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I. Shapes: Identification, Construction, and Transformation

Understanding the properties of basic shapes and being able to identify or construct them in complex figures is the first step in problem-solving.

Assessment Area	Core Knowledge & Essential Properties	Example Problem Types	Problem-Solving Routines & Steps
Basic Plane Shapes	Triangles: sum of interior angles = 180°; isosceles triangles have equal base angles and equal legs; equilateral triangles have each angle = 60°. Quadrilaterals: squares have four equal sides and four 90° angles; rectangles have opposite sides equal and four 90° angles; parallelograms have opposite sides parallel and equal. Regular polygons: all sides equal, all interior angles equal.	1. Find an unknown angle in a large figure composed of multiple identical equilateral triangles. 2. Determine the symmetry of a figure in a grid containing multiple squares.	1. Mark given information: Clearly label all given equal sides, equal angles, and right angles on the diagram. 2. Use "component units": Break down complex figures into basic units like triangles and squares, using their properties for reasoning. 3. Look for symmetry: Axial or central symmetry can provide key clues for discovering equal relationships (e.g., equal sides, equal angles).
Shape Transformations	Translation: size, shape, and orientation remain unchanged; position changes. Rotation: rotates around a point by a certain angle. Reflection (symmetry): flips over a line (axis of symmetry); corresponding points are equidistant from the axis.	1. Find the final position or coverage area of a figure after multiple translations or reflections. 2. Rotate a figure and combine it with another part to form a regular shape.	1. Step-by-step operation: Draw the transformation process step by step on scratch paper, especially tracking the trajectory of a key point. 2. Use invariants: In transformations, the side lengths, angles, and areas of figures remain unchanged — this is the foundation for establishing equal relationships.
Views & Nets	Cube nets: familiar with common forms like "141 type" and "231 type"; understand the positional relationship of opposite faces.	1. Determine whether a given 2D net can be folded into a cube. 2. Given a cube net, find the relative position of two marked points.	1. Remember "opposite across a gap": In a net, two faces that are "separated by one cell in the same row or column" are opposite. 2. Simulate folding: Mentally simulate the folding process, or draw arrows to mark the adjacency relationships of faces.

II. Angles: Calculation, Deduction, and Construction

Angle problems are the foundation of JMC geometry, often requiring the comprehensive use of properties from various shapes for deduction.

Assessment Area	Core Theorems & Relationships	Example Problem Types	Problem-Solving Routines & Steps
Polygon Interior & Exterior Angles	Sum of interior angles of an n-gon = (n-2) × 180°. Sum of exterior angles of a polygon = 360° (independent of the number of sides).	1. Find the interior angle of a regular dodecagon. 2. Find an unknown angle in a "star" or complex pattern formed by multiple polygons.	1. Direct formula: Interior angle of a regular n-gon = [(n-2)×180°] / n. 2. Break down: View the complex figure as a combination of several simple polygons, using shared sides or angles to establish relationships. 3. Use exterior angles: When interior angles are difficult to calculate, observe the exterior angles and use the fact that the sum of exterior angles is 360° to form equations.
Parallel Lines & Intersecting Lines	If two lines are parallel, then: corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.	Find an unknown angle in a figure containing parallel lines (e.g., fences, grids, staircase figures).	1. Find or construct parallels: The problem may not state it explicitly; actively discover them (e.g., opposite sides of a rectangle, equally spaced parallel lines). 2. "Move" angles: Use substitution to transfer unknown angles near known angles or into easily calculated triangles.
Special Angles & Isosceles Triangles	Isosceles triangle: two base angles are equal. Equilateral triangle: each angle is 60°. Right triangle: the two acute angles are complementary.	1. Find angles in a figure composed of multiple isosceles right triangles. 2. Combine with parallel lines to construct isosceles triangles in a figure to find angles.	1. Mark equal angles: Once an isosceles relationship is found, immediately mark the two base angles with the same symbol. 2. Equation approach: Set the unknown angle as x, and use relationships such as the sum of angles in a triangle being 180° or a straight angle being 180° to form equations and solve.

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III. Area: Formulas, Proportions, Decomposition/Recomposition, and Equivalence

Area problems test not just memory but "clever thinking" — how to find the area of complex figures using simple methods.

