For Scobol Solo yesterday, which is an all tossup tournament, we had 2/0 Math. In order to do that, we had to write 40 math tossups. Historically, this tournament had 3/0 Math, almost all of which was computational, but due to all the complaints with computational tossups, this year was all noncomputational. This is different than writing 3/3 for a 12 (or however many) round tournament IMSA is writing, but it is only a little bit smaller in scope. I think it's probably easier to write a bunch of math bonuses because you don't have to worry about transparency issues, you can go with answers that only have one or two interesting clues, and you can expand the canon more with hard parts of bonuses. These questions are a joint effort between Jonah, Bardoe, and me, with a little help from Brad Fischer and some feedback from a group led by Mike Cheyne.
Here are the questions. To save myself time right now, they are not in any particular order and include in the two large groups questions that ended up as tiebreakers, replacements, and in the Championship Match, and I'm not showing exactly where the underlining was. In the Championship Match, there is nothing wrong with including questions that most students don't know, because it's not supposed to be average high school difficulty. Keep in mind that conversion rates for a solo tournament are generally lower than for a regular tournament for obvious reasons, and overall conversion rate at this tournament generally is in the low 60s. We also use power matching, which lowers conversion a little because the decrease in conversion rate between two weak students is more than the increase in conversion rate between two strong students.
The first group, which we called Curricular Math, had a conversion rate of 80%:
Scobol Solo wrote:The Cayley-Hamilton theorem applies to these entities when they are part of an abelian (uh-BEE-lee-un) group, stating that each one satisfies its own characteristic equation. That characteristic equation is used to find scalars which multiply certain vectors to give the same result as multiplying by these entities; those scalars are called eigenvalues. Transition ones of these entities are used in Markov chains, and Gauss-Jordan elimination can be applied to these structures to convert them into reduced row-echelon form. An adjoint of one of these equals its inverse times its determinant. Name these mathematical structures that usually consist of rectangular arrays of numbers.
ANSWER: square matrices [or matrix; accept linear transformations before they are mentioned; prompt on transformations]
This operation often is combined with transposition for matrices, and matrices which do not change under the combined operations are called Hermitian (hur-MI-shun). Performing this operation on a number, multiplying the result by the number itself, and then taking the square root finds the complex modulus of the number, which is also that number’s absolute value or norm. When complex numbers are represented as vectors, this operation reflects over the real axis. Simplifying fractions with complex numbers is often accomplished by multiplying by the denominator after this operation has been done. Name this operation which transforms a plus b i to a minus b i.
ANSWER: conjugate [accept word forms, e.g. conjugation]
The polar equation theta equals this function of r produces a spira mirabilis, or golden spiral. The nth term of this function’s Maclaurin series equals the quantity x minus one, quantity to the nth power, over n. This function can be integrated by multiplying it by x and subtracting x, and its derivative equals the reciprocal of x. Multiplying two numbers together and then applying this function has the same result as applying this function separately to the two numbers and adding the results, and exponents inside these functions can instead be used to multiply outside of them. This function can be transformed to other bases by dividing it by itself applied to a constant, and its discovery is often credited to John Napier. Name this inverse of the exponential function.
ANSWER: natural logarithm of x [or logarithm (to the base e); accept ln; accept lg]
This term is used to describe experiments in which at least two factors are tested at all of their possible levels. It more commonly refers to a function which for large values can be estimated by the Stirling approximation. For positive integers, this function shifted to the right by one unit is equal to the gamma function. Most often it is defined recursively with the initial value at 0 equaling 1 and the ratio of this function for x divided by this function for the quantity x minus one equaling x. This function is often used in combinatorics in the formulas for combinations and permutations. Give the name of the function that, for a natural number n, equals the product of all the numbers from 1 to n.
This property does not hold in the rings of integers modulo a composite number, and whether it holds distinguishes integral domains with respect to rings. A short proof by contradiction of this property consists of multiplying both sides of an equation by the multiplicative inverse of any factor on one side of the equation. Often used to find solutions to polynomials of degree greater than one, this property makes the solutions obvious once the polynomial has been factored. Name this property that states that if quantities multiply to the additive identity then at least one of the quantities is equal to the additive identity—that is, zero.
