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Actual Circles
(1 problems)
Year 2022
(1 problems)
And angles
(0 problems)
Measurement of segments and angles. Adjacent angles.
(3 problems)
Three lines intersecting in the same point
(1 problems)
And secants
(0 problems)
Chords and secants (other)
(2 problems)
And skeletons
(0 problems)
Skeletons of multifaceted figures
(3 problems)
Covers and tilings
(0 problems)
Covers
(18 problems)
Dissections with certain properties
(8 problems)
Features of dissection pieces
(3 problems)
Paritions of figures in segments
(4 problems)
Partition rearrangement
(2 problems)
Plane dissected by lines
(5 problems)
Tilings with ordinary and domino tiles
(21 problems)
Various dissection problems
(6 problems)
Descriptions
(26 problems)
Empty Category
(63 problems)
Graph theory
(27 problems)
Problems
(2384 problems)
Algebra
(41 problems)
Inequalities
(14 problems)
Algebra and arithmetics
(759 problems)
Absolute value
(5 problems)
Absolute value of a number (other)
(1 problems)
Equations containing absoute values
(1 problems)
Inequalities containing absolute values
(3 problems)
Algebra and arithmetics (other)
(28 problems)
Functional equations
(11 problems)
Algebraic equations and systems of equations
(59 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 geometry
(6 problems)
Cryptography
(15 problems)
Number theory
(41 problems)
Divisibility
(1 problems)
Numeral systems
(14 problems)
Algebraic inequalities and systems of inequalities
(22 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)
Arithmetic operations. Number identities
(58 problems)
Arithmetics. Mental maths
(62 problems)
Complex numbers
(3 problems)
Cartesian form of a complex number, complex conjugate, Argand diagram, etc
(3 problems)
Exponential functions and logarithms
(6 problems)
Exponential functions and logarithms (other)
(5 problems)
Exponential inequalities
(1 problems)
Fractions
(3 problems)
Simple fractions
(3 problems)
Graphs and locus of points on the Cartesian plane
(18 problems)
Mean values
(55 problems)
Number systems
(57 problems)
Binary number system
(4 problems)
Decimal number system
(51 problems)
Number systems (other)
(1 problems)
Number theory. Divisibility
(230 problems)
Arithmetic functions
(5 problems)
Sum and number of divisors of a numberM
(4 problems)
Divisibility of a number. General properties
(53 problems)
Divisibility rules
(42 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
(54 problems)
Arithmetic of remainders
(12 problems)
Division with remainder
(32 problems)
Euler's theorem
(4 problems)
Fermat's little theorem
(1 problems)
Equations in integer numbers
(12 problems)
Number theory. Divisibility (other)
(1 problems)
Odd and even numbers
(59 problems)
Prime numbers
(2 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
(13 problems)
Polynomials
(35 problems)
Algebraic identities for polynomials
(12 problems)
Algebraic identities for polynomials (other)
(1 problems)
Factoring polynomials
(6 problems)
Identical transformations
(7 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)
Polynomilas of degree n has not more than n roots
(1 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)
Quadratic polynomilal (other)
(3 problems)
Symmetrical polynomials
(1 problems)
Vieta's Formulas
(1 problems)
Rational functions
(5 problems)
Inequalities. Method of Intervals.
