本次 2025 數學天才大競賽，題目範圍廣泛，全面覆蓋數學領域多個重要板塊，從基礎知識考查到高階的技術分析及難題解讀，旨在充分挖掘參賽者的數學潛能，展現數學知識的深度與廣度，具體內容如下：

一、基礎知識

（一）數與代數

1. 數的運算與進制：熟練掌握整數、小數、分數的四則運算，包括混合運算中的運算順序與簡便演算法。深入理解二進位、十進位、十六進制等數制，能夠精准實現不同數制間的轉換，如將十六進制數高效轉換為二進位和十進位，並能在實際情境中靈活運用數制知識，解決如電腦編碼、數據存儲等相關問題。

2. 代數式與方程：清晰辨析整式、分式、根式等代數式的概念與運算規則，能夠準確進行代數式的化簡、求值。深刻理解方程與方程組的內涵，熟練運用一元一次方程、二元一次方程組、一元二次方程等常見方程模型解決實際應用問題，例如通過構建方程求解行程問題、工程問題、銷售利潤問題中的未知量。

3. 函數基礎：精准把握函數的基本概念，包括定義域、值域、對應關係等要素。深入學習一次函數、二次函數、反比例函數的運算式、圖象特徵及性質，能夠根據函數的性質分析實際問題中的數量變化關係，如利用二次函數的最值性質解決優化問題，通過一次函數的單調性分析線性變化趨勢。

（二）幾何圖形

1. 平面幾何基礎：透徹理解點、線、面、角的基本概念與性質，熟練掌握相交線、平行線的判定定理與性質定理。深入學習三角形、四邊形、圓等常見平面圖形的性質與判定方法，例如三角形全等與相似的判定及應用，四邊形中平行四邊形、矩形、菱形、正方形的特性與相互關係，圓的周長、面積公式以及圓周角、圓心角的相關定理。

2. 立體幾何初步：熟悉常見立體圖形，如正方體、長方體、圓柱、圓錐、球體的結構特徵與表面積、體積計算公式。能夠在空間中想像圖形的位置關係，解決簡單的立體幾何問題，如計算組合體的體積、表面積，分析立體圖形中異面直線的位置關係。

（三）概率與統計

1. 統計基礎：熟練掌握數據的收集、整理與描述方法，能夠準確繪製頻數分佈直方圖、頻率分佈折線圖等統計圖表，並通過圖表分析數據的分佈特徵。深入理解平均數、中位數、眾數、方差、標準差等統計量的概念與意義，能運用這些統計量對數據的集中趨勢和離散程度進行有效分析，為決策提供數據支持。

2. 概率初步：清晰理解隨機事件、必然事件、不可能事件的概念，掌握概率的基本定義與計算方法，包括古典概型（如擲骰子、摸球等簡單等可能事件的概率計算）和幾何概型（如在一定區域內隨機取點的概率問題）。能夠運用概率知識對實際生活中的不確定現象進行分析與預測，如計算抽獎活動的中獎概率、評估風險發生的可能性。
    

二、數學方法與思想

（一）邏輯推理

1. 演繹推理：嚴格遵循演繹推理的規則，從已知的定義、定理、公理出發，通過逐步推導得出結論。能夠準確運用三段論等推理形式進行證明，如在幾何證明中，根據給定的條件和已有的幾何定理，有條理地推導出所需證明的命題。

2. 歸納推理與類比推理：善於通過對一系列具體事例的觀察、分析，歸納總結出一般性的規律或結論，如通過對多個等差數列的觀察，歸納出等差數列的通項公式。同時，能夠運用類比推理的方法，根據兩類對象在某些方面的相似性，推測它們在其他方面也可能具有相似的性質，如將平面幾何中三角形的面積公式類比到立體幾何中三棱錐的體積公式。

（二）數學建模

1. 模型構建：面對實際問題，能夠敏銳地分析問題中的數量關係和空間形式，選擇合適的數學模型進行刻畫，如利用函數模型描述經濟增長趨勢、用方程模型解決資源分配問題、通過幾何模型分析建築結構的穩定性。

