英国初级数学奥林匹克竞赛（JMO）真题解析与备考指南 | UK Junior Mathematical Olympiad (JMO) — Solutions & Study Guide

引言

英国初级数学奥林匹克竞赛（Junior Mathematical Olympiad，简称 JMO）由英国数学信托基金会（UKMT）主办，面向英格兰和威尔士 8 年级及以下（苏格兰 S2 及以下，北爱尔兰 9 年级及以下）的学生。JMO 是 UKMT 竞赛体系中极具挑战性的一环——只有在全国 Junior Mathematical Challenge 中取得优异成绩的学生才有资格受邀参加。与侧重速度的选择题竞赛不同，JMO 要求学生在两小时内完成 16 道题目，其中 A 部分 10 道填空题（只需答案），B 部分 6 道解答题（需要完整推理过程）。本文将以 2017 年真题为蓝本，深入剖析 JMO 的核心考点与解题策略，帮助有志于挑战数学奥林匹克的学生系统备考。

The UK Junior Mathematical Olympiad (JMO), organized by the UK Mathematics Trust (UKMT), is an invitational competition for students in Year 8 and below in England and Wales (S2 and below in Scotland, Year 9 and below in Northern Ireland). It represents the pinnacle of the UKMT junior competition ladder — only top performers in the nationwide Junior Mathematical Challenge are invited to participate. Unlike multiple-choice speed rounds, the JMO demands that students tackle 16 problems in two hours: 10 short-answer questions in Section A (answer only) and 6 full-solution problems in Section B (requiring complete logical reasoning). Using the 2017 paper as our reference, this article dissects the core topics and problem-solving strategies of the JMO, providing a systematic preparation guide for aspiring Olympiad mathematicians.

一、数论与模运算 — Number Theory and Modular Arithmetic

数论是 JMO 中出镜率最高的知识板块之一。数论问题通常不依赖复杂的公式，而是考察学生对整数性质、整除性、余数规律的直觉把握。以 2017 年 JMO A4 题为例：”2017 枚书签，每 7 枚一组售价 6 元，单买 1 枚售价 1 元，求购买 2017 枚书签的最小费用。”这道题的核心在于理解”批量优惠”的数学本质——尽可能多地使用 7 枚一组的优惠价。由于 2017 = 7 × 288 + 1，购买 288 组（每组 6 元）加上 1 枚单买（1 元），总费用 = 6 × 288 + 1 = 1729 元。数字 1729 在数学史上赫赫有名——哈代-拉马努金数，最小的可以用两种方式表示为两个立方数之和的正整数（1729 = 1³ + 12³ = 9³ + 10³）。

Number theory is one of the most frequently tested domains in the JMO. Number theory problems rarely rely on complex formulas; instead, they assess a student’s intuitive grasp of integer properties, divisibility, and remainder patterns. Consider Problem A4 from the 2017 JMO: “2017 bookmarks, sold in packs of 7 for £6 or individually at £1 each. Find the minimum cost to buy 2017 bookmarks.” The crux of the problem lies in understanding the mathematical essence of bulk discounts — maximize the use of the 7-pack deal. Since 2017 = 7 × 288 + 1, buying 288 packs (at £6 each) plus 1 singly (at £1), the total cost = 6 × 288 + 1 = 1729 pence. The number 1729 is famous in mathematical history as the Hardy-Ramanujan number — the smallest positive integer expressible as the sum of two cubes in two different ways (1729 = 1³ + 12³ = 9³ + 10³).

在备考过程中，学生应重点掌握以下数论工具：(1) 质因数分解——将整数分解为质数的乘积是理解整数结构的根本方法；(2) 整除性规则——2、3、4、5、8、9、11 的整除判定法则可以大幅加速运算；(3) 模运算——理解同余关系可以帮助学生在不进行大数计算的情况下判断余数模式；(4) 最大公因数与最小公倍数——在涉及分数化简和周期性问题的题目中至关重要。建议每天做 3-5 道数论专项练习，从 UKMT 官网的历年 Intermediate 和 Junior Mathematical Challenge 题目入手，逐步过渡到 JMO 真题。

When preparing for the JMO, students should master the following number theory tools: (1) Prime factorization — decomposing integers into products of primes is the foundational method for understanding integer structure; (2) Divisibility rules — knowing the tests for divisibility by 2, 3, 4, 5, 8, 9, and 11 dramatically accelerates computation; (3) Modular arithmetic — understanding congruence relations allows students to determine remainder patterns without performing large-number calculations; (4) Greatest common divisor and least common multiple — essential for problems involving fraction simplification and periodic phenomena. We recommend practicing 3-5 dedicated number theory problems daily, starting with past Intermediate and Junior Mathematical Challenge papers from the UKMT website, then graduating to JMO past papers.

