The Buffon's needle experiment is one way to empirically estimate the value of this number. This isthe number of radians in 180 degrees. The ratio between the (*) circumference and diameter of a circle isequal to this number. For 10 points, what number is multiplied by the radius squared to give a circle’s areaand is approximately equal to 3.14?
Only numbers congruent to 1 or 5 mod 6 can have this property. Numbers of this type which areone less than a power of two are named for Mersenne. According to the fundamental theorem ofarithmetic, every positive integer can be expressed uniquely as a (*) product of these numbers. The Sieveof Eratosthenes can be used to find, for 10 points, what numbers only divisible by one and themselves?
Groups are examples of these mathematical objects endowed with a binary operation fulfillingcertain properties. A symbol that looks like a circle with a line through it is commonly used to denotean "empty" one of these objects which has no (*) elements. The intersection and union operations can beperformed on, for 10 points, what collections of objects?
Along with variance, this value is affected by Bessel's correction. T and z scores have this number intheir denominator. This value, symbolized by a lowercase sigma, is equal to the square root of the (*)variance. 63% of the data in a normally distributed set is within this value of the mean. For 10 points, namethis value that measures the "spread" of a set of data.
One common proof of this theorem places four congruent triangles inside each side of a square,forming a smaller square. The law of cosines is a generalization of this theorem, which can be appliedto find the Euclidean (*) distance between two points. The length of a triangle's hypotenuse may be foundwith, for 10 points, what theorem about the squares of the side lengths of right triangles?
The Basel problem deals with the sum of the reciprocals of these numbers, which Euler proved isequal to pi squared over 6. The differences between these numbers are increasing consecutive (*) oddnumbers. These are the only integers with an odd number of factors. For 10 points, name these numbers thatare equal to another number times itself.
The derivative of a function at a point is this property of a line tangent to the graph at that point. Iftwo lines are parallel, they have the same value for this property, and this property is undefined for (*)vertical lines. Rise over run is equal to, for 10 points, what value which measures the "steepness" of a line onthe coordinate plane?
This function is generalized to the complex plane by the gamma function. The number of ways tochoose r objects out of n objects, if order matters, is equal to this function of n divided by this functionof r. This function of n gives the number of ways to (*) order n objects. An exclamation point denotes, for10 points, what function equal to the product of all integers up to the input?
This number is the only number that is used in the expression of the golden ratio through continuedfractions. Raising i to the fourth power gives this number, while raising anything to the (*) zerothpower also gives this number. This number is the only positive integer that is neither prime nor composite.The multiplicative identity is, for 10 points, what smallest positive integer?
This quantity times one-half the apothem gives the area of any regular polygon. A value equal toone-half of this quantity appears in Heron's formula, which is used to find the areas of (*) triangles. Fora rectangle, this quantity equals twice the width plus twice the height. Circumference is the value for a circleof, for 10 points, what quantity equal to the sum of the side lengths of a polygon?
For two vectors, this operation can be performed on them by using the tip-to-tail or parallelogrammethod. Repeated applications of this operation can be denoted using a capital (*) sigma. Performingthis operation on two adjacent numbers in Pascal's triangle gives the number below them. Multiplication isequivalent to repetition of, for 10 points, what operation which gives the sum of two numbers?
If a matrix has determinant zero, then it cannot have one of these things. Functions that fail thehorizontal line test do not have one of these things, and reflecting the graph of a function across theline y=x gives the graph of this thing for it. For the function f(x)=x ("f of x equals x squared"), thesquare (*) root function serves as this thing. For 10 points, identify these functions which "undo" anotherfunction.
This is the number of elements in the smallest non-cyclic group. A hypercube in this manydimensions is also known as a tesseract. The regular polyhedron with the fewest faces has this manyfaces. The surface area of a (*) sphere is equal to this number times pi times the radius squared.Tetrahedrons have, for 10 points, what number of faces which is also the number of sides of a trapezoid?
[10] Name these expressions which combine variables and coefficients using basic arithmetic operations. Quadratics and cubics are special cases of these expressions.
[10] This term refers to the largest exponent in a polynomial. The fundamental theorem of algebra says that a polynomial can have at most as many roots as this value for it.
[10] A technique named for this man, his "rule of signs", can be used to find the number of positive real roots of a polynomial. This French mathematician is also the namesake of the standard coordinate plane.
[10] Name this set of numbers, which are denoted by a double-struck Z. This set of numbers includes all of the whole numbers and their negatives.
[10] This theorem concerns a general class of Diophantine equations. It states that if you know the remainders when a number is divided by several integers, you can uniquely determine the remainder when that number is divided by their product.
[10] One of the most famous Diophantine equations is the subject of this French mathematician’s “last theorem”, which states that that the equation "a to the n plus b to the n equals c to the n" has no solutions in the integers if n is greater than 2 and was finally proved by Andrew Wiles in 1995.