2. 模型求解與驗證：熟練運用所學數學知識對構建的模型進行求解，得到問題的理論解。並能夠將理論解代入實際情境中進行驗證，檢驗模型的合理性與有效性，根據驗證結果對模型進行調整和優化，確保模型能夠準確反映實際問題。

（三）數形結合

1. 以形助數：充分利用圖形的直觀性來輔助解決代數問題，如通過繪製函數圖象來直觀地分析函數的性質，利用數軸來理解實數的大小關係、絕對值的幾何意義，借助幾何圖形求解方程或不等式，將抽象的代數問題轉化為形象的幾何問題，降低解題難度。

2. 以數解形：運用代數方法精確地研究幾何圖形的性質和位置關係，如通過建立平面直角坐標系，將幾何圖形中的點用座標表示，利用代數運算求解幾何圖形的長度、角度、面積等問題，實現幾何問題的代數化處理，使問題的解決更加嚴謹和精確。
    

三、高階知識與應用

（一）數列與級數

1. 數列通項與求和：熟練掌握等差數列、等比數列的通項公式與求和公式的推導過程及應用，能夠靈活運用這些公式解決相關問題。同時，掌握一些常見數列的通項公式求法，如通過累加法、累乘法、構造法等求數列的通項公式，以及數列求和的多種方法，如裂項相消法、錯位相減法、分組求和法等，用於求解非等差、等比數列的和。

2. 級數初步：瞭解級數的基本概念，如無窮級數的斂散性判斷方法（如比較判別法、比值判別法、根值判別法等），能夠對一些簡單的級數進行斂散性分析。初步認識冪級數的收斂區間與和函數的求解方法，為進一步學習高等數學中的級數理論奠定基礎。

（二）解析幾何

1. 直線與圓的方程：深入理解直線的點斜式、斜截式、兩點式、一般式等方程形式，能夠根據已知條件準確選擇並確定直線方程。熟練掌握圓的標準方程與一般方程，以及直線與圓的位置關係的判定方法（如通過圓心到直線的距離與半徑的比較），解決相關的計算與證明問題，如求直線與圓的交點座標、計算圓的切線方程。

2. 圓錐曲線：全面學習橢圓、雙曲線、拋物線的定義、標準方程、幾何性質（如離心率、焦點、准線等）。能夠運用圓錐曲線的知識解決實際問題，如利用拋物線的光學性質設計探照燈的反射鏡，通過橢圓的定義解決行星軌道問題，借助雙曲線的性質進行導航定位。

（三）數學在實際中的應用

1. 優化問題：運用數學知識解決各種優化問題，如在生產生活中，通過建立函數模型，利用導數等工具求函數的最值，以實現資源的最優配置、成本的最小化或利潤的最大化，如確定產品的最優生產數量、最佳銷售價格，設計最節省材料的包裝形狀。

2. 數據擬合與預測：面對大量的實際數據，能夠運用統計方法進行數據擬合，選擇合適的函數模型（如線性回歸模型、二次函數擬合等）來描述數據的變化趨勢，並利用擬合模型對未來的數據進行預測，為決策提供依據，如根據歷史銷售數據預測未來市場需求，通過氣象數據擬合預測天氣變化趨勢。
    

四、挑戰題與創新思維

（一）開放性問題

1. 問題探索：設置具有開放性的數學問題，要求參賽者從不同角度進行思考與探索，提出多種可能的解決方案或結論。例如，給定一個數學情境，探索其中可能存在的數學規律、關係或模型，鼓勵參賽者發揮創新思維，不拘泥於傳統的解題思路，嘗試新的方法和途徑。

2. 方案設計：提出實際應用場景下的開放性任務，如設計一個城市交通流量優化方案、規劃一個生態園區的佈局等，參賽者需要運用數學知識和綜合素養，構建合理的數學模型，制定詳細的實施方案，並對方案的可行性、有效性進行分析與評估。