二、几何与空间推理 — Geometry and Spatial Reasoning

几何题在 JMO 中占有重要地位。2017 年 JMO A6 题就是一个典型的几何面积推理题：”六个完全相同的正方形组成一个 2×3 的矩形，其中一块灰色区域被画上阴影，求阴影面积占整个矩形面积的比例。”这类题目不需要复杂的公式——无需三角函数、也无需解析几何——但要求学生能够将复杂图形分解为基本几何形状。解题关键在于将阴影区域分解为两个三角形、一个矩形和一个正方形。设每个小正方形的边长为 2，六个正方形的总面积为 24。通过计算各个组成部分的面积并求和，可以得到阴影面积为 11，因此阴影占比为 11/24。这类”化整为零”的策略是 JMO 几何题的核心思想。

Geometry features prominently in the JMO. Problem A6 from the 2017 JMO is a classic geometric area reasoning question: “Six identical squares form a 2×3 rectangle. A grey shaded region is drawn inside. Find the shaded area as a fraction of the total area.” This type of problem requires no complex formulas — no trigonometry, no coordinate geometry — but demands that students decompose complex shapes into basic geometric building blocks. The key insight is to split the shaded region into two triangles, one rectangle, and one square. Let each small square have side length 2, giving a total area of 24. Summing the areas of the component shapes yields a shaded area of 11, so the fraction is 11/24. This “divide and conquer” strategy is the core philosophy behind JMO geometry problems.

备考几何题时，重点培养以下能力：(1) 面积分解与组合——学会将不规则图形切分为三角形、矩形、梯形等规则图形，或通过”补全法”从大图形中减去空白区域；(2) 对称性识别——利用图形的对称性简化计算，往往可以将工作量减半；(3) 比例与相似性——在涉及线段分割或平行线的题目中，相似三角形的比例关系是常用工具；(4) 格点多边形——在坐标网格上，学会利用皮克定理（Pick’s Theorem）快速计算多边形的面积。推荐使用网格纸进行几何练习，并尝试用不同方法求解同一道题，以培养”一题多解”的灵活思维。

To prepare for geometry questions, focus on developing the following competencies: (1) Area decomposition and composition — learn to split irregular shapes into triangles, rectangles, and trapezoids, or use the “completion method” to subtract blank regions from a larger shape; (2) Symmetry recognition — exploiting symmetry can halve the computational workload in many problems; (3) Proportion and similarity — in problems involving segment division or parallel lines, similarity ratios are a frequently used tool; (4) Lattice polygons — on coordinate grids, mastering Pick’s Theorem allows rapid area calculation for polygons with integer-coordinate vertices. We recommend practicing geometry on grid paper and attempting multiple solution methods for the same problem, cultivating the mental flexibility of “one problem, many solutions.”

三、比例、分数与速度问题 — Ratio, Fractions, and Rate Problems

分数与比例的灵活运用是 JMO 的基础功。2017 年 JMO A3 题是一个典型的分数叠加问题：”Howard 跑完了全程的 1/2，此时他正处于后半程的 3/5 处，问他一共跑了全程的几分之几？”这题的陷阱在于学生容易混淆”全程”和”后半程”这两个参照系。正确的思路是：先把全程分成两半，前半程跑了 1/2（即全程的 1/2），后半程又跑了后半程的 3/5，即全程的 (1/2) × (3/5) = 3/10。因此总共跑了 1/2 + 3/10 = 5/10 + 3/10 = 8/10 = 4/5。这道题的本质是训练学生同时处理多个参照框架的能力，而这种能力在物理和工程问题中同样至关重要。

Flexible manipulation of fractions and ratios is a fundamental skill assessed in the JMO. Problem A3 from the 2017 JMO is a classic fraction-composition problem: “Howard has completed 1/2 of a run. He is now 3/5 of the way through the second half. What fraction of the whole run has he completed?” The trap lies in confusing two reference frames: “the whole run” versus “the second half.” The correct approach: split the run into two halves. The first half contributes 1/2 of the whole. Within the second half, Howard has covered 3/5 of it, which is (1/2) × (3/5) = 3/10 of the whole run. Total: 1/2 + 3/10 = 5/10 + 3/10 = 8/10 = 4/5. At its core, this problem trains students to manage multiple reference frames simultaneously — a skill equally critical in physics and engineering contexts.