Assessment Area	Core Principles & Techniques	Example Problem Types	Problem-Solving Routines & Steps
Direct Formula Application	Triangle: A = ½ × base × height. Square/Rectangle: A = length × width. Parallelogram: A = base × height. Trapezoid: A = ½ × (base1 + base2) × height.	1. Directly calculate the area of a regular shape (e.g., a composite rectangle). 2. Given area and some side lengths, find the missing side length.	1. Identify corresponding elements: Especially for triangles, be clear about the "base" and its corresponding "height". 2. Consistent units: Ensure all length units are the same before calculating.
Decomposition/Recomposition & Area Equivalence	Decomposition/Recomposition: Split an irregular shape into regular shapes, or complete it into a regular shape and subtract the excess. Area equivalence: Shapes with equal area can be transformed into each other (e.g., parallelogram into rectangle).	1. Find the area of an irregular shape (e.g., "L-shape", "plus sign") or the shaded part of a polygon. 2. Determine the change in area after cutting, shearing, or translating a shape.	1. Try splitting/filling: Experiment with a few different split options using dashed lines, choosing the one that is easiest to calculate. 2. Look for "half" relationships: In complex figures, there is often a relationship where one area is half or a multiple of another area. 3. "Whole minus parts": When the shaded part is irregular, calculate the area of the entire regular figure and subtract the area of the blank parts.
Area Ratios & Equal Height Model	Equal height triangles: If two triangles have the same height, their area ratio equals the ratio of their bases.	1. In a figure, connect points to form new triangles and compare the area relationship between the new triangle and the original figure. 2. Given the area of some regions, find the area of another region.	1. Find common heights: Observe which triangles share the same height. 2. Express area using ratios: Set the smallest unit area as 1 part, use base ratios to deduce the number of parts for other regions, and finally sum them.

IV. Loci & Optimization: Dynamic Thinking and Models

This is the difficult part of JMC geometry, testing the ability to find invariant patterns and extreme values amid change.

Assessment Area	Core Concepts & Models	Example Problem Types	Problem-Solving Routines & Steps
Locus of a Point (Path)	Points equidistant from a fixed point → the locus is a circle (or arc). Points equidistant from two fixed points → the locus is the perpendicular bisector of the segment.	1. A moving point always maintains a constant distance from a fixed point; ask for the shape of its path. 2. A bug crawls on the surface of a cube; find its shortest path.	1. Understand the definition: Return to the basic geometric definition of the locus. 2. Hands-on experiment: Take two or three special points that satisfy the condition on the diagram and connect them to see the trend. 3. Flatten spatial problems: Unfold the surface of a 3D shape into a plane, turning curves into straight lines.
Geometric Optimization (Shortest Path)	The shortest path between two points is a straight line. "General drinking water" model: Transform a broken line path into a straight line by constructing symmetric points.	1. There are two villages on the same side of a river; where should water be drawn from the river to minimize the total distance? 2. In a rectangle or grid, find the length of the shortest path from one point to another and back.	1. Identify the model: Recognize whether the problem is about finding the point on a line that minimizes the sum of distances to two points on the same side. 2. Construct symmetric points: Reflect one point across the line to its mirror image. The shortest path is the straight line connecting the other point to the symmetric point.
Figure Movement & Swept Area	The area covered by the boundary of a figure as it moves (translating, rotating).	1. The path length of a vertex of a square as it rolls in a straight line for one full revolution. 2. The sector area swept by a rotating pointer.	1. Segment analysis: Decompose the continuous motion into several standard motion phases (e.g., translation, rotation around a point). 2. Determine the locus: The trajectory of a key point is usually a straight segment or an arc. 3. Calculate separately, then sum: Calculate the path length or swept area for each segment, then add them together.

V. Comprehensive Problem-Solving Framework

When facing an unfamiliar JMC geometry problem, you can follow this "four-step thinking method" to crack it:

Step	Self-Questioning	Specific Actions
Step 1: Observe & Mark	What is in the figure? Which sides are equal? Which angles are right angles? Are there parallels or symmetry? What numbers are given?	Use a pen to clearly mark all given equal sides, equal angles, right angles, and parallel lines on the diagram (or scratch paper). Write the data in the corresponding positions.
Step 2: Associate & Connect	What does this figure remind me of? Is it a combination of several basic shapes? Are there hidden isosceles or right triangles? Which theorem could be applied?	Starting from the marked conditions, recall learned properties. For example, seeing a "square" immediately brings to mind four equal sides, four right angles, and perpendicularly bisecting diagonals.
Step 3: Attempt & Transform	If a direct solution isn't possible, can I draw an auxiliary line (e.g., connecting two points, drawing a height, drawing a parallel line) to turn the figure into something familiar?	Try common auxiliary lines on scratch paper. For instance, connect diagonals to split a quadrilateral into triangles; draw heights to construct right triangles.
Step 4: Check & Verify	Is my answer reasonable? Did I miss another case? Are my calculations accurate?	Plug the answer back into the diagram to see if all conditions are satisfied. Quickly verify using another approach. Check your calculation steps.

