Categories

Descriptions (48 problems)
Graph theory (12 problems)
Problems (2764 problems)
- Algebra (374 problems)
  - Absolute value (7 problems)
    - Absolute value of a number (other) (1 problems)
    - Equations containing absoute values (1 problems)
    - Inequalities containing absolute values (4 problems)
    - Proof by exhaustion (1 problems)
      - Inequalities (1 problems)
  - Algebra (2 problems)
  - Algebraic equations and systems of equations (60 problems)
    - Algebraic equations and systems of equations (other) (3 problems)
    - Binomial theorem (2 problems)
    - Diophantine equations (8 problems)
    - Higher order equations. Palindromic polynomial equations (37 problems)
      - Equations of higher order (other) (37 problems)
    - Systems of algebraic non-linear equations (1 problems)
    - Systems of linear equations (2 problems)
  - Algebraic inequalities and systems of inequalities (25 problems)
    - Algebraic inequalities (other) (5 problems)
    - Classical inequalities (3 problems)
    - Linear inequalites and systems of inequalities (6 problems)
    - Number inequalities. Number comparison (3 problems)
    - Quadratic inequalites (several variables) (6 problems)
  - Complex numbers (4 problems)
    - Cartesian form of a complex number, complex conjugate, Argand diagram, etc (3 problems)
  - Discrete Mathematics (1 problems)
    - Combinatorics (1 problems)
  - Exponential functions and logarithms (6 problems)
    - Exponential functions and logarithms (other) (4 problems)
    - Exponential inequalities (1 problems)
  - Fractions (5 problems)
    - Simple fractions (3 problems)
  - Functional equations (12 problems)
  - Graphs and locus of points on the Cartesian plane (18 problems)
  - Inequalities (18 problems)
  - Mean values (54 problems)
  - Polynomials (34 problems)
    - Algebraic identities for polynomials (11 problems)
      - Factoring polynomials (6 problems)
      - Identical transformations (5 problems)
    - Cubic polynomials (1 problems)
    - Division of polynomials with remainders. The greatest common divisor (GCD) and the least common multiple (LCM) for polynomilas. (3 problems)
    - Polynomial remainder theorem (little Bezout's theorem). Factorisation. (2 problems)
    - Polynomials(other) (3 problems)
    - Polynomials with integer coefficients and integer values (3 problems)
    - Quadratic polynomials (11 problems)
      - Investigating the quadratic polynomial (4 problems)
      - Quadratic equations and systems of equations (2 problems)
      - Quadratic equations. Formula for the roots. (1 problems)
      - Quadratic equations. Vieta's rule. (2 problems)
    - Symmetrical polynomials (1 problems)
    - Vieta's Formulas (1 problems)
  - Rational functions (5 problems)
    - Identical transformations (2 problems)
    - Inequalities. Method of Intervals. (1 problems)
    - Rational equations (2 problems)
  - Sequences (66 problems)
    - Fibonacci (11 problems)
    - Progressions (12 problems)
      - Arithmetic progression (7 problems)
      - Geometric progression (5 problems)
    - Recurrent relations (30 problems)
      - Fibonacci numbers (10 problems)
      - Linear recurrent relations (6 problems)
      - Recurrent relations (other) (14 problems)
    - Sums of number sequences and difference series (2 problems)
  - Surds. Rational powers (12 problems)
    - Irrational equations. (7 problems)
    - Irrational inequalities. (3 problems)
    - Square roots (1 problems)
  - Trigonometry (19 problems)
    - Identical transformations (trigonometry) (3 problems)
    - Inverse trigonometric functions (2 problems)
    - Systems of trigonometric equations and inequalities (1 problems)
    - Trigonometrical equations (4 problems)
    - Trigonometrical inequalities (2 problems)
    - Trigonometry (other) (5 problems)
- Algebra and arithmetic (74 problems)
  - Algebraic geometry (5 problems)
    - Cryptography (5 problems)
  - Arithmetics. Mental maths (61 problems)
  - Rational and irrational numbers (1 problems)
- Arithmetic (5 problems)
- Arithmetic puzzles (3 problems)
- Calculus (220 problems)
  - Derivative (27 problems)
    - Calculating a derivative (1 problems)
    - Convexity and concavity (2 problems)
    - Derivative and extrema (2 problems)
    - Derivative and multiple roots (12 problems)
    - Derivative and tangent line (2 problems)
    - Increasing and decreasing. Researching functions (10 problems)
  - Functions of one variable. Continuity (41 problems)
    - Certain properties of a function and recurrence relations. (12 problems)
    - Continuity and compactness (1 problems)
    - Continuous functions (general properties) (4 problems)
    - Functions. Continuity (other) (5 problems)
    - Intermediate value theorem. Connectedness. (2 problems)
    - Monotonicity, boundedness (8 problems)
    - Periodicity and aperiodicity (9 problems)
  - Functions of several variables (4 problems)
  - Integral (3 problems)
    - Applications of integration (1 problems)
      - Area by integration (1 problems)
    - Calculating integrals (1 problems)
    - Existence of a definite integral (1 problems)
  - Number sequences (31 problems)
    - Boundedness, monotonicity (6 problems)
    - Limit of a sequence, convergence (12 problems)
    - Number e (2 problems)
    - Number sequences (other) (11 problems)
  - Real numbers (116 problems)
    - Countable and uncountable subsets (1 problems)
    - Integer and fractional parts. Archimedean property (86 problems)
    - Rational and irrational numbers (28 problems)
- Combinatorics (1 problems)
- Constructions (6 problems)
  - Dissections (2 problems)
  - Geometry (2 problems)
  - Mathematical logic (1 problems)
- Cryptography (9 problems)
  - Number theory (9 problems)
- Discrete Mathematics (848 problems)
  - Algorithm Theory (226 problems)
    - Balance puzzles (19 problems)
    - Cooperative algorithms (2 problems)
    - Game theory (105 problems)
      - Fun games (5 problems)
      - Game theory (other) (34 problems)
      - Symmetric strategies (22 problems)
      - Winning and loosing positions (37 problems)
    - Game Theory (1 problems)
    - Theory of algorithms (other) (92 problems)
  - Combinatorics (269 problems)
    - Colorings of the board (18 problems)
    - Discrete geometry (other) (2 problems)
    - Dissections, partitions, covers and tilings (57 problems)
      - Covers (17 problems)
      - Dissections with certain properties (8 problems)
      - Features of dissection pieces (3 problems)
      - Paritions of figures in segments (2 problems)
      - Tilings with ordinary and domino tiles (20 problems)
      - Various dissection problems (6 problems)
    - Geometry on grid paper (16 problems)
    - Integer lattices (14 problems)
      - Integer lattices (other) (12 problems)
      - Polygons and polyhedra with vertices in lattice points (2 problems)
    - Number Theory (1 problems)
    - Packing problems (2 problems)
    - Painting problems (33 problems)
    - Permutations (3 problems)
    - Systems of points and line segments (10 problems)
      - Center of mass (1 problems)
        