（二）跨學科問題

1. 數學與物理結合：設置涉及數學與物理知識交叉的問題，如利用數學方法解決物理中的運動學、力學、電磁學問題，通過物理現象建立數學模型，如運用微積分知識分析物體的變速運動、利用向量知識解決力的合成與分解問題，實現數學與物理學科的相互滲透與融合。

2. 數學與其他學科融合：探索數學與化學、生物、經濟、電腦科學等其他學科的聯繫，設計跨學科問題，如在化學中利用數學方法分析化學反應速率與化學平衡，在生物學中運用數學模型研究種群增長規律，在經濟學中借助數學工具進行成本效益分析、風險評估，在電腦科學中利用數學演算法優化程式性能、進行數據加密等，培養參賽者跨學科運用知識的能力和綜合素養。

 

參賽者需扎實掌握數學基礎知識，熟練運用各類數學方法與思想，具備將數學知識應用於實際問題的能力，並勇於挑戰創新，突破思維定式。大賽通過設置具有挑戰性的題目，激發學生的創新思維，鼓勵學生不斷探索數學新領域，力求在有限的競賽資源下，全面、高效地檢驗學生的數學素養和綜合能力，避免考查內容的繁瑣與重複，做到精准、有效地考核。
 

The scope of the questions in the 2025 Mathematics Genius Competition is extensive, comprehensively covering multiple important areas of the mathematics field. It ranges from the examination of basic knowledge to advanced technical analysis and the interpretation of difficult problems, aiming to fully tap into the mathematical potential of the participants and showcase the depth and breadth of mathematical knowledge. The specific content is as follows:

I. Basic Knowledge

(A) Numbers and Algebra

1. Operations of Numbers and Number Systems: Master the four arithmetic operations of integers, decimals, and fractions proficiently, including the operation order and convenient algorithms in mixed operations. Have an in-depth understanding of number systems such as binary, decimal, and hexadecimal. Be able to accurately convert between different number systems, for example, efficiently convert a hexadecimal number into a binary and decimal number. Also, be able to flexibly apply number system knowledge in practical situations to solve related problems such as computer coding and data storage.

2. Algebraic Expressions and Equations: Clearly distinguish the concepts and operation rules of integral expressions, fractional expressions, radical expressions, etc. Be able to accurately simplify and evaluate algebraic expressions. Have a profound understanding of the connotations of equations and systems of equations. Skillfully use common equation models such as linear equations with one unknown, systems of linear equations with two unknowns, and quadratic equations with one unknown to solve practical application problems. For example, solve for unknowns in problems related to travel, engineering, and sales profit by constructing equations.

3. Fundamentals of Functions: Precisely grasp the basic concepts of functions, including elements such as the domain, range, and correspondence. Study in-depth the expressions, graph characteristics, and properties of linear functions, quadratic functions, and inverse proportional functions. Be able to analyze the quantitative change relationships in practical problems according to the properties of functions. For example, use the maximum and minimum value properties of quadratic functions to solve optimization problems, and analyze the linear change trend through the monotonicity of linear functions.

(B) Geometric Figures

1. Fundamentals of Plane Geometry: Thoroughly understand the basic concepts and properties of points, lines, planes, and angles. Master the determination theorems and properties theorems of intersecting lines and parallel lines proficiently. Study in-depth the properties and determination methods of common plane figures such as triangles, quadrilaterals, and circles. For example, the determination and application of triangle congruence and similarity, the characteristics and interrelationships of parallelograms, rectangles, rhombuses, and squares among quadrilaterals, the formulas for the circumference and area of a circle, and the relevant theorems of circumferential angles and central angles.

2. Introduction to Solid Geometry: Be familiar with the structural characteristics and formulas for surface area and volume of common solid figures such as cubes, cuboids, cylinders, cones, and spheres. Be able to visualize the positional relationships of figures in space and solve simple solid geometry problems, such as calculating the volume and surface area of composite solids and analyzing the positional relationships of skew lines in solid figures.