The world of JMC geometry is full of cleverness and fun. Mastering these core routines is not about rote memorization but about being able to quickly summon the right "thinking tools" during the exam. Real improvement comes from practice: after solving each problem, ask yourself which "routine" you used and whether there is another solution.

I. Understanding the Rules of the Game: In-depth Analysis of IMC Scoring

The IMC employs a carefully designed scoring mechanism that encourages thoughtful answering rather than blind guessing. A thorough understanding of the rules is the first step in developing a scoring strategy.

Question Zone	Question Range	Points per Question	Penalty for Wrong Answer	Characteristics & Strategy Core
Foundation Zone	Questions 1-15	5 points	0 points	Tests core concepts and basic skills. Relatively lower difficulty. Strategy: Aim for nearly 100% accuracy. This is the foundation of your score. Solve quickly and accurately to reserve time for later questions. Aim to spend less than 1 minute per question.
Advanced Zone	Questions 16-20	6 points	-1 point	Increased difficulty, requiring the integration of multiple knowledge points and some mental flexibility. Strategy: Progress steadily and answer cautiously. Only answer if you have some confidence. If you have no clue, skipping is better than guessing. Aim to solve 3-4 questions correctly.
Challenge Zone	Questions 21-25	6 points	-2 points	The most difficult part of the paper, involving complex logical reasoning, innovative thinking, and knowledge beyond the standard curriculum (e.g., advanced number theory). Strategy: Be selective to avoid penalties. Only attempt if you have a clear path to the solution. Consistently solving 1-2 questions is excellent. Blind guessing may lower your score.

Core Principle: The IMC is not a test to be "finished," but a test of "score maximization." The key to decision-making is weighing the expected score against the risk of penalties.

II. Target Setting: Analysis of Historical Gold Medal Cut-off Scores and Trends

Gold medal cut-off scores fluctuate each year based on question difficulty and global performance. Understanding historical data helps set a reasonable target.

Year	Gold Medal Cut-off	Silver Medal Cut-off	Bronze Medal Cut-off	Analysis of Difficulty & Characteristics
2022	81+	65+	50+	Overall stable questions, with a relatively higher proportion of geometry problems.
2023	71+	55+	42+	Significant increase in difficulty, especially the number theory final problem which had a very high error rate. The scoring rule for Q21-25 (a -2 point penalty) led to more conservative strategies and a drop in average scores.
2024	77+	61+	47+	Difficulty eased, but the proportion of innovative thinking questions increased, placing greater emphasis on practical application and cross-disciplinary thinking.
2025	72+	52+	37+	The impact of the penalty mechanism persisted, with questions demanding greater logical rigor and trick awareness.

Trend Insights:

Score Volatility: Gold medal cut-offs fluctuate between 71 and 81 points, corresponding to an accuracy rate of approximately 53%-60%. This means you don't need to be perfect, but your margin for error is limited.

Difficulty and Strategy Go Hand in Hand: Since the scoring rules became widely known in 2023, test-takers have become more cautious. Relying solely on "guessing" makes it difficult to achieve a high score. Solid ability and wise decision-making are crucial.

Target Setting: Aim for 80 points as a solid target for a Gold medal. This requires near-perfect or perfect scores in the Foundation Zone (Qs 1-15, 70-75 points) and consistently scoring 10-15 points in the Advanced and Challenge Zones.

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III. Gold Medal Target Score Breakdown and Question Strategy

Based on the scoring rules and cut-off scores, we can create a specific scoring blueprint.

Score Section	Target Score	Strategy for Specific Questions	Time Allocation Advice
Foundation Guarantee	70-75 points	Questions 1-15, must get all correct. This is the baseline for a Gold medal; errors are not acceptable.	15-20 minutes. Solve quickly and accurately to save time for later.
Advanced Striving	12-24 points	Questions 16-20, aim to solve 3-4 correctly. Analyze carefully, using techniques like elimination and substitution of special values to improve accuracy.	20-25 minutes. Give each question ample thought, but mark and skip if you have no clue after 3 minutes.
Challenge Breakthrough	0-12 points	Questions 21-25, aim to solve 1-2 correctly. Prioritize questions with shorter stems and more familiar knowledge points. For completely unfamiliar question types, decisively skip.	15-20 minutes. Use this time to tackle 1-2 promising difficult questions and review previously marked questions.
Total Target	80-105 points	With the above combination, a total score of 80 points or more will place you securely in the Gold medal range.	Total duration is 60 minutes, allocate 5 minutes for filling in the answer sheet and final checks.

Golden Rule for Exam Time Allocation: "20-25-15". Spend the first 20 minutes solving Foundation questions, the next 25 minutes on Advanced questions, and the final 15 minutes tackling Challenge questions and reviewing.