        Center of mass formula (1 problems)
      - Systems of points (1 problems)
      - Systems of points and line segments (counterexamples) (1 problems)
      - Systems of points and line segments (other) (3 problems)
      - Systems of points, lines, and line segments (3 problems)
  - Covers and tilings (2 problems)
    - Covers (1 problems)
  - Game Theory (5 problems)
  - Graph Theory (14 problems)
  - Mathematical logic (6 problems)
  - Set theory and logic (338 problems)
    - Linguistics (2 problems)
    - Logic and set theory (33 problems)
    - Mathematical logic (254 problems)
      - Mathematical logic (other) (135 problems)
      - Paradoxes (14 problems)
      - Puzzles (52 problems)
      - Weightings (1 problems)
    - Order relations (10 problems)
    - Set theory (34 problems)
      - Inclusion-exclusion principle (6 problems)
      - Set cardinality. One-to-one correspondence (1 problems)
      - Set theory (other) (2 problems)
      - Union, intersection, and set difference (4 problems)
      - Unusual constructions (2 problems)
- Dissections (16 problems)
  - Joke problems (1 problems)
  - Puzzles (1 problems)
- Fun Problems (175 problems)
  - Arithmetic Puzzles (4 problems)
  - Covers (1 problems)
  - Dissections (20 problems)
    - Dissections with certain properties (1 problems)
  - Joke Problems (33 problems)
  - Linguistics (1 problems)
  - Tilings (1 problems)
  - Word Problems (100 problems)
    - Combined labour (work) problems (1 problems)
    - Motion problems (7 problems)
    - Percentages and ratios (8 problems)
    - Solving problems by inequalities. Going through the cases. (5 problems)
    - Tables and tournaments (67 problems)
      - Chessboards and chess pieces (21 problems)
      - Number tables and its properties (13 problems)
      - Tables and tournaments (other) (22 problems)
      - Tournament tables (11 problems)
    - Tournament tables (1 problems)
- Geometry (456 problems)
  - Areas (39 problems)
  - Combinatorial geometry (1 problems)
  - Constructions (1 problems)
  - Dissections (2 problems)
  - Euclidean geometry (2 problems)
  - Geometric inequalities (4 problems)
    - Triangle inequality (4 problems)
  - Geometry (other) (3 problems)
  - Plane geometry (240 problems)
    - Area (44 problems)
      - Area of a circle, sector, segment (9 problems)
      - Area of a parallelogram (3 problems)
      - Area of a polygon (2 problems)
      - Area of a trapezium (2 problems)
      - Area of curvilinear shapes (1 problems)
      - Area of quadrilateral (2 problems)
      - Area (other) (7 problems)
      - Area ratio (8 problems)
        
