STEP 练习 · 75 道题带完整解答

这 75 道题全部改编自历年 STEP 真题，由剑桥大学 Stephen Siklos 博士选编并逐题写了提示与完整解答。
75 STEP-based problems, each with the author's comments and a full solution.

先知道这两件事。
① 难度分三档，是作者标的：✓ 入门22 题 ✓✓ 常规38 题 ✓✓✓ 难题15 题。题号顺序不等于难度顺序，新手从 ✓ 开始。
② 这些题大多来自 1987–2002 年的旧 STEP Paper I。STEP 1 自 2021 年起已不再举行，现行考试只有 STEP 2 和 STEP 3；这批题难度大致相当于旧 STEP 1、略低于现行 STEP 2，适合打基础和练思路。要练现行真卷，去官方真题。

从 P1 开始做 →先读「STEP 是什么、怎么准备」

只看：跳到：纯数学力学概率统计

纯数学 · Pure Mathematics

✓ 做过

An integer equation

✓ 入门STEP 1993 Paper I ✓ 做过

Partitions of 10 and 20

✓ 入门STEP 1997 Paper I ✓ 做过

Mathematical deduction

✓ 入门STEP 1994 Paper I ✓ 做过

Divisibility

✓ 入门STEP 1999 Paper I ✓ 做过

The modulus function

✓ 入门STEP 1997 Paper I ✓ 做过

The regular Reuleaux heptagon

✓ 入门STEP 1987 Specimen Paper I ✓ 做过

Chain of equations

✓ 入门STEP 1997 Paper II ✓ 做过

Trig. equations

✓✓ 常规STEP 1997 Paper II ✓ 做过

Integration by substitution

✓ 入门STEP 1998 Paper I ✓ 做过

P10

True or false

✓✓ 常规STEP 1998 Paper I ✓ 做过

P11

Egyptian fractions

✓ 入门STEP 2000 Paper II ✓ 做过

P12

Maximising with constraints

✓ 入门STEP 1998 Paper I ✓ 做过

P13

Binomial expansion

✓ 入门STEP 1998 Paper II ✓ 做过

P14

Sketching subsets of the plane

✓✓ 常规STEP 1999 Paper I ✓ 做过

P15

More sketching subsets of the plane

✓✓ 常规STEP 1995 Paper I ✓ 做过

P16

Non-linear simultaneous equations

✓ 入门STEP 1996 Paper II ✓ 做过

P17

Inequalities

✓✓ 常规STEP 2001 Paper I ✓ 做过

P18

Inequalities from cubics

✓✓ 常规STEP 2001 Paper I ✓ 做过

P19

Logarithms

✓✓✓ 难题STEP 2000 Paper I ✓ 做过

P20

Cosmological models

✓✓ 常规STEP 2001 Paper I ✓ 做过

P21

Melting snowballs

✓✓ 常规STEP 1991 Paper I ✓ 做过

P22

Gregory’s series

✓✓ 常规STEP 1991 Paper II ✓ 做过

P23

Intersection of ellipses

✓ 入门STEP 2002 Paper I ✓ 做过

P24

Sketching x^m (1 − x)ⁿ

✓✓✓ 难题STEP 2002 Paper I ✓ 做过

P25

Inequalities by area estimates

✓✓✓ 难题STEP 2002 Paper I ✓ 做过

P26

Simultaneous integral equations

✓✓ 常规STEP 2002 Paper I ✓ 做过

P27

Relation between coefficients of quartic for real roots

✓✓ 常规STEP 1997 Paper III ✓ 做过

P28

Fermat numbers

✓✓ 常规STEP 2002 Paper II ✓ 做过

P29

Telescoping series

✓✓ 常规STEP 1998 Paper II ✓ 做过

P30

Integer solutions of cubics

✓✓✓ 难题STEP 1998 Paper II ✓ 做过

P31

The harmonic series

✓✓✓ 难题STEP 1999 Paper I ✓ 做过

P32

Integration by substitution

✓✓ 常规STEP 1999 Paper II ✓ 做过

P33

More curve sketching

✓✓ 常规STEP 1999 Paper II ✓ 做过

P34

Trig. sum

✓✓✓ 难题STEP 1999 Paper II ✓ 做过

P35

Roots of a cubic equation

✓✓✓ 难题STEP 1999 Paper III ✓ 做过

P36

Root counting

✓✓✓ 难题STEP 1999 Paper III ✓ 做过

P37

Irrationality of e

✓✓ 常规STEP 1997 Paper III ✓ 做过

P38

Discontinuous integrands

