National High School Mathematics League Test
The national senior high school mathematics league exam (the second test) is in line with the international mathematics Olympics, expanding the knowledge; Appropriate to add some content outside the syllabus, the added content is:
1. Plane geometry
Several important theorems: Menelius Theorem, Seva Theorem, Ptolemy Theorem and siemsen Theorem.
Several Special Points in Triangle: Imitation Center, fermat point and Euler Line.
Geometric inequality.
Geometric extremum problem.
Transformation in geometry: symmetry, translation and rotation.
Power and root axis of a circle.
Area method, complex number method, vector method, analytic geometry method.
2. Algebra
Periodic function, a function with absolute value.
Trigonometric formula, trigonometric identity, trigonometric equation, trigonometric inequality, inverse trigonometric function.
Recursion, recursive sequences and their properties, general formulas of first-order and second-order linear recursive sequences with constant coefficients.
The second mathematical induction.
Mean inequality, Cauchy inequality, rank inequality, Chebyshev inequality, univariate convex function.
Complex number and its exponential form, triangular form, Euler formula, Dimov theorem, unit root.
Polynomial division theorem, factorization theorem, polynomial equality, rational root of integer coefficient polynomial *, polynomial interpolation formula *.
The number of roots of polynomials of degree n, the relationship between roots and coefficients, and the virtual root pairing theorem of polynomials with real coefficients.
Function iteration, simple function equation *
3. Elementary number theory
Congruence, Euclid division, Pei tree theorem, complete residue class, quadratic residue, indefinite equations and equations, Gaussian function [x], Fermat's last theorem, lattice point and its properties, infinite descent method, euler theorem *, Sun Tzu's theorem *.
4. Combination problem
Cyclic permutation, permutation and combination of repeated elements, combinatorial identity.
Combinatorial counting, combinatorial geometry.
Dove cage principle
Exclusion principle.
Extreme principle.
Graph theory problems.
Division of sets.
Cover.
Planar convex set, convex hull and their applications.
Note: Contents marked with * will not be tested in additional tests, but may be tested in winter camps.
In addition, the topic of the initial test will also involve the content of the second test, so it is more difficult to prepare than the previous test.