ANSWER: zero product property [accept logical equivalents for “property”; accept answers logically equivalent to nonexistence of zero divisors]
This shape can be generated by drawing an infinite number of lines that are perpendicular to segments coming from a fixed point where those segments hit a fixed line. This shape also can be generated by finding all of the points so that a right angle is formed at the y-axis with one ray going through a fixed point on the x-axis and one ray going through the point on this shape with twice the y-coordinate as the point on the y-axis. Any ray coming from the fixed point on the x-axis that bounced off this shape the way light bounces off a mirror would travel parallel to the x-axis. This shape is also defined as the set of points equidistant from a fixed line, called a directrix, and a fixed point, called a focus, and this shape also is the conic section with an eccentricity of one. Name this shape generated by quadratic functions such as y equals x squared.
ANSWER: parabolas [prompt on quadratics]
This unit must be used for one of the quantities so that the small oscillation restoring force of a pendulum can be approximated as mass times little g times theta. This unit also must be used for the common Taylor series approximation x minus x cubed over six plus x to the fifth over one hundred twenty, etcetera. One way to define this unit is arc length divided by radius, which is why numbers measured in these units can be multiplied by numbers in other units without changing those other units. This unit can be thought of as the number of radii that would fit along the arc inscribed in an angle, and it equals approximately 57.3 degrees. Give this unit that can be converted to by multiplying pi over one hundred eighty times the number of degrees.
A group action has this property if it is possible to find an element of the group that takes any element of the set to any other. This adjective precedes the word “closure” to describe a graph that contains edges between any two vertices that are connected. Like reflexivity, this property of binary relations is an essential quality to both preorders and partially ordered sets. One of the three properties of equivalence relations, it also applies to relations such as implication, set containment, and inequality; the other two properties of equivalence relations are reflexivity and symmetry. Name this property, an example of which is the statement that if x equals y and y equals z, then x equals z.
ANSWER: transitive property [accept word forms, e.g. transitivity]
In a circle, this distance equals the radius minus the sagitta of a chord, which equals the distance from the center of the circle to the midpoint of that chord. For other shapes, this equals the side length divided by twice the tangent of the quantity pi over n, where n is the number of sides, so, for example, it equals the side length times root three over two for a regular hexagon. For any regular polygon, this distance is twice the area divided by the perimeter. Contrasted with the radius of a polygon, name this distance, or segment, from the center of a regular polygon to the midpoint of one of its edges.
ANSWER: apothem [accept inradius; do not accept or prompt on “radius”]
The fixed point of this function is known as the Dottie number, and this function equals the real component of the value of e raised to the power of i times x. The derivative of the inverse of this function equals negative one over the square root of the quantity one minus x squared, and the graph of its hyperbolic version is known as a catenary (KAA-tuh-nehr-ee). Its namesake law can be used to determine the length of the third side of a triangle given the two other sides and the included angle and is a generalization of the Pythagorean theorem to any triangle. Give the name of this function equal to both the x-coordinate on a unit circle and the length of the adjacent leg divided by the hypotenuse of an angle in a right triangle.
ANSWER: cosine [accept addition of “of x” or any other variable]
This theorem is reversed in Minkowski space, and in Euclidean geometry it is sometimes confused with a statement involving inner products which can be used to prove it. When applied to vectors, this theorem is equivalent to the statement that the sum of the lengths of two vectors is greater than the length of the sum of two vectors. Simplistically, it guarantees that there are no shortcuts between two points in space, that instead the shortest path between two points is a straight line. This theorem can be used to show that the number of non-congruent three-sided figures with integer side lengths and perimeter less than five is one. Name this statement, sometimes confused with the Cauchy-Schwarz (koh-SHEE SHWARZ) inequality, that the sum of the lengths of any two sides of a three-sided figure must be larger than the third side.
ANSWER: triangle inequality [accept answers augmented by anything similar to “theorem” or “rule”]
This shape’s namesake numbers are also known as polite numbers, and when this figure’s diagonals break it down into four triangles, the result is exactly one pair of triangles with equal areas; dividing that equal area by the areas of the other triangles gives reciprocals. Approximations using this shape generally give a worse result than Simpson’s Rule but a better result than Riemann sums when approximating integrals. Its area is usually given by the formula half the sum of the bases times the height, and it has a median which is parallel to its bases and whose length is the average of them. Name this quadrilateral with one pair of parallel sides.
