子，每个孩子至少得到1个礼物。有多少种不同的分配方式？</p>
<div class="flex flex-wrap gap-2 mb-4">
<span class="bg-gray-100 text-gray-800 text-xs font-medium px-2.5 py-0.5 rounded">高难度</span>
<span class="bg-indigo-100 text-indigo-800 text-xs font-medium px-2.5 py-0.5 rounded">分配问题</span>
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<h4 class="font-semibold text-gray-800 mb-2">解析：</h4>
<p class="text-gray-700 mb-3">1. 这是一个典型的"将n个不同的物品分配到k个不同的盒子，每个盒子至少一个"的问题。</p>
<p class="text-gray-700 mb-3">2. 可以使用容斥原理计算：总分配方式减去至少一个孩子没得到礼物的方式。</p>
<p class="text-gray-700 mb-3">3. 总分配方式：3^6 = 729（每个礼物有3种选择）</p>
<p class="text-gray-700 mb-3">4. 减去只有一个孩子得到礼物：C(3,1)×1^6 = 3</p>
<p class="text-gray-700 mb-3">5. 减去只有两个孩子得到礼物：C(3,2)×2^6 = 3×64=192</p>
<p class="text-gray-700 mb-4">6. 但是第二步多减了，需要加回：3^6 - 3×2^6 + 3×1^6 = 729 - 192 + 3 = 540</p>
<h4 class="font-semibold text-gray-800 mb-2">答案：</h4>
<p class="text-green-600 font-bold">540</p>
</div>
</div>
</div>
<!-- 题目卡片4 -->
<div class="problem-card bg-white rounded-lg overflow-hidden shadow-md" data-year="2020" data-difficulty="medium" data-topic="algebra">
<div class="p-6">
<div class="flex justify-between items-start mb-2">
<span class="bg-blue-100 text-blue-800 text-xs font-semibold px-2.5 py-0.5 rounded">2020年</span>
<span class="bg-orange-100 text-orange-800 text-xs font-semibold px-2.5 py-0.5 rounded">代数</span>
</div>
<h3 class="text-xl font-bold text-gray-800 mb-3">第18题</h3>
<p class="text-gray-600 mb-4">解方程：2^(x+3) - 2^x = 224</p>
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<span class="bg-gray-100 text-gray-800 text-xs font-medium px-2.5 py-0.5 rounded">中等难度</span>
<span class="bg-teal-100 text-teal-800 text-xs font-medium px-2.5 py-0.5 rounded">指数方程</span>
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<h4 class="font-semibold text-gray-800 mb-2">解析：</h4>
<p class="text-gray-700 mb-3">1. 首先将方程左边因式分解：2^x(2^3 - 1) = 224</p>
<p class="text-gray-700 mb-3">2. 计算括号内：8 - 1 = 7，所以方程变为：7×2^x = 224</p>
<p class="text-gray-700 mb-3">3. 两边除以7：2^x = 32</p>
<p class="text-gray-700 mb-4">4. 32是2的5次方，所以x=5。</p>
<h4 class="font-semibold text-gray-800 mb-2">答案：</h4>
<p class="text-green-600 font-bold">5</p>
</div>
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<!-- 题目卡片5 -->
<div class="problem-card bg-white rounded-lg overflow-hidden shadow-md" data-year="2019" data-difficulty="easy" data-topic="probability">
<div class="p-6">
<div class="flex justify-between items-start mb-2">
<span class="bg-blue-100 text-blue-800 text-xs font-semibold px-2.5 py-0.5 rounded">2019年</span>
<span class="bg-red-100 text-red-800 text-xs font-semibold px-2.5 py-0.5 rounded">概率</span>
</div>
<h3 class="text-xl font-bold text-gray-800 mb-3">第10题</h3>
<p class="text-gray-600 mb-4">一个袋子里有3个红球和2个蓝球。随机取出2个球，都是红球的概率是多少？</p>
<div class="flex flex-wrap gap-2 mb-4">
<span class="bg-gray-100 text-gray-800 text-xs font-medium px-2.5 py-0.5 rounded">简单难度</span>
<span class="bg-purple-100 text-purple-800 text-xs font-medium px-2.5 py-0.5 rounded">组合概率</span>
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<h4 class="font-semibold text-gray-800 mb-2">解析：</h4>
<p class="text-gray-700 mb-3">1. 总球数：3红 + 2蓝 = 5个球</p>
<p class="text-gray-700 mb-3">2. 总的取法数：C(5,2) = 10</p>
<p class="text-gray-700 mb-3">3. 取到两个红球的取法数：C(3,2) = 3</p>
<p class="text-gray-700 mb-4">4. 概率 = 有利事件数 / 总事件数 = 3/10</p>
<h4 class="font-semibold text-gray-800 mb-2">答案：</h4>
<p class="text-green-600 font-bold">C) 3/10</p>
</div>
</div>
</div>
<!-- 题目卡片6 -->
<div class="problem-card bg-white rounded-lg overflow-hidden shadow-md" data-year="2023" data-difficulty="hard" data-topic="geometry">
<div class="p-6">
<div class="flex justify-between items-start mb-2">
<span class="bg-blue-100 text-blue-800 text-xs font-semibold px-2.5 py-0.5 rounded">2023年</span>
<span class="bg-green-100 text-green-800 text-xs font-semibold px-2.5 py-0.5 rounded">几何</span>
</div>
<h3 class="text-xl font-bold text-gray-800 mb-3">第24题</h3>
<p class="text-gray-600 mb-4">在△ABC中，AB=5，BC=12，CA=13。点D在BC上，使得AD平分∠BAC。求BD的长度。</p>
<div class="flex flex-wrap gap-2 mb-4">
<span class="bg-gray-100 text-gray-800 text-xs font-medium px-2.5 py-0.5 rounded">高难度</span>
<span class="bg-blue-100 text-blue-800 text-xs font-medium px-2.5 py-0.5 rounded">角平分线定理</span>
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<h4 class="font-semibold text-gray-800 mb-2">解析：</h4>
<p class="text-gray-700 mb-3">1. 首先验证△ABC是直角三角形：5² + 12² = 13² (25 + 144 = 169)。</p>
<p class="text-gray-700 mb-3">2. 应用角平分线定理：BD/DC = AB/AC = 5/13</p>
<p class="text-gray-700 mb-3">3. 设BD=5k，DC=13k，则BD+DC=BC=12 → 18k=12 → k=2/3</p>
<p class="text-gray-700 mb-4">4. 因此BD=5×(2/3)=10/3</p>
<h4 class="font-semibold text-gray-800 mb-2">答案：</h4>
<p class="text-green-600 font-bold">10/3</p>
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