IV. High-Scoring Mindset: Breaking Through the Four Core Modules

The IMC tests not just knowledge, but also ways of thinking. Here is a guide to developing a high-scoring mindset for the core modules.

Module	Weight	Core Assessment Focus & High-Scoring Mindset	Typical Question Types & Breakthrough Points for Preparation
Number Theory & Algebra	~40%	Logical rigor and constructive ability. Number theory often involves knowledge beyond the curriculum, such as primes, divisibility, congruences, and Diophantine equations. Algebra emphasizes expression manipulation and modeling.	Number Theory: Master divisibility rules (e.g., divisibility by 3, 9, 11), properties of congruences, and the uniqueness of prime factorization. Algebra: Be proficient in polynomial operations, adept at translating word problems into equations or inequalities, and pay attention to constraints like integer solutions.
Geometry & Spatial Thinking	~35%	Spatial imagination and transformation ability. Tests not only plane geometry (triangles, circles, quadrilaterals) but also spatial problems like nets of solids, orthographic projections, and shortest paths.	Memorize basic geometric theorems (Pythagoras, similarity, circle power). For spatial problems, you must draw diagrams or use simple physical models to aid thinking. Train your ability to view shapes from different perspectives.
Combinatorics & Logic	Integrated throughout	Systematic thinking and the ability to discuss cases. Includes counting, probability, logic puzzles, and strategy games.	Master basic counting tools such as enumeration, the addition and multiplication principles, and the Inclusion-Exclusion principle. For logic problems, use tables and hypothesis methods to reason.
Innovative Thinking & Application	~25%	Real-world problem modeling and cross-disciplinary thinking. Problems often originate from everyday scenarios (e.g., traffic planning, resource allocation, game strategies), requiring you to abstract a mathematical model and solve it.	Broadly expose yourself to various fun math and logic puzzles. Practice distilling lengthy problem statements into concise mathematical relationships or diagrams. Keep an open mind and try multiple solution paths.

Common High-Scoring Mindset:

Pattern Recognition: Quickly identify the mathematical model behind the problem (Is it number theory, geometry, or combinatorics?).

Reverse Thinking: Work backwards from the answer choices or the target conclusion, especially effective for multiple-choice questions.

Extremization & Specialization: Use extreme cases (e.g., taking boundary values) or special values (e.g., taking 0, 1, primes) to test options or discover patterns.

Checking & Verification: After getting an answer, take a few seconds to verify it using another method or by plugging it back into the problem to avoid careless mistakes.

V. Sprint Phase Preparation Plan

The 1-2 months before the exam are a critical period for improving your score.

Time Phase	Core Task	Specific Actions & Resources
8-4 weeks before exam (Systematic Strengthening)	Module-based thematic breakthrough	1. Focus on weak modules: Concentrate on practicing questions from past papers in your weaker areas like number theory or geometry. 2. Summarize and conclude: Create a "problem-solving toolbox" for each high-frequency topic (e.g., prime determination, geometric theorems, counting methods). 3. Analyze mistakes: Keep a mistake notebook, analyzing the reasons for each error (knowledge gap, incorrect approach, carelessness).
4-2 weeks before exam (Full Paper Simulation)	Full-length mock exams and strategy refinement	1. Timed mocks: Complete 2-3 recent past papers weekly, strictly timed to 60 minutes, using an answer sheet. 2. Optimize strategy: Practice the "20-25-15" time allocation method to find the answering rhythm and selection criteria that work best for you. 3. Score analysis: After each mock, analyze your score in each section against the target score breakdown to clarify the next focus area.
1 week before exam (Fill Gaps & Adjust Mindset)	Return to fundamentals and adjust mindset	1. Review mistakes: Focus on reviewing all questions in your mistake notebook to ensure similar errors are not repeated. 2. Review formulas & theorems: Recite common geometry formulas, number theory properties, and algebraic identities. 3. Maintain problem-solving feel: Do a small number of medium-difficulty problems daily to keep your mind active, but stop studying very difficult or unusual questions. 4. Adjust schedule: Ensure adequate sleep to approach the exam in the best possible state.

Competing for the IMC Gold medal is a dual contest of knowledge and strategy. It requires solid mathematical skills to quickly conquer the first 15 questions, a clear head to make wise decisions on mid-to-late questions, and a flexible mind to crack the cleverly designed innovative problems. By deeply understanding the scoring rules, precisely grasping historical cut-off scores, and engaging in targeted high-scoring mindset training, you are fully capable of standing out in this intellectual contest of 700,000 participants worldwide and winning the Gold medal.