ANSWER: trapezoid [or trapezium]
In hyperbolic geometry there are two forms of this geometric configuration, ultra and limiting. In Euclidean geometry it is the degenerate conic section found by the limiting case as the focal distance goes to infinity while the vertices remain constant of a hyperbola. This degenerate conic can also be seen as the intersection of a plane and double cone when the vertex of the cone is moved to a point at infinity making the cone into a cylinder. In projective geometry, this configuration cannot exist. Solutions in the coordinate plane to the equation y squared equals one provide an algebraic description to this configuration. This configuration also exists in the graph of an inconsistent system of two linear equations--that outcome arises when the two lines have the same slope. Identify this relationship between two lines in the same plane that never intersect.
ANSWER: parallel lines
This point is collinear with the Nagel Point, incenter, and Spieker center. It is always twice as far from the orthocenter as from the circumcenter, and it is collinear with those points together with the nine-point circle center. Its barycentric coordinates never change, and it always is located in the interior of the triangle. This point is a point of concurrency of three cevians (CHAY-vee-uns), and the ratio of each of the smaller segments of those cevians to point of concurrency is 2 to 1. If a triangle is made of uniform density, this point is located at the center of mass. Name this point of intersection of the medians of a triangle.
ANSWER: centroids [prompt on center of mass; prompt on center of gravity]
In image processing, this statistic is used to reduce salt-and-pepper noise. It can be estimated by Pareto interpolation, and for continuous distributions this statistic cannot be more than one standard deviation from the mean. For a distribution, this is the point that minimizes the average of the absolute deviations. It can be defined as the spot where the cumulative distribution function equals one half. A measure of central tendency, it is less sensitive to outliers than the mean. Name this value that can be found by finding the value at the fiftieth percentile or, for finite sets with an odd number of elements, the middle number of the set.
The Marcinkiewicz–Zygmund (mar-sihn-KEE-vich ZIG-moond) inequality concerns the sum of the variances of variables with this property. The General Poisson (pwah-SOHN) Approximation Theorem holds for events with the asymptotic type of this property, and events can have this property as well as mutual exclusivity only if at least one of them has probability zero. If the covariance, or equivalently the correlation coefficient, of two events is zero, then they have this property; and the multiplication rule in probability applies to any number of events that all have this property. Formally, events A and B have this property if the probability of A given B equals the probability of A. Name this property of statistical events that means that the outcome of one is unrelated to the outcome of the other.
ANSWER: event independence [or independent events, etc.]
The existence of this function for two elements of a ring ensures that the ring is a principal ideal domain. This value can be found by beginning a sequence with the two given numbers with the larger one first and repeatedly adding to the sequence with the difference between the second to last number and the largest multiple of the last number that will leave a positive result. That process is known as the Euclidean algorithm, and if this value is one for two numbers, then the numbers are said to be relatively prime. It can also be calculated by dividing the product of two numbers by their least common multiple. Given the prime factorization of two numbers, this value can be found by multiplying together the set of their common prime factors. Name this value which is the largest value that divides two numbers.
ANSWER: greatest common factor [or greatest common divisor; or GCF; or GCD]
When these structures have three consecutive perfect squares, they use a multiple of twenty-four, and a proof has been submitted that these structures cannot have four consecutive perfect squares. In 2004, Ben Green and Terrence Tao proved that there exist arbitrarily long lists of prime numbers that are this type of structure. These structures can be thought of as linear functions from the natural numbers to the real numbers or complex numbers. By pairing up entries from these structures, Gauss famously found a fast method to sum them when he was young. Examples include the numbers equal to two mod five, even numbers, odd numbers, and the multiples of any integers, but not the Fibonacci numbers or the powers of two. Give the name of this type of sequence whose general form is given by a sub n equals a sub one plus d times the quantity n minus one.
ANSWER: arithmetic (aa-rith-MAA-tik, but be lenient) sequence [or arithmetic progression; prompt on sequence]
One type of these objects consists of all finite words built from a set; that is the free variety. They are partitioned by their orbits under an action, and these mathematical structures can be represented by Cayley graphs or Cayley tables. The types of these objects that represent the symmetries of a regular polygon are known as dihedral ones, and those that can be generated by products of a single element are called cyclic. If they are commutative, they are known as abelian. Name these mathematical structures that consist of a set closed under a binary operation that is associative and possesses inverses and an identity.