[10] Name this mathematician who formulated five postulates of a certain field of math in his book Elements, in which he also proved the infinitude of the primes.
[10] Euclid's Elements was primarily concerned with proofs in this kind of math. This field of math concerns points, lines, shapes, areas, and objects like circles and triangles.
[10] Non-Euclidean geometry violates this one of Euclid's postulates, which states that given a line L and a point P not on that line, exactly one line through P has the namesake property with respect to L.
[10] Name these pieces of infrastructure in a city now called Kaliningrad. Leonhard Euler formulated, and solved negatively, a famous problem asking whether it was possible to cross each of these things exactly once.
[10] The Seven Bridges of Königsberg problem was the first famous problem in a field of math named for these things. Another kind of these things are used to visually represent functions on the Cartesian plane, and advanced calculators are sometimes named for being able to produce these.
[10] The eighth-most famous bridge in mathematics, meanwhile, is the Broom Bridge in Dublin, Ireland, into which W. R. Hamilton carved the formula for multiplying quaternions, an extension of this number system. This set contains both real and imaginary numbers and is symbolized with a double-struck C.
[10] Name this quantity computed by adding all the values in a data set and then dividing by the number of values.
[10] The arithmetic mean, or average, of two values is always greater than this other mean, which is calculated by multiplying two numbers by each other and taking the square root.
[10] This other statistical measure of an "average"-like value is the most common value in a data set.
[10] Analytic geometry uses the coordinate plane, which is centered at this point. This point is the intersection of the x- and y-axes and has coordinates (0,0) (“zero comma zero”).
[10] Pick’s theorem states that for a polygon on the coordinate plane, this quantity is equal to the number of lattice points contained inside the polygon plus half the number of lattice points on the boundary. For an ellipse, this quantity is equal to pi times the product of the semimajor and semiminor axes.
[10] The plane can also be described using this coordinate system, which describes points in terms of a radius r and an angle θ (theta). Graphing circles in this coordinate system involves setting the radius to a constant.
[10] Identify this number equal to the value of an extremely large exponentiation tower, which was popularized in a 1977 column by Martin Gardner, at which point it was the largest number ever used in a serious mathematical proof.
[10] Graham's number is the upper bound to a problem from Ramsey theory, a subfield of graph theory which involves doing this process to graphs in certain ways. A famous computer-assisted proof by Haken and Appel showed that any map can have this process done to it with only four different labels.
[10] Specifically, the problem asked for the smallest dimension n in which a complete graph on the vertices of an n-dimensional hypercube, if each edge were colored red and blue, must contain a mono-colored one of these shapes. These shapes have four equal sides and four right angles.
[10] Name these self-similar figures which include the Sierpiński triangle, the Cantor set, the dragon curve, and others.
[10] A fractal named for Niels von Koch and this kind of figure can be constructed by starting with a triangle, then adding triangles with smaller bases onto the straight lines, and repeating infinitely.
[10] Fractals have a "Hausdorff" variety of this concept which can be a non-integer and measures the complexity of a fractal. Shapes like circles and squares exist in a plane, which has two of these, whereas solid objects like cubes have three of them.
[10] Name these collections of constraints, which can be solved using elimination, substitution, or Cramer's rule.
[10] Cramer's rule is a method for solving systems of equations using these mathematical objects. These rectangular arrays of numbers can have determinants, and their multiplication is not commutative.
[10] An elimination method named for this man performed on matrices can be used to assist in solving systems of linear equations. As a child, this man amazed his teachers by summing the numbers from 1 to 100 in his head.
[10] Name this property of sets that contain more elements than the value of any integer and can be said to go on "forever". It is represented by a symbol resembling a sideways eight.
[10] Cantor proved that this set of numbers is not countably infinite, making it a "larger" infinity than the set of integers. This set contains all the numbers on the number line, including rational and irrational numbers, and is represented with a double-struck R.
[10] A thought experiment attributed to David Hilbert explains the equivalence of seemingly differently-sized infinite sets by considering one of these buildings with an infinite number of rooms and showing that it can accommodate more people even if every room is full.
[10] Name these polygons with three sides. Their interior angles always sum to 180 degrees.
[10] This term refers to the line segment connecting a vertex to the midpoint of the opposite side of a triangle.
[10] This point is the intersection of the three medians of a triangle. It is also the triangle's center of gravity.
[10] There are this many possible such numbers, equal to the value of 4 choose 2.
[10] Two of the numbers, 7887 and 8778, are this kind of number which is the same written both forwards and backwards.
[10] All six numbers must be divisible by this smallest odd prime, because any number whose digits sum to a multiple of this number, like 12 or 39, must be divisible by this number.