Math 55a Lecture Notes - Evan Chen

X0 g? f 2-f 1 Lemma 2.2 In the above commutative diagram, if gis surjective then f 1 = f 2. Proof. Just use gto get everything equal. Yay. §2.3Groups De nition 2.3. A semi-group is a set Gendowed with an associative1 binary operation: G2!G. Lots of groups work. Example 2.4 (Star sh) Let Gbe an arbitrary set and x a g 0 2G. Then let ab= g 0 for ...

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Math Olympiad Hardness Scale (MOHS) - Evan Chen

2 Speci cation Here is what each of the possible ratings means.1 Rating 0M: Sub-IMO. Problems rated 0 are too easy to use at IMO. I can often imagine such a problem could be solved by a strong student in an honors math class, even without olympiad training. Rating 5M: Very easy. This is the easiest rating which could actually appear while

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A Brief Introduction to Olympiad Inequalities

Evan Chen (April 30, 2014) A Brief Introduction to Olympiad Inequalities Example 1.2 Prove that a2 + b2 + c2 ab+ bc+ caand a4 + b4 + c4 a2bc+ b2ca+ c2ab. Proof. By AM-GM, a2 + b2 2 aband 2a4 + b4 + c4 4 a2bc: Similarly, b2 + c2 2 bcand

Barycentric Coordinates in Olympiad Geometry

problems with a formula sheet beside them. I’ve included Appendix B to facilitate these needs; I also have a version which includes just the formula sheet and the problems, which should be oating around. Happy bashing! 1For ABC counterclockwise, this is positive when P, Q and R are in counterclockwise order, and negative other-wise.

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The OTIS Excerpts - Evan Chen

Introduction The book is divided into algebra, combinatorics, and number theory. We do not cover geometry, for which Euclidean Geometry in Mathematical Olympiads [Che16] already serves the role of “comprehensive book”. The twelve main chapters in this book are structured in to four sections.

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The Incenter/Excenter Lemma - Evan Chen

midpoint of arc BC. Show that L is the center of a circle through I, I A, B, C. A BC I L I A Proof. This is just angle chasing. Let A = \BAC, B = \CBA, C = \ACB, and note that A, I, L are collinear (as L is on the angle bisector). We are going to show that LB = LI, the other cases being similar. First, notice that \LBI = \LBC + \CBI = \LAC ...

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Finding Real Roots of Polynomial Equations

Polynomial Equations The Irrational Root Theorem say that irrational roots come in conjugate pairs. For example, if you know that 1 + is a root of x3 –x2 –3x –1 = 0, then you know that 1 – is also a root. Recall that the real numbers are made up of 2 the rational and irrational numbers. You can use the Rational Root Theorem and the

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Math 231L Calculus co-req - University of North Carolina ...

9. Writing the equations of vertical and horizontal lines through a given point 10. Finding slopes of lines parallel and perpendicular to a line given in the form Ax + By = C 11. Writing equations of lines parallel and perpendicular to a given line through a point 12. Graphing a rational function: Constant over linear 13.

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Rational Exponents and Radical Equations - Math Plane

Solving radical (exponent) equations 4 Steps: 1) Isolate radical 2) Square both sides 3) Solve 4) Check (for extraneous answers) 4 Steps for fractional exponents ... Rational Exponent Equations Domain Restrictions: A Comparison Examples : 2(x + 4) Since it is a 1/5 root, a negative is permitted. 8) 3 -125 2(x) +21

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L EQUATIONS IN O VARIABLE Linear Equations in One ... - …

2.2 Solving Equations which have Linear Expressions on one Side and Numbers on the other Side Let us recall the technique of solving equations with some examples. Observe the solutions; they can be any rational number . Example 1: Find the solution of 2 x – 3 = 7 Solution: Step 1 Add 3 to both sides. 2x – 3 + 3 = 7 + 3 (The balance is not ...

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Unit 4 PacketMPLG - State College Area School District

Rational Exponents 7. I can convert from rational exponents to radical expressions (and vice versa). 8. I can simplify numbers with rational exponents. Solving Radical Equations 9. I can solve equations with roots. 10. I can solve equations with …

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Solving Cubic Polynomials - SHSU

q is a rational solution to the polynomial equation f(x) = 0 then qx pis a factor of the polynomial f(x) and so we can use long division to write f(x) = (qx p)g(x) where g(x) is a polynomial of smaller degree. We teach a version of this method in high school when students learn to solve quadratic equations by factoring.

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New Jersey Student Learning Standards for Mathematics

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, …

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