Numbers from this set are multiplied to generate Beatty sequences, and Apéry’s theorem states that the sum as n goes from one to infinity of one over n cubed is one of these numbers. The reals can be constructed by exploiting the fact that this set’s complement lacks the least upper bound property; that procedure is known as creating Dedekind (DEH-duh-kend) cuts. A classic proof for the existence of this set constructs an example by considering the equation x squared equals two y squared, deriving a contradiction because y must be both odd and even. Examples of these kinds of numbers include e, the square root of two, and pi. Give the name of this set of numbers that cannot be written as a simple fraction of two integers.
ANSWER: irrational numbers [accept J]
The second group, which we called Other Math, had a conversion rate of 40%, ranging from 3% for Test (which is all that was underlined) to 97% for Convergence.
Scobol Solo wrote:Proofs of this result sometimes make use of Liouville’s Theorem that a bounded entire complex function must be constant. An early proof of this by Argand was based on earlier work by D’Alembert, while the most famous proof of it was actually faulty because it assumed the Jordan curve theorem. Early arguments over it were faulty because they did not use complex numbers or, in Leibniz’s case, did not realize that the square root of i could be expressed in the form a + b i, and thus failing in the case of x to the fourth power plus one. Nearly proven in Gauss’s dissertation, name this theorem that states that every polynomial of degree n with complex coefficients has n complex roots.
ANSWER: Fundamental Theorem of Algebra [prompt on partial answer]
Hipparchus wrote a work criticizing this person’s three-volume work Geographica, which described how to map the inhabited world. This person was the first cartographer to use both parallels and meridians, centering them at Rhodes. In a famous experiment, this person measured the shadow of a vertical stick at noon at Alexandria during the summer solstice and approximated the distance from Alexandria to Syene, which was almost due south near the Tropic of Cancer, to approximate the circumference of the Earth. This person is also credited with devising a system in which all of the multiples of each number, other than multiplication by one, are crossed out from a table. Name this ancient Greek who developed a technique for finding prime numbers, his eponymous sieve.
ANSWER: Eratosthenes of Cyrene [or Erastosthenes]
The dihedral angles in this solid have an angle equal to the arccosine of negative root five over three, which is about 138 degrees. All of its vertices can be sorted into groups of four that form golden rectangles. One of these solids with side length two can be placed in rectangular coordinates so that each point has one coordinate equal to zero, one coordinate equal to plus or minus one, and one coordinate equal to the golden ratio. It contains thirty edges and twelve vertices, and each vertex brings together five triangles. Name this platonic solid, the dual of the dodecahedron, whose name is derived from the Greek for twenty because it has twenty sides.
This term can be used as an adjective to describe a set that represents the state of a dynamical system after an infinite amount of time or to describe a point that can also be called an accumulation point. This term also is applied to values whose existence can be proven using the squeeze theorem or a method referred to as delta-epsilon. These values are often evaluated by taking the derivatives of both the numerator and denominator of a rational function, a method known as l’Hôpital’s rule. These values are equal to the values of continuous functions, and they are commonly used to find derivatives. Give this value that the output of a function may approach when the input approaches a specified value.
In the Euclidean metric, compactness is equivalent to being bounded and having this property, which is known as the Heine–Borel theorem. Sets with this property retain it under intersection with other such sets even if the number of sets intersected is uncountable. There exists a neighborhood around every point outside a set of this type that is disjoint from this set, and sets with this property can be defined as containing all of their limit points. An interval with this property contains a maximum and minimum and in interval notation is denoted with square brackets. Name this property possessed by sets that contain their own boundary and whose complements are open.
ANSWER: topological closure [or closed set, etc.; do not accept “algebraic closure”]
One sequence of polynomials with this property begins one, then x, then three-halves x squared minus one half; the integral of any product of two of those from negative one to one equals zero. This term also refers to a specialization of unitary matrices, namely, matrices whose transposes equal their inverses. A set with this property can be produced by recursively subtracting vector projections; that is a basis produced by the Gram-Schmidt process. This term also describes vectors whose inner product, or dot product, is zero; as well as two lines whose slopes are opposite reciprocals. Give this property which applies to adjacent sides of a rectangle.