        Area ratio of similar figures (1 problems)
        
        Area ratio of triangles with common base and height (1 problems)
        
        Area ratio (other) (4 problems)
      - Finding the area by rearrangment (5 problems)
      - Triangle area (5 problems)
    - Circles (41 problems)
      - Central angle. Arc length and circumference (4 problems)
      - Circle, sector, segment, etc (1 problems)
      - Circles (other) (4 problems)
      - Connection between the arc length, the angle, and the chord (4 problems)
      - Diameter, chords, and secants (2 problems)
        
        Chords and secants (other) (2 problems)
      - Four points on a circle (2 problems)
      - Inscribed angle (10 problems)
        
        An angle between a secant and a chord (1 problems)
        
        An angle between two chords or two secants (3 problems)
        
        Angles supported by congruent arcs and chords (4 problems)
        
        Inscribed angle (other) (2 problems)
      - Tangent lines and tangent circles (4 problems)
        
        Tangent circles (2 problems)
        
        Tangent lines to circles (2 problems)
    - Constructions (13 problems)
      - Apollonian circles (1 problems)
      - Circles (construction) (1 problems)
      - Construction of a triangle given different points (2 problems)
      - Constructions (other) (6 problems)
      - Similar triangles and homothetic transformations (construction) (1 problems)
      - Unusual constructions (1 problems)
        
        Constructing with right angle (1 problems)
    - Convex and non-convex figures (4 problems)
      - Convex polygons (4 problems)
    - Dissections (1 problems)
    - Geometrical inequalities (34 problems)
      - Area inequalities (2 problems)
      - Geometric inequalities (other) (4 problems)
      - Inequalities in a triangle (10 problems)
        
        A perpendicular distance is the shortest. Inequalities for right triangles (1 problems)
        
        A segment inside the triangle is smaller than the largest side ,Problems (5 problems)
        
        Geometry (2 problems)
        
        Plane geometry (2 problems)
        
        Polygons (1 problems)
        
        Hexagons (1 problems)
        
        Triangles (1 problems)
        
        Types of triangles (1 problems)
        
        Right-angled triangles (1 problems)
        
        Pythagorean theorem and its converse (1 problems)
        
        Methods (2 problems)
        
        Pigeonhole principle (2 problems)
        
        Pigeonhole principle (angles and lengths) (1 problems)
        
        Pigeonhole principle (finite number of poits, lines etc.) (1 problems)
        
        Probability and statistics (1 problems)
        
        Probability theory (1 problems)
        
        Discrete distribution (1 problems)
        
        Inequalities in a triangle involving heights (1 problems)
        
        Inequalities in a triangle involving medians (1 problems)
        