(C) Probability and Statistics

1. Fundamentals of Statistics: Master the methods of data collection, organization, and description proficiently. Be able to accurately draw statistical charts such as frequency distribution histograms and frequency distribution broken line graphs, and analyze the distribution characteristics of data through these charts. Have an in-depth understanding of the concepts and meanings of statistical quantities such as the mean, median, mode, variance, and standard deviation. Be able to effectively analyze the central tendency and dispersion degree of data using these statistical quantities to provide data support for decision-making.

2. Introduction to Probability: Clearly understand the concepts of random events, certain events, and impossible events. Master the basic definitions and calculation methods of probability, including classical probability models (such as calculating the probability of simple equally likely events like rolling a dice or drawing a ball) and geometric probability models (such as the probability problem of randomly selecting a point within a certain area). Be able to analyze and predict uncertain phenomena in real life using probability knowledge, such as calculating the winning probability of a lottery activity and assessing the possibility of a risk occurring.
    

II. Mathematical Methods and Thoughts

(A) Logical Reasoning

1. Deductive Reasoning: Strictly follow the rules of deductive reasoning, starting from known definitions, theorems, and axioms, and gradually deriving conclusions through reasoning. Be able to accurately use reasoning forms such as syllogisms for proof. For example, in geometric proof, based on the given conditions and existing geometric theorems, systematically derive the proposition to be proved.

2. Inductive Reasoning and Analogical Reasoning: Be good at observing and analyzing a series of specific examples, and summarize general laws or conclusions through induction. For example, by observing multiple arithmetic sequences, summarize the general term formula of arithmetic sequences. At the same time, be able to use the method of analogical reasoning. According to the similarities of two types of objects in certain aspects, infer that they may also have similar properties in other aspects. For example, analogize the area formula of a triangle in plane geometry to the volume formula of a triangular pyramid in solid geometry.

(B) Mathematical Modeling

1. Model Construction: When facing practical problems, be able to keenly analyze the quantitative relationships and spatial forms in the problems and select appropriate mathematical models to depict them. For example, use function models to describe the economic growth trend, use equation models to solve resource allocation problems, and use geometric models to analyze the stability of building structures.

2. Model Solving and Verification: Proficiently use the learned mathematical knowledge to solve the constructed models and obtain the theoretical solutions to the problems. Then, be able to substitute the theoretical solutions into the practical situations for verification, check the rationality and effectiveness of the models, and adjust and optimize the models according to the verification results to ensure that the models can accurately reflect the practical problems.

(C) The Combination of Numbers and Shapes

1. Using Shapes to Assist Numbers: Make full use of the intuitiveness of figures to assist in solving algebraic problems. For example, analyze the properties of functions intuitively by drawing function graphs, understand the magnitude relationships of real numbers and the geometric meaning of absolute values using the number axis, and solve equations or inequalities with the help of geometric figures. Transform abstract algebraic problems into vivid geometric problems to reduce the difficulty of problem-solving.

2. Using Numbers to Solve Shapes: Use algebraic methods to accurately study the properties and positional relationships of geometric figures. For example, establish a plane rectangular coordinate system, represent the points in geometric figures with coordinates, and use algebraic operations to solve problems such as the length, angle, and area of geometric figures. Realize the algebraic processing of geometric problems to make the problem-solving process more rigorous and accurate.
    

III. Advanced Knowledge and Applications

(A) Sequences and Series

1. General Terms and Summation of Sequences: Master the derivation processes and applications of the general term formulas and summation formulas of arithmetic sequences and geometric sequences proficiently. Be able to flexibly use these formulas to solve related problems. At the same time, master the methods of finding the general term formulas of some common sequences, such as finding the general term formulas of sequences through methods like the method of cumulative addition, the method of cumulative multiplication, and the construction method. Also, master various methods of sequence summation, such as the method of splitting terms and canceling, the method of dislocation subtraction, and the method of grouping summation, to solve the summation problems of non-arithmetic and non-geometric sequences.

2. Introduction to Series: Understand the basic concepts of series, such as the methods for judging the convergence and divergence of infinite series (such as the comparison test, the ratio test, the root test, etc.). Be able to conduct convergence and divergence analysis on some simple series. Have a preliminary understanding of the methods for solving the convergence interval and the sum function of power series, laying a foundation for further studying the series theory in higher mathematics.