ANSWER: orthogonality [accept perpendicularity; accept normality; accept word forms of any of the answer choices; do not accept “right”]
This operation creates equivalence classes based on the ideal of a ring, and can be used to efficiently express Euler’s (OY-lurz) Theorem and Fermat’s (fair-MAHZ) Little Theorem without using subtraction and divisibility. It also explains why casting out nines can be used to check calculations. Single-bit Boolean algebra is equivalent to this operation using two, and though it is not a type of number base, this operation is a type of arithmetic. Also known as clock arithmetic, this operation involves residue classes and is often written using congruence notation. Name this type of arithmetic based on division remainders.
ANSWER: modular arithmetic [or modulo; prompt on clock arithmetic before it is mentioned; accept word forms like modulo or modulus]
The best-known form of this process is equivalent to the Darboux (DAR-boo) type, while a generalization of it is named for Lebesgue (luh-BAY). Quadrature methods approximate it, and iterations of this process on well-behaved functions are independent of order according to Fubini’s Theorem. Executing this process may be made easier by decomposing a given function into the product of two simpler functions, which is called performing this process “by parts”. With polynomials, this operation involves adding one to all of the exponents and dividing by the new exponents. Name this process which, by the Fundamental Theorem of Calculus, is equivalent to antidifferentiation.
ANSWER: Riemann integration [or integrals; accept answers modified by either “definite” or “indefinite”; accept antiderivatives or antidifferentiation before the last word; do not accept or prompt on “derivative” or “differentiation”]
Inter-event times for this distribution follow the exponential distribution, and its moment-generating function is proportional to e to the t, quantity minus one. The sum of variables with this distribution itself follows this distribution; that result is its reproductive property. This distribution applies when the question of how many events have not occurred has no meaning, and when the product of the number of trials with the probability remains fixed as the number of trials goes to infinity, it approximates the binomial distribution; that approximation is good for many trials with small probability of success. Its mean and variance are both equal to its single parameter, and its mass function at x for a parameter mu is e to the minus mu times mu to the x, over x factorial. Name this distribution used to model waiting times for arrivals, named for a Frenchman.
ANSWER: Poisson distribution [accept Poisson processes]
When this function is applied to the natural logarithm of the golden mean it returns one half. The derivative of this function’s inverse is the reciprocal of the square root of the quantity x squared plus one, and any solution with initial condition y equals zero to the differential equation of y double-prime equals y must be a multiple of this function. For small values of x, this function can be approximated by x plus x cubed over six. This function itself is given by the formula one half times the quantity e to the x, minus e to the negative x. Name this function that, when it is inputted half of the area confined by the x-axis, the right half of the curve x squared minus y squared equals one, and a segment connecting the origin to that curve, returns the y-coordinate of the point of intersection.
ANSWER: hyperbolic sine function [do not accept or prompt on a partial answer; accept sinh; accept sinch; accept shine]
This mathematical system can be modeled in the conformal disk model, or for the two-dimensional case by the half-plane model, both of which are named for Poincaré (poin-kar-AY). This system incorporates the ultraparallel theorem and possesses negative curvature, allowing Saccheri (suh-CAIR-ee) quadrilaterals in it to have acute summit angles. That is a consequence of allowing multiple lines through a particular point to be parallel to a given line, which also means that triangles in this type of system have degree sum less than one hundred eighty degrees. Developed by János (YAH-noash) Bolyai and Nikolai Lobachevsky, name this type of geometric system that violates Euclid’s (YOO-klid’s) fifth postulate, contrasted with Euclidean and elliptical geometries.
ANSWER: hyperbolic geometry [prompt on non-Euclidean geometry/ies; accept Bolyai-Lobachevskian geometry before “Lobachevsky”]
This operation is combined with the del operator to find the limiting value of circulation per unit area for certain fields; that is the curl operation. This operation is used to find binormal vectors, the intersection of two planes, and the area of a parallelogram. Often found by taking a three-by-three determinant, the result of this operation is perpendicular to the two input vectors, and its direction can be determined using the right hand rule. Name this vector operation whose result is equal in magnitude to the product of the magnitude of the two vectors times the sine of the angle between them, and that in the scalar triple product is combined with the dot product.