        The longest side of the triangle lies opposite the largest angle (2 problems)
      - Polygons (inequalities) (2 problems)
      - Quadrilaterals (inequalities) (4 problems)
      - Triangle inequality (11 problems)
        
        Triangle inequality algebra problems (3 problems)
        
        Triangle inequality (other) (4 problems)
    - Lines, segments, rays, and angles (6 problems)
      - Measurement of segments and angles. Adjacent angles. (3 problems)
    - Locus (8 problems)
      - Line or line segment as a locus (1 problems)
      - Loci with nonzero area (1 problems)
      - Locus and auxiliary triangles (1 problems)
      - Locus (other) (4 problems)
    - Plane transformations (16 problems)
      - Isometry of the plane (3 problems)
        
        Rotations (3 problems)
      - Parallel translation (4 problems)
      - Projection onto a line (1 problems)
        
        Orthogonal projection (1 problems)
      - Similarity (7 problems)
        
        Homothetic transformations, rotations of homothetic transformations (7 problems)
    - Polygons (17 problems)
      - Inscribed and circumscribed polygons (1 problems)
      - Pentagons (1 problems)
      - Polygons (other) (6 problems)
      - Regular polygons (8 problems)
      - Sum of internal and external angles of polygons (1 problems)
    - Quadrilateral (14 problems)
      - Arbitrary quadrilaterals (1 problems)
      - Circumscribed quadrilaterals (1 problems)
      - Inscribed quadrilaterals (1 problems)
      - Parallelogram (9 problems)
        
        Special cases (8 problems)
        
        Rectangles and squares. Criteria and properties (8 problems)
      - Trapeziums (1 problems)
    - Triangles (48 problems)
      - Incircle and circumcircle of a triangle (1 problems)
      - Relationship of side lengths and angles of a triangle. Solving triangles. (6 problems)
        
        Ceva's theorem and Menelaus's theorem (1 problems)
        
        Law of sines (3 problems)
        
        Relations between sides and angles of a triangle (other) (2 problems)
      - Remarkable points and lines in a triangle (1 problems)
      - Similar triangles (16 problems)
        
        Auxiliary similar triangles (1 problems)
        
        The triangle generated by bases of two altitudes and a vertex (2 problems)
      - Sum of angles of a triangle. Exterior angle theorem (2 problems)
      - Types of triangles (17 problems)
        
        Equilateral triangle (4 problems)
        
        Right-angled triangles (13 problems)
        
        Pythagorean theorem and its converse (13 problems)
    - Vectors (4 problems)
      - Law of polygon of vectors (1 problems)
      - The sum and the difference of vectors, the multiplication of a vector by a scalar. Properties (2 problems)
    - Visual geometry (5 problems)
  - Regular polygons (2 problems)
  - Solid geometry (82 problems)
    - Convex solids (1 problems)
    - Lines and planes in space (3 problems)
    - Parallelepipeds (5 problems)
      - Special cases of parallelepipeds (5 problems)
        
        Cube (5 problems)
    - Polyhedra and polygons in space (1 problems)
    - Regular polyhedra (1 problems)
      - Main features and definitions of regular polyhedra (1 problems)
    - Skeletons of multifaceted figures (3 problems)
    - Space transformations (10 problems)
      - Isometry (7 problems)
        
        Isometry helps solve the problem (1 problems)
        
        Point symmetry (4 problems)
        
        Reflectional symmetry (5 problems)
      - Projection (2 problems)
        
        Parallel projection (1 problems)
        
        Orthographic projection (1 problems)
        
        Area and orthographic projection (1 problems)
        
        Projection helps solve the problem (1 problems)
      - Similarity (1 problems)
    - Spheres (2 problems)
      - Circle on a sphere (2 problems)
    - Tetrahedron and pyramid (2 problems)
      - Particular cases of tetrahedra (1 problems)
        
        Regular tetrahedron (1 problems)
      - Pyramid (1 problems)
        