(B) Analytic Geometry

1. Equations of Lines and Circles: Have an in-depth understanding of the point-slope form, slope-intercept form, two-point form, general form, and other equation forms of lines. Be able to accurately select and determine the equation of a line according to the known conditions. Master the standard equation and general equation of a circle proficiently, as well as the determination method of the positional relationship between a line and a circle (such as comparing the distance from the center of the circle to the line with the radius). Solve related calculation and proof problems, such as finding the intersection coordinates of a line and a circle and calculating the tangent equation of a circle.

2. Conic Sections: Comprehensively study the definitions, standard equations, and geometric properties (such as eccentricity, foci, directrices, etc.) of ellipses, hyperbolas, and parabolas. Be able to use the knowledge of conic sections to solve practical problems. For example, design the reflector of a searchlight using the optical properties of a parabola, solve the problem of planetary orbits through the definition of an ellipse, and carry out navigation and positioning with the help of the properties of a hyperbola.

(C) Applications of Mathematics in Practice

1. Optimization Problems: Use mathematical knowledge to solve various optimization problems. For example, in production and daily life, by establishing function models and using tools such as derivatives to find the maximum and minimum values of functions, achieve the optimal allocation of resources, minimize costs, or maximize profits. For example, determine the optimal production quantity of products, the best sales price, and design the packaging shape that saves the most materials.

2. Data Fitting and Prediction: When facing a large amount of actual data, be able to use statistical methods for data fitting. Select appropriate function models (such as linear regression models, quadratic function fitting, etc.) to describe the change trends of the data, and use the fitting models to predict future data, providing a basis for decision-making. For example, predict future market demand according to historical sales data and predict the weather change trend through meteorological data fitting.
    

IV. Challenge Questions and Innovative Thinking

(A) Open-ended Questions

1. Problem Exploration: Set open-ended mathematical problems, requiring participants to think and explore from different angles and propose multiple possible solutions or conclusions. For example, given a mathematical situation, explore the possible mathematical laws, relationships, or models within it. Encourage participants to give full play to their innovative thinking, not be limited by traditional problem-solving ideas, and try new methods and approaches.

2. Scheme Design: Propose open-ended tasks in practical application scenarios, such as designing an urban traffic flow optimization scheme or planning the layout of an ecological park. Participants need to use mathematical knowledge and comprehensive literacy, construct reasonable mathematical models, formulate detailed implementation plans, and analyze and evaluate the feasibility and effectiveness of the plans.

(B) Interdisciplinary Problems

1. Combination of Mathematics and Physics: Set questions involving the intersection of mathematical and physical knowledge, such as using mathematical methods to solve problems in kinematics, mechanics, and electromagnetism in physics, and establishing mathematical models through physical phenomena. For example, use calculus knowledge to analyze the variable-speed motion of objects and use vector knowledge to solve the composition and decomposition of forces, realizing the mutual penetration and integration of mathematics and physics disciplines.

2. Integration of Mathematics with Other Disciplines: Explore the connections between mathematics and other disciplines such as chemistry, biology, economics, and computer science, and design interdisciplinary questions. For example, use mathematical methods to analyze the reaction rate and chemical equilibrium in chemistry, use mathematical models to study the population growth law in biology, use mathematical tools for cost-benefit analysis and risk assessment in economics, and use mathematical algorithms to optimize program performance and carry out data encryption in computer science. Cultivate participants' ability to apply knowledge across disciplines and their comprehensive literacy.

 

Participants are required to have a solid grasp of basic mathematical knowledge, proficiently use various mathematical methods and thoughts, possess the ability to apply mathematical knowledge to practical problems, and be brave enough to challenge and innovate, breaking through the limitations of traditional thinking patterns. By setting challenging questions, the competition aims to stimulate students' innovative thinking and encourage them to continuously explore new areas of mathematics. It strives to comprehensively and efficiently test students' mathematical literacy and comprehensive abilities within the limited competition resources, avoiding cumbersome and repetitive examination content, and achieving accurate and effective assessment.