✓✓ 常规STEP 2000 Paper I ✓ 做过

P39

A difficult integral

✓✓✓ 难题STEP 1996 Paper II ✓ 做过

P40

Estimating the value of an integral

✓✓ 常规STEP 2000 Paper I ✓ 做过

P41

Integrating the modulus function

✓✓ 常规STEP 2000 Paper I ✓ 做过

P42

Geometry

✓✓ 常规STEP 2015 Paper II ✓ 做过

P43

The t substitution

✓✓ 常规STEP 2000 Paper II ✓ 做过

P44

A differential-difference equation

✓✓ 常规STEP 1990 Paper II ✓ 做过

P45

Lagrange’s identity

✓✓✓ 难题STEP 1987 Paper II ✓ 做过

P46

Bernoulli polynomials

✓✓ 常规STEP 1987 Paper III ✓ 做过

P47

Vector geometry

✓✓ 常规STEP 2000 Paper II ✓ 做过

P48

Solving a quartic

✓✓ 常规STEP 2000 Paper III ✓ 做过

P49

Areas and volumes

✓✓ 常规STEP 1987 Paper II ✓ 做过

P50

More curve sketching

✓✓✓ 难题STEP 2001 Paper II

力学 · Mechanics

✓ 做过

P51

Spherical loaf

✓✓✓ 难题STEP 2001 Paper I ✓ 做过

P52

Snowploughing

✓✓✓ 难题STEP 1987 Specimen Paper III ✓ 做过

P53

Tortoise and hare

✓✓ 常规STEP 1999 Paper I ✓ 做过

P54

How did the chicken cross the road?

✓✓ 常规STEP 1997 Paper I ✓ 做过

P55

Hank’s gold mine

✓✓ 常规STEP 1998 Paper I ✓ 做过

P56

A chocolate orange

✓ 入门STEP 1987 Specimen Paper II ✓ 做过

P57

Lorry on bend

✓ 入门STEP 2002 Paper I ✓ 做过

P58

Fielding

✓✓ 常规STEP 1998 Paper II ✓ 做过

P59

Equilibrium of rod of non-uniform density

✓✓ 常规STEP 2002 Paper II ✓ 做过

P60

Newton’s cradle

✓✓ 常规STEP 1999 Paper II ✓ 做过

P61

Kinematics of rotating target

✓✓ 常规STEP 1999 Paper II ✓ 做过

P62

Particle on wedge

✓✓✓ 难题STEP 1998 Paper II ✓ 做过

P63

Sphere on step

✓ 入门STEP 1997 Paper II ✓ 做过

P64

Elastic band on cylinder

✓ 入门STEP 2001 Paper I

概率与统计 · Probability and Statistics

✓ 做过

P65

A knock-out tournament

✓ 入门STEP 1987 Specimen Paper II ✓ 做过

P66

Harry the calculating horse

✓✓ 常规STEP 1997 Paper II ✓ 做过

P67

PIN guessing

✓ 入门STEP 2002 Paper I ✓ 做过

P68

Breaking plates

✓ 入门STEP 2001 Paper I ✓ 做过

P69

Lottery

✓✓ 常规STEP 2001 Paper II ✓ 做过

P70

Bodies in the fridge

✓✓ 常规STEP 1987 Paper II ✓ 做过

P71

Choosing keys

✓ 入门STEP 2000 Paper II ✓ 做过

P72

Commuting by train

✓✓ 常规STEP 2000 Paper II ✓ 做过

P73

Collecting voles

✓ 入门STEP 2000 Paper II ✓ 做过

P74

Breaking a stick

✓✓ 常规STEP 1999 Paper II ✓ 做过

P75

Random quadratics

✓✓✓ 难题STEP 1988 Paper II

Problems and solutions are reproduced from Stephen Siklos, Advanced Problems in Mathematics: Preparing for University, Second Edition (Cambridge, UK: Open Book Publishers, 2019), https://doi.org/10.11647/OBP.0181, licensed under CC BY 4.0. STEP questions are reproduced in that book by kind permission of Cambridge Assessment Group Archives.

题目与解答出自 Stephen Siklos《Advanced Problems in Mathematics》（2019 第二版，开放许可 CC BY 4.0）。