MathProblemsBank Math Problems Bank
  • Home
  • Forum
  • About Us
  • Contact Us
  • Login
  • Register
  • language
 MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Mathematics sections
  • Algebra
    • Vector Algebra
    • Determinant calculation
    • Permutation group
    • Matrix transformations
    • Linear transformations
    • Quadratic forms
    • Fields, Groups, Rings
    • Systems of algebraic equations
    • Linear spaces
    • Polynomials
    • Tensor calculus
    • Vector analysis
  • Analytic geometry
    • Curves of the 2-nd order
    • Surfaces of the 2-nd order
    • Lines on a plane
    • Line and plane in space
    • Tangents and normals
  • Complex analysis
    • Operations with complex numbers
    • Singular points and residues
    • Integral of a complex variable
    • Laplace transform
    • Conformal mappings
    • Analytic functions
    • Series with complex terms
    • Calculating integrals of a real variable using residues
  • Differential equations
    • Ordinary differential equations
      • First order differential equations
      • Second order differential equations
      • Higher order differential equations
      • Geometric and physical applications
    • Systems of ordinary differential equations
    • Stability
      • Stability of the equations
      • Stability of the systems of equations
    • Operating method
      • Differential equations
      • Systems of differential equations
  • Differential geometry
  • Discret mathematics
    • Boolean algebra
    • Set theory
    • Combinatorics
    • Graph theory
    • Binary relations
    • Propositional algebra
      • Propositional calculus
      • Sequent calculus
    • Predicate calculus
    • Theory of algorithms and formal languages
    • Automata theory
    • Recursive functions
  • Functional analysis
    • Metric spaces
      • Properties of metric spaces
      • Orthogonal systems
      • Convergence in metric spaces
    • Normed spaces
      • Properties of normed spaces
      • Convergence in normed spaces
    • Measure theory
      • Lebesgue measure and integration
      • Measurable functions and sets
      • Convergence (in measure, almost everywhere)
    • Compactness
    • Linear operators
    • Integral equations
    • Properties of sets
    • Generalized derivatives
    • Riemann-Stieltjes integral
  • Geometry
    • Planimetry
      • Transformations on the plane
      • Construction problems
      • Complex numbers in geometry
      • Various problems on the plane
      • Locus of points
    • Stereometry
      • Construction of sections
      • Various problems in the space
    • Affine transformations
  • Mathematical analysis
    • Gradient and directional derivative
    • Graphing functions using derivatives
    • Plotting functions
    • Fourier series
      • Trigonometric Fourier series
      • Fourier integral
    • Number series
    • Function extrema
    • Power series
    • Function properties
    • Derivatives and differentials
    • Functional sequences and series
    • Calculation of limits
    • Asymptotic analysis
  • Mathematical methods and models in economics
  • Mathematical physics
    • First order partial differential equations
    • Second order partial differential equations
      • d'Alembert method
      • Fourier method
      • With constant coefficients
      • With variable coefficients
      • Mixed problems
    • Convolution of functions
    • Nonlinear equations
    • Sturm-Liouville problem
    • Systems of equations in partial derivatives of the first order
  • Mathematical statistics
  • Numerical methods
    • Golden section search method
    • Least square method
    • Sweep method
    • Simple-Iteration method
    • Approximate calculation of integrals
    • Approximate solution of differential equations
    • Approximate numbers
    • Function Interpolation
    • Approximate solution of algebraic equations
  • Olympiad problems
    • Olympic geometry
    • Number theory
    • Olympic algebra
    • Various Olympiad problems
    • Inequalities
      • Algebraic
      • Geometric
    • Higher mathematics
  • Probability theory
    • One dimensional random variables and their characteristics
    • Theory of random processes
    • Markov chains
    • Queuing systems
    • Two-dimensional random variables and their characteristics
    • Definition and properties of probability
    • Limit theorems
  • Real integrals
    • Integrals of functions of a single variable
      • Indefinite integrals
      • Definite Integrals
      • Improper integrals
    • Double integrals
    • Triple integrals
    • The area of a region
    • Volume of a solid
    • Volume of a solid of revolution
    • Flux of the vector field
    • Surface integrals
    • Curvilinear integrals
    • Potential and solenoidal fields
    • Vector field circulation
    • Integrals depending on a parameter
  • Topology
  • USE problems
  • Variational calculus
Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: Convert the following formulas to clausal form: \[ R(t, w) \vee \neg \exists x \forall w[\neg(P(w) \vee S(x)) \rightarrow \neg Q(w)] \text {. } \]

6.5.7 Predicate calculus

3.08 $

Problem: For the surface given parametrically, find: 1. The unit normal vector at the point \( \left(u=u_{0}, v=v_{0}\right) \); 2. The equation of the tangent plane and the normal at the point \( \left(u=u_{0}, v=v_{0}\right) \) 3. The volume of the tetrahedron formed by a tangent plane at the point \( \left(u=u_{0}, v=v_{0}\right) \) to the given surface and coordinate planes; 4. The normals parallel to coordinate planes; 5. The first quadratic form of the surface; 6. The second quadratic form of the surface; 7. The angle between the coordinate lines of the surface at the point \( \left(u=u_{0}, v=v_{0}\right) \); 8. Gaussian and mean curvature of the surface; 9. The elliptic, hyperbolic and parabolic points on the given surface.

