Category: Blog

I. Number Theory Question Types: Concept Extension and Problem-Solving Techniques

Number theory questions account for about 20% of the IMC competition and are one of the main points where marks are lost. The difficulty of these questions lies in the fact that the content tested, such as the study of prime numbers, the Chinese Remainder Theorem, and Diophantine equations, usually does not appear in the standard secondary school mathematics syllabus and requires additional study by students.

Common Points of Mark Loss:

Unfamiliarity with the properties of special numbers, especially the application of prime factorisation, divisibility rules, and congruence theorems.

Failure to identify implicit number theory relationships in the problem, such as modular arithmetic patterns.

Insufficient understanding of complex number theory theorems, leading to inflexible application.

Targeted Problem-Solving Techniques:

The special value substitution method is a powerful tool for solving number theory multiple-choice questions. When the problem involves general integer properties, try substituting simple special values (such as 2, 3, 5 and other small prime numbers) into the options to test them. This can often quickly eliminate incorrect answers.

For problems involving large numbers, prioritise prime factorisation, converting the problem into exponential form for comparison and analysis. Problems related to the Chinese Remainder Theorem can be transformed into systems of congruences and then solved step by step.

II. Geometry Question Types: Spatial Imagination and Practical Strategies

Geometry questions account for as much as 35% of the IMC, making them the largest proportion and a major area where marks are lost. These questions mainly test the properties of triangles, quadrilaterals, circles, and the ability to determine the orientation of 3D net diagrams.

Common Points of Mark Loss:

Incorrect addition of auxiliary lines in complex diagrams, leading to an erroneous problem-solving path.

Unfamiliarity with the application of core theorems such as the Power of a Point theorem and triangle similarity properties.

Errors in judging the orientation of 3D net diagrams and insufficient spatial imagination.

Targeted Problem-Solving Techniques:

For plane geometry problems, use the graphical aid method by accurately redrawing the diagram on scrap paper. Add auxiliary lines (such as perpendicular bisectors, angle bisectors, parallel lines, etc.) as needed to reveal hidden relationships.

For 3D net problems, use the relative position memory method. Determine a reference face and then judge the relative positions of other faces to the reference face. Familiarity with the various possible net diagrams of regular polyhedra is key to improving accuracy.

In IMC geometry problems, when the diagram is complex, consider using algebraic methods to solve geometry problems, such as establishing a coordinate system and converting geometric problems into algebraic calculations.

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III. Algebra Question Types: Manipulation Techniques and Strategic Choices

Algebra questions account for about 20% and mainly test polynomial operations, multivariate higher-order indeterminate equations, and quadratic function extrema. The difficulty of these questions lies in the choice of manipulation techniques and solution strategies.

Common Points of Mark Loss:

Insufficient skills in simplifying multivariate higher-order equation systems, leading to complicated calculations.

Lack of sensitivity in identifying symmetry and cyclic symmetry in expressions.

Incomplete analysis of parameter ranges, leading to missed solutions.

Targeted Problem-Solving Techniques:

For multivariate higher-order indeterminate equations, prioritise the factorisation method, converting the equation into the product of several factors and then analysing the possibility of integer solutions.

For complex polynomial problems, observe the relationships between coefficients and look for symmetry or cyclic symmetry, which can often greatly simplify the problem.

The special value method is also applicable in algebra problems. Substituting specific values (such as 0, 1, -1, etc.) into equations or inequalities can quickly test options or reveal patterns.

IV. Creative Thinking Questions: Abstracting Real-World Problems and Model Building

Creative thinking questions account for about 15% and involve practical application problems such as Manhattan distance optimisation and game theory strategy deduction. These questions are highly flexible, closely related to real-life scenarios, and are a type of question that the IMC has been emphasising more in recent years.

Common Points of Mark Loss:

Inability to accurately abstract a mathematical model from the description of a real-world problem.

Insufficient understanding of new definitions in the problem, leading to an incorrect solution direction.

For optimisation problems, incomplete enumeration, overlooking a better solution.

Targeted Problem-Solving Techniques:

When faced with a creative problem, first identify the problem context and determine whether it is an optimisation problem, a game problem, or a logical reasoning problem.

For Manhattan distance problems, convert them into coordinate system problems and use the properties of absolute values to simplify calculations.

For game strategy problems, start with simple cases and gradually generalise to complex cases, looking for cycles or patterns of winning strategies. Reverse thinking is particularly effective in game problems, deducing winning strategies backwards from the final state.

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V. Word and Logic Questions: Trap Identification and Key Points for Reading Comprehension

Word and logic questions account for about 10%. Although the proportion is not high, the mark loss rate is relatively high because these questions often contain traps that require careful reading.

Common Points of Mark Loss:

Overlooking qualifiers and key details in the problem, such as words like "at least", "at most", "not".

Inability to clearly analyse chains of logical relationships, leading to reasoning errors.

Unreasonable time allocation, spending too much time on complex logic questions.

Targeted Problem-Solving Techniques:

When reading the problem, circle key words, especially logical connectives (such as "and", "or", "not") and quantitative qualifiers, to avoid overlooking important conditions.

For complex logical relationships, convert them into logical symbols to make the relationships clearer.

The option reverse deduction method is an effective way to solve logic problems: starting from the options and working backwards to verify the problem conditions can quickly eliminate options that do not meet the conditions.

VI. General Test-Taking Strategies and Time Management

The unique scoring mechanism of the IMC competition requires students to have good time management and decision-making skills. Since questions 21-25 deduct 2 points for incorrect answers, skipping questions you are unsure about cautiously may be a wiser choice.

The table below summarises the time allocation strategy for the IMC competition:

During the exam, do not panic if you encounter difficult problems. Make decisive decisions and avoid spending too much time on a single question. If you get stuck on multiple questions in a row, you can initiate a "Mark → Skip → Return" process: first complete the questions you are confident about, then go back to think about the difficult ones.

Students preparing for the competition can download IMC competition past papers + answer analysis for free.