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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

3.13 $

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

26.11 $

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

13.06 $

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

9.14 $

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

6.53 $

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

5.22 $

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

3.13 $

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

2.61 $

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

6.53 $

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

1.31 $

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

2.61 $

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

3.13 $

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

2.61 $

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

0.78 $

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

3.92 $

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