        Square pyramid (1 problems)
    - Visual geometry in space (55 problems)
    - Volume (5 problems)
      - Ratio of volumes (2 problems)
      - Volume of a parallelepiped (1 problems)
      - Volume of a prism (1 problems)
      - Volume of a tetrahedron and pyramid (1 problems)
  - Triangle inequality (11 problems)
  - Triangles (4 problems)
  - Visual geometry in space (1 problems)
- Graph theory (28 problems)
- Mathematical logic (67 problems)
  - Invariants (18 problems)
  - Liars and Knights (1 problems)
  - Liars and knights (1 problems)
  - Logic and set theory (1 problems)
  - Weightings (1 problems)
- Methods (617 problems)
  - Algebraic methods (78 problems)
    - Counting in two ways (21 problems)
    - Iterations (5 problems)
    - Partitions into pairs and groups; bijections (22 problems)
    - Processes and operations (12 problems)
    - Proof by exhaustion (20 problems)
    - Symmetry and involutions (2 problems)
    - Working backwards (1 problems)
  - Colouring (30 problems)
    - Chessboard colouring (10 problems)
    - Colouring (other) (7 problems)
  - Examples and counterexamples (3 problems)
  - Examples and counterexamples. Constructive proofs (68 problems)
  - Extremal principle (18 problems)
    - Extremal principle (other) (12 problems)
    - Smallest or largest distance (length) (1 problems)
    - Sorting in increasing (decreasing) order (4 problems)
  - Invariants and semi-invariants (35 problems)
    - Invariants (24 problems)
  - Mathematical induction (41 problems)
    - Mathematical induction in geometry (3 problems)
    - Mathematical induction (other) (11 problems)
  - Methods for solving problems with parameter (1 problems)
  - Methods from geometry (2 problems)
    - Geometric interpretations in algebra (2 problems)
  - Methods of calculus (3 problems)
    - Continuity considerations (2 problems)
    - Monotonicity and boundedness (1 problems)
  - Pigeonhole principle (405 problems)
    - Pigeonhole principle (angles and lengths) (61 problems)
    - Pigeonhole principle (area and volume) (13 problems)
    - Pigeonhole principle (finite number of poits, lines etc.) (49 problems)
    - Pigeonhole principle (other) (246 problems)
  - Proof by contradiction (58 problems)
  - Trial and error (2 problems)
- Number theory (31 problems)
  - Numeral systems (13 problems)
- Number Theory (363 problems)
  - Arithmetic (1 problems)
  - Arithmetic operations. Number identities (58 problems)
  - Divisibility (242 problems)
    - Arithmetic functions (4 problems)
    - Divisibility of a number. General properties (52 problems)
    - Divisibility rules (42 problems)
      - Divisibility (1 problems)
      - Divisibility tests for 2 and 4 (3 problems)
      - Divisibility tests for 3 and 9 (16 problems)
      - Divisibility tests (other) (1 problems)
    - Division with remainders. Arithmetic of remainders (58 problems)
      - Arithmetic of remainders (12 problems)
      - Division with remainder (37 problems)
      - Euler's theorem (4 problems)
      - Fermat's little theorem (1 problems)
    - Equations in integer numbers (13 problems)
    - Odd and even numbers (55 problems)
    - Prime numbers (3 problems)
    - Products and factorials (4 problems)
    - The fundamental theorm of arithmetic. Prime factorisation. (10 problems)
    - The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers (17 problems)
  - Equations in integer numbers (2 problems)
  - Number systems (1 problems)
    - Decimal number system (1 problems)
  - Number theory. Divisibility (1 problems)
    - Divisibility rules (1 problems)
  - Numeral systems (58 problems)
    - Binary number system (5 problems)
    - Decimal number system (51 problems)
    - Number systems (other) (1 problems)
- Number theory. Divisibility (1 problems)
- Numeral systems (1 problems)
- Pigeonhole principle (1 problems)
- Probability and statistics (142 problems)
  - Probability theory (132 problems)
    - Conditional probability (21 problems)
    - Continuous distribution (12 problems)
    - Discrete distribution (100 problems)
    - Probability theory (other) (14 problems)
  - Statistics (10 problems)
- Puzzles (1 problems)
- Topology (25 problems)
  - Knots (1 problems)
  - Torus (1 problems)
- Visual geometry (1 problems)
- Visual geometry in space (1 problems)