Advanced problems in mathematics : preparing for university

Description

Contents About this book STEP What is STEP? What is the purpose of STEP? STEP vs A-level STEP vs Olympiad STEP Questions Advice to candidates Worked Problems Worked problem 1 Worked problem 2 Problems Problem 1: An integer equation Problem 2: Partitions of 10 and 20 Problem 3: Mathematical deduction Problem 4: Divisibility Problem 5: The modulus function Problem 6: The regular Reuleaux heptagon Problem 7: Chain of equations Problem 8: Trig. equations Problem 9: Integration by substitution Problem 10: True or false Problem 11: Egyptian fractions Problem 12: Maximising with constraints Problem 13: Binomial expansion Problem 14: Sketching subsets of the plane Problem 15: More sketching subsets of the plane Problem 16: Non-linear simultaneous equations Problem 17: Inequalities Problem 18: Inequalities from cubics Problem 19: Logarithms Problem 20: Cosmological models Problem 21: Melting snowballs Problem 22: Gregory’s series Problem 23: Intersection of ellipses Problem 24: Sketching x^m(1 - x)^n Problem 25: Inequalities by area estimates Problem 26: Simultaneous integral equations Problem 27: Relation between coefficients of quartic for real roots Problem 28: Fermat numbers Problem 29: Telescoping series Problem 30: Integer solutions of cubics Problem 31: The harmonic series Problem 32: Integration by substitution Problem 33: More curve sketching Problem 34: Trig. sum Problem 35: Roots of a cubic equation Problem 36: Root counting Problem 37: Irrationality of e Problem 38: Discontinuous integrands Problem 39: A difficult integral Problem 40: Estimating the value of an integral Problem 41: Integrating the modulus function Problem 42: Geometry Problem 43: The t substitution Problem 44: A differential-difference equation Problem 45: Lagrange’s identity Problem 46: Bernoulli polynomials Problem 47: Vector geometry Problem 48: Solving a quartic Problem 49: Areas and volumes Problem 50: More curve sketching Problem 51: Spherical loaf Problem 52: Snowploughing Problem 53: Tortoise and hare Problem 54: How did the chicken cross the road? Problem 55: Hank’s gold mine Problem 56: A chocolate orange Problem 57: Lorry on bend Problem 58: Fielding Problem 59: Equilibrium of rod of non-uniform density Problem 60: Newton’s cradle Problem 61: Kinematics of rotating target Problem 62: Particle on wedge Problem 63: Sphere on step Problem 64: Elastic band on cylinder Problem 65: A knock-out tournament Problem 66: Harry the calculating horse Problem 67: PIN guessing Problem 68: Breaking plates Problem 69: Lottery Problem 70: Bodies in the fridge Problem 71: Choosing keys Problem 72: Commuting by train Problem 73: Collecting voles Problem 74: Breaking a stick Problem 75: Random quadratics Syllabus