ANSWER: cross product [or Gibbs vector product; prompt on vector product]
For a ring, this quantity is the least upper bound of its prime ideals and is named after Krull. For a vector space, this value equals the cardinality of a basis. Another version of this quantity is given by the quotient of the logarithms of the space-scaling rate and the log of the linear scaling rate; that form of this quantity is named for Felix Hausdorff, and gives non-integer values for fractals. This value is equal to the number of coordinates required to locate a point in a given space, which is why spherical and cylindrical coordinate systems require three values, while polar coordinates require two. Give this value which equals zero for a point and one for a line.
ANSWER: dimensions [or dimensionality; accept rank]
The uniform and pointwise types of this property may be contrasted for sequences, and Cauchy sequences in a complete space always have this property. A limit approaching zero is necessary but not sufficient for this property, which exists when the root test gives a value less than one or the integral test gives a finite number. The harmonic series lacks this property, but the alternating version thereof as well as p-series for p greater than one do have this property. This term also applies to geometric series with ratio between negative one and one. Give this term that describes series whose sum approaches a finite number, contrasted with divergence.
ANSWER: convergence [accept word forms]
One type of this procedure is a triply-eponymous method that counts how many times somebody from a stronger sample beats somebody from a weaker sample, quantified as capital U. In addition to the Wilcoxon-Mann-Whitney type, the Wald type of this procedure determines whether or not a parameter exists, and the Yates type corrects the Pearson chi-square type, which, like the type that uses the Student’s t-distribution, requires finding the degrees of freedom to calculate the p-value necessary to reject the null hypothesis. Give the name common to these procedures which determine whether or not a result is statistically significant.
ANSWER: statistical hypothesis testing of significance
This shape is the cissoid of a single circle using a point located root two times the radius from the center, and it is the inversion of a hyperbola. This shape also is the envelope of circles centered on a hyperbola and passing through the center of that hyperbola, and it also is the locus of points such that the product of the distances to two fixed points is constant. A specific case of a Cassini oval, this shape can be expressed in polar form as r squared equals a squared times the cosine of two theta. It is the shape of the slice of a torus if the slice has a point in the center. Name this shape which looks like a figure eight or an infinity sign.
ANSWER: lemniscate of Bernoulli [prompt Cassini oval]
This mathematician proposed a problem which became the first of Hilbert’s problems and is known as the continuum hypothesis. This mathematician developed the concept of numbers that are larger than all finite numbers but not necessarily infinite, which he called transfinite numbers; and his eponymous fractal is created by taking the closed interval zero, one and repeatedly deleting open middle thirds. This mathematician attempted to write every number in its binary representation but derived a contradiction to show that the set of reals is uncountable—his namesake diagonal argument. Name this German founder of set theory.
ANSWER: Georg Ferdinand Ludwig Philipp Cantor
When a sample size equals one mod four, this value is equal to the H-Spread, or the difference between the hinges. In a normal distribution, this value is supposed to equal approximately 1.35 times the standard deviation, covering about two-thirds of a standard deviation both above and below the mean, and, like the median absolute deviation, this statistic is robust. This value is represented as the width of the box in a boxplot. Name this statistic which is calculated by subtracting the the median of the lower half of a data set from the median of the upper half.
ANSWER: interquartile range [or IQR, do not accept or prompt on range]
One of these methods using Markov chains and sometimes classified as either an independence chain or a random walk is the Metropolis-Hastings algorithm, which is also considered a type of simulated annealing. Often used in stochastic modeling, this method was first used to study radiation shielding by John von Neumann and Stanisław Ulam. Often used to approximate integrals, this mathematical method is used to study situations such as Brownian motion, economic modeling, and games of chance when deterministic solutions are unknown or unhelpful. Identify this type of solution that repeatedly uses random numbers which is named after a European city.
ANSWER: Monte Carlo algorithm or method [accept annealing or stochastic before they are mentioned]
This result can be calculated by taking the inverse of the quantity A transpose A, end quantity, times A transpose B, where matrix A has a column of ones on its left or right side. That equation finds what is often considered the “optimal” solution to an overdetermined system. The accuracy of this result is measured by the square of the correlation coefficient, which equals one for perfect linear fits. Name this process of producing a line that approximately goes through a set of given points.
ANSWER: ordinary linear least-squares regression line [accept either or both parts, or anything containing “regression”; accept line of best-fit; accept OLSRL]