2017 年 JMO A5 题则将分数概念延伸到了速度与时间的关系：”一艘游艇在 13 天 22 小时内航行了 3000 海里，求平均速度（精确到整数）”。首先将时间统一换算为小时：13 × 24 + 22 = 334 小时。平均速度 = 3000 ÷ 334 ≈ 8.982 海里/小时，四舍五入为 9 海里/小时。这道题看似简单，但考察的是学生对”单位统一”的意识——在竞赛的紧张氛围中，将”天”和”小时”混在一起计算是极其常见的失分原因。养成”先统一单位，再做计算”的习惯，可以避免大量不必要的粗心错误。

Problem A5 from the 2017 JMO extends fraction concepts to speed-time relationships: “A yacht travels 3000 nautical miles in 13 days and 22 hours. Find its average speed in nautical miles per hour, to the nearest integer.” First, convert all time units to hours: 13 × 24 + 22 = 334 hours. Average speed = 3000 ÷ 334 ≈ 8.982 knots, rounded to 9. This problem appears straightforward, but it tests students’ awareness of unit consistency — under the pressure of competition, mixing days and hours in calculations is a remarkably common cause of lost marks. Cultivating the habit of “unify units first, compute second” can prevent a vast number of avoidable careless errors.

四、组合数学与逻辑推理 — Combinatorics and Logical Reasoning

组合数学是 JMO B 部分解答题的常客，要求学生不仅算出答案，还要用清晰的语言呈现推理过程。JMO 的组合题通常围绕以下主题展开：(1) 排列与计数——例如”用数字 1-9 组成满足特定条件的多位数有多少种”；(2) 鸽巢原理——”如果 n+1 个物体放入 n 个盒子，则至少有一个盒子包含至少两个物体”，这个看似简单的原理在竞赛中有惊人的应用广度；(3) 游戏策略与必胜法——分析两人游戏中的制胜策略，通常需要逆向推理（从终局倒推至开局）；(4) 逻辑谜题与真值表——”A 说 B 在说谎，B 说 C 在说谎…”这类循环语句需要系统化的真值分析。

Combinatorics is a staple of the JMO Section B, where students must not only compute answers but also present their reasoning in clear, logical prose. JMO combinatorics problems typically revolve around these themes: (1) Permutations and counting — e.g., “How many multi-digit numbers using digits 1-9 satisfy a given condition?”; (2) The Pigeonhole Principle — “If n+1 objects are placed into n boxes, at least one box contains at least two objects.” This deceptively simple principle has astonishing breadth of application in Olympiad contexts; (3) Game strategy and winning positions — analyzing two-player games for winning strategies, often requiring backward induction (reasoning from the end state to the start); (4) Logic puzzles and truth tables — statements like “A says B is lying, B says C is lying…” demand systematic truth-value analysis.

在备考组合数学时，建议采用”从特例到一般”的方法：先手动枚举小规模情况（n=1,2,3,4），观察规律，再推导出通项公式。这种方法不仅适用于 JMO，也是更高级别数学竞赛（如 BMO、IMO）的标准思维范式。同时，要学会用简洁的英语（或母语）书写解题过程——JMO B 部分的评分标准不仅看答案是否正确，更看重推理过程的完整性和清晰度。一篇好的解答应该让任何有同等数学水平的读者都能不加思索地跟随你的思路。

When preparing for combinatorics, we recommend the “from specific to general” approach: manually enumerate small cases (n = 1, 2, 3, 4), observe patterns, and then derive the general formula. This method is not only applicable to the JMO but is a standard thinking paradigm for higher-level competitions such as the BMO and IMO. Additionally, learn to write solutions in concise English — JMO Section B marking criteria weigh the completeness and clarity of reasoning as heavily as the correctness of the final answer. A well-written solution should enable any reader of equivalent mathematical maturity to follow your train of thought without hesitation.

五、学习建议与备考路线图 — Study Tips and Preparation Roadmap

第一阶段：夯实基础（考前 3-6 个月）。在这一阶段，学生应系统梳理小学数学竞赛的核心知识体系。建议使用 UKMT 官方出版的《A Mathematical Olympiad Primer》和《A Primer for Mathematics Competitions》作为自学教材。每天投入 30-45 分钟，按主题（数论→代数→几何→组合）循环学习。每学完一个主题后，完成 5 道对应难度的练习题。UKMT 官网提供了 1999 年至今的全部 JMO 真题及详细解答，这是最权威的备考资源。

Phase 1: Build foundations (3-6 months before the exam). During this phase, students should systematically consolidate the core knowledge framework of junior mathematics competitions. We recommend using UKMT’s official publications — “A Mathematical Olympiad Primer” and “A Primer for Mathematics Competitions” — as self-study textbooks. Dedicate 30-45 minutes daily, cycling through topics (Number Theory → Algebra → Geometry → Combinatorics). After completing each topic, solve 5 practice problems at the appropriate difficulty level. The UKMT website provides all JMO past papers from 1999 to the present with detailed solutions — this is the most authoritative preparation resource available.

