數學競賽AMC必備公式整理！

作為(wei) AIME晉級賽之一，AMC10/12競賽向來備受初高中考生關(guan) 注。AMC10/12備賽倒計時隻剩3個(ge) 月不到，同學們(men) 要抓緊學起來！

AMC美國數學競賽

AMC是由美國數學協會(hui) 舉(ju) 辦的美國數學競賽。作為(wei) 美國國家隊的唯一選拔平台，每年僅(jin) 北美地區就有超多30萬(wan) 次學生報名參加。 

從(cong) 小學到高中，全球有近幾十個(ge) 國家和3000所學校的學生角逐於(yu) 這場競賽。可以說無論是參賽人數、範圍還是年齡，AMC比其他數學競賽更具流行性和權威性。

AMC10適合學生：

針對10年級及以下且17.5歲以下學生的數學測試，建議8-10年級學生首選。

AMC12適合學生：

針對12年級及以下且19.5歲以下的學生的數學測驗，建議10年級學生首選。

AMC10/12報名截止日期：

A卷：2023年10月30日9：00

B卷：2023年11月5日9：00

AMC10/12準考證下載：

A卷：2023年11月5日10：00至考試開始前

B卷：2023年11月11日10：00至考試開始前

AMC10/12賽事時間：

A卷：2023年11月9日

B卷：2023年11月15日

官方網站：https://www.maa.org/math-competitions

獎項設置：

全球個(ge) 人獎項：

滿分獎Perfect Score：獲得滿分150 分

全球卓越獎Distinction Honor Roll：全球成績排名前1%

全球優(you) 秀獎Honor Roll：全球排名前5%

全球榮譽獎Achievement Roll：8 年級及以下年級在AMC10 中獲得90 分以上

AMC必備公式

01、Geometry

Angles

Acute Angle: Angle measures less than 90 degrees

Right Angle: Angle measure that is 90 degrees

Obtuse Angle: Angle measure that is greater than 90 degrees

Straight Angle: Angle that is 180 degrees

Complementary Angles: Two angles that up to 90 degrees

Supplementary Angles: Two angles that add up to 180 degrees

Transversals and Parallel Lines with Angles

If there are two parallel lines, and they are cut by a transversal (just a line interesting both of the parallel lines), then the alternate interior angles are congruent. 

If two parallel lines are cut by a transversal, then corresponding angles are congruent. 

(In this diagram above, both of the angles that have X in them are congruent and have the same angle measure)

If two lines intersect, then vertical angles will be congruent.

In the diagram below, angle 1 is congruent to angle 3 while angle 2 is congruent to angle 4. 

Special Right Triangles 

30-60-90 Right Triangles 

45-45-90 Right Triangles

Angles in ANY Polygon

The sum of the interior angles of any polygon is (n-2) x 180. 

The sum of the exterior angles of ANY shape is 360 degrees. The formula to find the exterior angle of a REGULAR polygon is 360/n. (N represents the number of shapes.)

Ptolemy’s Theorem

Ptolemy’s Theorem states that if you multiply the diagonals, then that will be equivalent to the sum of the multiplication of the opposite sides. If you don’t understand what I mean, then the picture and example below should help you understand it. 

E x F = (C x A) + (B x D)

02、Ptolemy’s Theorem states that if you multiply the diagonals, then that will be equivalent to the sum of the multiplication of the opposite sides. If you don’t understand what I mean, then the picture and example below should help you understand it. 

Algebra 

Vieta’s Formula

Vietas is a great way to find the sum of the roots or the multiplication of it. In any polynomial that is in the form of  ax2+bx+c, Vieta says that the sum of the roots is -b/a. The multiplication of the roots in this polynomial is just c/a. (Don’t forget that in vietas you don’t include the variables. You just use coefficients.) 

Polynomial Remainder Theorem

If you have any polynomial f(x), and you’re dividing it by x-a, then the remainder can be found by replacing the variable x with the value of a throughout the entire polynomial. If that expression is equivalent to 0, then the polynomial is indeed divisible by x-a. 

Conjugate Root Theorem 

The AMC 10 kids can ignore this short section of conjugate roots if they want. 

Theorem: If a polynomial p(x) that only has real number coefficients has a root of a+bi, then a-bi is also a root of that same polynomial. This can be extremely useful if you have a complex number root given in a polynomial. 

Quadratics

The quadratic formula is usually used when you have an equation in the form ax2+bx+c. 

The formula above can help you factoring a quadratic. However, remember that sometimes you might see quartic polynomials that might be in the form ax4+bx2+c. The clever way of factoring that is to substitute x2 with another variable. Pretend that variable is y. After you do so, you get ay2+bx2+c. 

Geometric Sequences

A geometric sequence has a property that for any two consecutive terms, the ratioses of the numbers will always be the same. An example is 2, 4, 8, 16, 32. As you can see, the numbers are multiplying by 2. 

In a geometric sequence, if the first term is a and the common ratio is r, then the nth term of the sequence is arn-1. 

If you have to use algebra in your geometric sequence, then a way to label the terms is a, ar, ar2, ar3, etc. On the left, a represents the first term while r represents the number you multiply by.

The sum of a geometric series is in the picture below. 

03

Number Theory

Euclidean Algorithm

If you’re asked to find the GCD (Greatest Common Divisor) or two numbers like 2 and 4, then you’re easily gonna say 2. However, what if the numbers are huge? That’s when this algorithm will help. 

Pretend I have two numbers a and b in which a is less than b. THe GCD of (a, b) is equivalent to (a, b-a). You keep subtracting the numbers. 

Pretend you want to find the GCD of 48 and 880. The Euclidean Algorithm states that the GCD of those two numbers is equivalent to (48, 880-48) which is equivalent to (48, 832). Instead of just subarcting this again and again, we can simply divide and use the remainder. If you divide 880 and 48, you get 18 with a remainder of 16. You disregard the 18 and only use the remainder. You then find the GCD of the remainder and the divisor which was 48 in this case. Now we need to find the GCD of 16 and 48 which is obviously 16. 

Modular Arithmetic Basics

People sometimes get confused by the word mod, but it’s time to clear it up. When you say 8 is divisible by 4, you can simply say 8 = 0 (mod 4). All this means is that when 8 is divisible by 4, the remainder is 0. For example, 4, 8, 12, 16, 20, 24 are all 0 in mod 4. This means that when you divide all of those numbers by 4, you always get a remainder of 0. (The equal sign that i’m using actually has another line below it like in the picture below.)

Euler’s Totient Theorem

If we have a number a and we’re dividing it by n, and a and n are both relatively prime, then 

a φ(n)1 (mod n). The φ(n)can be calculated using the euler totient function which is included in the handout. 

04

Probability and Combinations

Formula for Permutations and Combinations

Pascal’s Identity

The image above is an important identity to know because it can help you in simplifying combinatorial expressions in a short amount of time. 

Hockey Stick Identity

You should absolutely memorize this theorem because in some problems, you’ll have a lot of combinatorial expressions. Computing them will take a long time. Knowing this identity will make it super fast. 

Vandermonde’s Identity

Sum of a Row in Pascal’s Triangle

This is something very important to remember since it can make your computation a lot easier.