7.13 Differential geometry

25.69 $

Problem: Find the unit vectors of the tangent, principal normal, and binormal of the curve. Compose the equations of the contiguous plane, the normal plane, the rectifiable plane at the point \( M_{0} \). \[ x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad z=\cos 2 t . \]

7.5 Differential geometry

12.85 $

Problem: The expert company assesses \( n \) girls in accordance with 3 criteria: intelligence, beauty and kindness, in points (arbitrary natural numbers, they can be considered real numbers, it's not important). The ratings for each criterion are different for all girls, that is, they can be sorted by their intelligence, beauty or kindness. Prove or disprove that it is possible to choose three such girls, that for any of the unchosen ones, one of these three will exceed her in at least two criteria.

12.4.7 Various Olympiad problems

8.99 $

Problem: Using the vertex adjacency matrix, construct a graph diagram. Build flat layout. Create an adjacency matrix of the edges and an incidence matrix. Find the eccentricities of the vertices, the radius and diameter of the graph, peripheral, central vertices, diametrical chains. Find all cycles. Construct a cyclomatic matrix. Color the graph. \[ \left(\begin{array}{lllll} 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \end{array}\right) . \]

6.4.4 Graph theory

6.42 $

Problem: Using the vertex adjacency matrix, construct the graph diagram. Create an arc adjacency matrix, an incidence matrix and a reachability matrix. Arrange the vertices and arcs of the digraph. \[ \left(\begin{array}{llllll} 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \end{array}\right) . \]

6.4.5 Graph theory

5.14 $

Problem: Write the following sentences as predicate logic formulas and transform them into clausal form: «A student loves logic or philosophy if and only if there is a teacher who loves both logic and philosophy».

6.5.8 Predicate calculus

3.08 $

Problem: Determine the most general unifier and the corresponding general example for the following set of terms or show that the set is non-unifiable. \[ \left\{L_{i}\right\}=\{h(f(a), g(y, z), y), h(x, g(b, u), c)\} . \]

6.5.9 Predicate calculus

2.57 $

Problem: Solve the problem of finding the extrema of the functional reducing it to an optimal control problem with restrictions on the phase coordinates and controls: \[ \begin{array}{l} \int_{0}^{2} x d t \rightarrow \text { extr }, \quad|\ddot{x}| \leq 2, \quad x(0)+x(2)=0, \\ \dot{x}(0)=0 . \end{array} \]

4.20 Variational calculus

6.42 $

Problem: Perform skolemization of the following formulas presented in preliminary form: \[ \forall y \forall z \exists x \forall w[(T(x, w) \vee P(x, y) \& S(x, z)) \rightarrow R(x, w)] . \]

6.5.11 Predicate calculus

1.28 $

Problem: Convert the following formulas to clausal form: \[ Q(x, w) \vee \neg \exists x \forall w(P(x, w) \& Q(x, z)) \vee \neg S(z, w) \text {. } \]

6.5.12 Predicate calculus

2.57 $

Problem: Write the following sentences as predicate logic formulas and transform them into clausal form: "If there are no intelligent machines, and if every ideal machine is an intelligent machine, then an ideal machine does not exist".

6.5.13 Predicate calculus

3.08 $

Problem: Determine the most general unifier and the corresponding general example for the following set of terms or show that the set is non-unifiable. \[ \left\{L_{i}\right\}=\{g(f(b), f(x), a), g(y, v, b)\} . \]

6.5.14 Predicate calculus

2.57 $

Problem: An identically correct formula of predicate logic is the formula: 1. \( \exists x A(x) \Rightarrow \forall x B(x) \); 2. \( \forall x A(x) \Rightarrow \exists x A(x) \); 3. \( \forall x(A(x) \vee B(x)) \Rightarrow \forall x B(x) \); 4. \( \forall x A(x) \Rightarrow \exists x(A(x) \wedge B(x)) \).

6.5.15 Predicate calculus

0.77 $

Problem: Write the following argument in predicates and prove its validity using the resolution method. The premise: "no freshman likes sophomores. Everyone living in Vasyuki is a sophomore." Conclusion: "Not a single freshman likes anyone living in Vasyuki".

6.5.10 Predicate calculus

3.85 $

  • ‹
  • 1
  • 2
  • ...
  • 35
  • ...
  • 116
  • 117
  • ›

mathproblemsbank.net

Terms of use Privacy policy

© Copyright 2025, MathProblemsBank

Trustpilot
Order a solution
Order a solution to a problem?
Order a solution
Order a solution to a problem?
home.button.login