AoPS Volume 1 Flashcards

This is a (mostly) comprehensive flashcard set with all of the important formulas/definitions from AoPS Volume 1 contest math book. I have left out some that I already know (basic level).

When do you include the negative root?

If you introduce the root while solving the problem

Radicand

The number under the radical

Conjugate Radical (for two terms)

Found by changing the sign in front of one radical

What is (a+bi)(c+di)

(ac-bd)+(bc+ad)i

Imaginary part of a+bi

b

Procedure for Algebra Word Problems

Define all variables, write the equations described, solve the equations

Work Rate Equation

(time working)(fraction of work in one unit of time)=(fraction of work done)

Jointly Proportional

A quantity is directly proportional to more than one variable

What does ‘of ‘ mean in percent problems?

Multiply

What does ‘are/is’ mean is percent problems?

Equals

What does a | b mean?

a divides b

What does a | b mean if there is a slash through the vertical line?

a does not divide b

Can you multiply/use exponentials in modular arithmetic?

Yes

Can you divide in modular arithmetic?

No

When is a number divisible by 11?

The sum of its digits with alternating signs is

Difference of Squares

a²-b²=(a-b)(a+b)

Sum of Squares

a²+b²=(a+b)²-2ab

Difference of Cubes

a³-b³=(a-b)(a²+ab+b²)

Sum of Cubes

a³+b³=(a+b)(a²-ab+b²)

x^4+y^4+x²y²

(x²+y²-xy)(x²+y²+xy)

For roots r and s of a quadratic with no leading coefficient, r²+s² is…

b²-2c

For roots r and s of a quadratic with no leading coefficient, r-s is…

+-sqrt(b²-4c)

For roots r and s of a quadratic with no leading coefficient, r²s+s²r is…

-bc

For roots r and s of a quadratic with no leading coefficient, r^4+s^4 is…

b^4-4b²c+2c²

What are transcendental numbers?

Numbers that aren’t a solution to any nonzero polynomial with integer coefficients

Fundamental Theorem of Algebra

A polynomial of degree n, where n>0, with real or complex coefficients has n real/complex roots, including repeated roots

What is a secant (circle)?

A line that passes through the circle, intersectingit in two places

Sector

The area inside a circle cut off by two radii

Circular Segment

The area between a chord and the arc it intercepts

What is angular velocity?

How fast something is spinning (ex. revolutions per minute)

Which letter comes first when writing a ray?

The origin (where the ray starts)

What does it mean for an angle to subtend an arc?

The rays/lines forming the angle pass through its endpoints

How many minutes are in a degree?

60

How many seconds are in a degree?

360

Orthogonal

Lines that are perpendicular/at right angles to each other

Reflex Angle

An angle over 180 degrees

Vertical Angle

Opposite angles formed by two crossing lines

Transversal

A line with no specific orientation that crosses two parallel lines

Alternate Interior Angles

Angles within two parallel lines formed on the opposite sides of the transversal

Corresponding Angles

Angles on the corresponding locations of parallel lines

Same-Side Interior Angles

Angles within the two parallel lines on the same side of the transversal

Exterior Angle

The angle formed by extending the side of a triangle past the vertex

Remote Interior Angles

Interior angles of a triangle not adjacent to a specific exterior angle

Length of an Arc

Radius*theta (in radians)

Area of a Sector

radius²*theta/2 where theta is in radians

Central Angle

An angle whose vertex is at the center of a circle and whose sides are radii

Inscribed Angle

Angle formed by two chords with a common endpoint

Inscribed Angle Measure

½ the arc it intercepts

Angle formed by two secants/tangents/tangent and secant which intersect outside the circle measure

½ the difference of the arcs intercepted by the lines

Angles formed by a tangent and a chord

½ the arc is cuts off

Angles formed by two chords measure

½ the sum of the intercepted arcs

Base Side

The non-equal side in an isosceles triangle

Vertex Angle

The angle opposite the base in an isosceles triangle

Base Angles

The equal angles in an isosceles triangle

Median

A segment drawn from a vertex to the midpoint of the opposite side of a triangle

Centroid

Where the medians of a triangle meet

Concurrent

Lines that meet at one point

What ratio does the centroid divide each median into?

2 (distance from angle to centroid):1 (distance from centroid to side)

Angle Bisector

Line that divides an angle into two equal parts

Where are the angle bisectors concurrent?

The incenter

Incenter

The center of a circle inscribed in a polygon

Perpendicular Bisector

Line that is perpendicular to a side and splits it in half

Where are the perpendicular bisectors concurrent?

The circumcenter

Where are the altitudes concurrent?

The orthocenter

Common Pythagorean Triples

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17)

What is sufficient proof that two triangles are congruent?

SSS, SAS, ASA, AAS

When are two triangles similar?

AA

Angle Bisector Theorem

If AX bisects A of ABC, then AC/CX = AB/BX

sin(A)=

cos(90-A)

tan(A)=

cot(90-A)

sec(A)=

csc(90-A)

Pythagorean Identities

sin²(x)+cos²(x)=1, tan²(x)+1=sec²(x), cot²(x)+1=csc²(x)

Area of a Triangle

Side*altitude/2=a*b*sin(included angle)/2=inradius*semiperimeter

Ways to Show an Angle is 90 degrees

It is inscribed in a semicircle or show that two adjacent angles are equal

Ways to show a Triangle is A Right Triangle

Two of the angles are complementary, the sides satisfy the Pythagorean Theorem, or a median of the triangle is equal to half the side to which it is drawn

Ways to Show Two Lines are Perpendicular without Angles

One line passes through the center of a circle which is tangent to the other line where they intersect, a radius/diameter bisects a chord, or one segment is an altitude in a triangle and the other is the side to which the altitude is drawn

Convex Polygon

All angles are less than 180 degrees

Concave Polygon

One angle is more than 180 degrees

Orthodiagonal

The diagonals of a quadrilateral are perpendicular

What are the parallel sides of a trapezoid called?

Bases

What are the non-parallel sides of a trapezoid called?

Legs

What is the median of a trapezoid?

The segment that connects the midpoints of the legs

Ways to Determine if A Trapezoid is Isosceles

If the legs, base angles, or diagonals are congruent

12.3 Parallelograms