第二阶段：真题实战（考前 1-3 个月）。此阶段的目标是将零散的知识点整合为解题能力。每周完成一套完整的 JMO 真题（A 部分 30 分钟 + B 部分 90 分钟），严格按照考试时间计时。做完后，比照官方解答进行自我批改，重点关注以下三个方面：(1) 哪些题目做错了，错误的原因是什么（知识缺陷、审题不清、计算失误还是时间不足）；(2) 即使做对了，是否有更简洁或更优美的解法；(3) B 部分的解答书写是否完整——是否每一步推理都有清晰的依据，是否避免了逻辑跳跃。建议建立一个”错题本”，将每道错题按主题分类记录，并定期复习。

Phase 2: Past paper practice (1-3 months before the exam). The goal of this phase is to integrate fragmented knowledge into cohesive problem-solving ability. Complete one full JMO past paper per week (Section A: 30 minutes + Section B: 90 minutes), strictly timed. After finishing, self-mark against the official solutions, focusing on three aspects: (1) Which problems did you get wrong, and what was the root cause — knowledge gap, misreading, calculation error, or time pressure? (2) Even for problems you solved correctly, is there a more concise or more elegant approach? (3) Are your Section B solutions written completely — does every step of reasoning have clear justification, and are logical leaps avoided? We strongly recommend maintaining an “error logbook,” classifying every mistake by topic and reviewing them regularly.

第三阶段：冲刺模拟（考前 2-4 周）。最后阶段的核心任务是提升速度和准确率。每天至少完成 3-5 道 A 部分题目（限时每道 3-5 分钟），训练快速找到解题切入口的能力。对于 B 部分，选择 2-3 道难度适中的题目，模拟考试状态下的完整解答写作。此时应特别注意”时间管理”——JMO 两小时看似充裕，但 B 部分的 6 道题每道可能耗时 15-20 分钟。建议的策略是：先快速浏览 A 部分全部 10 题，优先完成有把握的题；然后转向 B 部分，不必按顺序，先做擅长的题型；最后回到剩余的 A 部分题目，用排除法和极限值检验不确定的答案。

Phase 3: Final sprint (2-4 weeks before the exam). The core task in the final phase is improving speed and accuracy. Complete at least 3-5 Section A problems daily (timed at 3-5 minutes each), training the ability to quickly identify the entry point to a solution. For Section B, select 2-3 moderately challenging problems and write full solutions under simulated exam conditions. Pay special attention to time management at this stage — two hours for the JMO may seem generous, but each of the six Section B problems can consume 15-20 minutes. A recommended strategy: first, quickly scan all 10 Section A problems and prioritize those you are confident about; then move to Section B, not necessarily in order, tackling your strongest question types first; finally, return to the remaining Section A problems, using elimination and limiting-case checks to verify uncertain answers.

总结

英国初级数学奥林匹克竞赛不仅仅是一场考试——它是通往更高层次数学思维的大门。JMO 的训练价值远超竞赛本身：它教会学生如何在面对陌生问题时保持冷静、如何将复杂问题分解为可管理的子问题、以及如何用精确的语言传达抽象的思维过程。无论最终是否获奖，为 JMO 所做的准备都将对学生的 GCSE、A-Level 数学乃至大学申请产生深远的积极影响。记住一句数学竞赛界的格言：”重要的不是你解出了多少道题，而是你在解题过程中成长了多少。”

The UK Junior Mathematical Olympiad is more than just an exam — it is a gateway to higher-level mathematical thinking. The training value of the JMO extends far beyond the competition itself: it teaches students to remain calm when facing unfamiliar problems, to decompose complex challenges into manageable sub-problems, and to communicate abstract reasoning with precise language. Whether or not a student ultimately wins a medal, preparation for the JMO has a profound positive impact on GCSE and A-Level mathematics performance and even university applications. Remember a well-known adage in the mathematics competition community: “It’s not about how many problems you solved, but how much you grew while solving them.”

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