(1 problems)
Rational equations
(2 problems)
Sequences
(43 problems)
Progressions
(12 problems)
Arithmetic progression
(7 problems)
Geometric progression
(5 problems)
Recurrent relations
(22 problems)
Fibonacci numbers
(2 problems)
Linear recurrent relations
(6 problems)
Recurrent relations (other)
(14 problems)
Sequnces
(5 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)
Square roots (other)
(1 problems)
Surds. Rational powers (other)
(1 problems)
Trigonometry
(18 problems)
Identical transformations ( trigonomery)
(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)
Word problems
(94 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
(69 problems)
Chessboards and chess pieces
(22 problems)
Number tables and its properties
(13 problems)
Tables and tournaments (other)
(22 problems)
Tournament tables
(12 problems)
Word problems (other)
(3 problems)
Arithmetic
(5 problems)
Arithmetic puzzles
(3 problems)
Calculus
(221 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
(117 problems)
Countable and uncountable subsets
(1 problems)
Integer and fractional parts. Archimedean property
(87 problems)
Rational and irrational numbers
(29 problems)
Combinatorics
(226 problems)
Colorings of the board
(15 problems)
Discrete geometry (other)
(3 problems)
Dissections
(20 problems)
Joke problems
(1 problems)
Dissections, partitions, covers and tilings
(78 problems)
Dissections (other)
(16 problems)
Euler characteristic
(3 problems)
Geometry on grid paper
(12 problems)
Integer lattices
(14 problems)
Integer lattices (other)
(12 problems)
Polygons and polyhedra with vertices in lattice points
(2 problems)
Packing problems
(2 problems)
Painting problems
(33 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)
Cryptography
(1 problems)
Number theory
(1 problems)
Euclidean geometry
(1 problems)
Examples and counterexamples
(3 problems)
Geometry
(403 problems)
Areas
(36 problems)
Combinatorial geometry
(1 problems)
Eucliden geometry
(2 problems)
Geometry (other)
(3 problems)
Plane geometry
(214 problems)
Area
(42 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
(3 problems)
Area ratio (other)
(4 problems)
Auxiliary area. The area helps solve the problem
(1 problems)
Finding the area by rearrangment
(5 problems)
The area of a composite figure equals to the sum of the areas it is divided into
(5 problems)
Triangle area
(5 problems)
Triangle area (via base and height)
(2 problems)
Circles
(33 problems)
Central angle. Arc length and circumference
(5 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)
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
(5 problems)
Tangent circles
(2 problems)
Tangent lines to circles
(3 problems)
Properties and criteria of a secant
(1 problems)
Tangent-secant theorem. Intersecting secant theorem.
(1 problems)
Two tangent lines to a circle, intersecting at a particular point
(1 problems)
Constructions
(20 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)
Convex and non-convex figures
(4 problems)
Convex polygons
(4 problems)
Geometrical inequalities
(34 problems)
Area. A figure lies inside another figure
(1 problems)
Area inequalities
(2 problems)
Geometric inequalities (other)
(4 problems)
Inequalities in a triangle
(10 problems)
A perpendicular distance is the shortest. Ineaqualities for right triangles
(1 problems)
A segment inside the triangle is smaller than the largest side ,Problems
(5 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)
The area of a triangle is less than half the product of two sides
(1 problems)
Triangle inequality
(11 problems)
Triangle inequality algebra problems
(3 problems)
Triangle inequality (other)
(4 problems)
Lines, segments, rays, and angles
(7 problems)
Lines, segments, rays, and angles (other)
(1 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)
(5 problems)
Plane transformations
(15 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
(8 problems)
Homothetic transformations, rotations of homothetic transformations
(7 problems)
Polygons
(16 problems)
Hexagons
(1 problems)
Inscribed and circumscribed polygons
(1 problems)
Pentagons
(1 problems)
Polygons (other)
(6 problems)
Regular polygons
(7 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)
Incribed quadrilaterals (other)
(1 problems)
Parallelogram
(9 problems)
Special cases
(8 problems)
Rectangles and squares. Criteria and properties
(8 problems)
Quadrlaterals (other)
(1 problems)
Trapeziums
(1 problems)
Trapeziums (other)
(1 problems)
Triangles
(36 problems)
Incirlce 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)
Remarkable points and lines in a triangle (other)
(1 problems)
Similar triangles
(5 problems)
Auxiliary simialr triangles
(2 problems)
Similar triangles (other)
(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)
Triangles (other)
(6 problems)
Types of triangles
(18 problems)
Equilateral triangle
(4 problems)
Right-angled triangles
(14 problems)
Pythagorean theorem and its converse
(14 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
(6 problems)
Solid geometry
(79 problems)
Convex solids
(1 problems)
Cross-sections, nets, and skeletons
(3 problems)
Lines and planes in space
(3 problems)
Lines and planes in space (other)
(2 problems)
Parallelepipeds
(5 problems)
Special cases of parallelepipeds
(5 problems)
Cube
(5 problems)
Polyhedra and polygons in space
(1 problems)
Polyhedra and polygons in space (other)
(1 problems)
Regular polyhedra
(1 problems)
Main features and definitions of regular polyhedra
(1 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)
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
(53 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
(9 problems)
Grid geometry
(1 problems)
Mathematical logic
(6 problems)
MAthematical logic
(1 problems)
Mathematical logic and set theory
(9 problems)
Set theory
(9 problems)
Methods
(578 problems)
Algebraic methods
(79 problems)
Change of variables
(0 problems)
Counting in two ways
(21 problems)
Iterations
(5 problems)
Partitions into pairs and groups; bijections
(22 problems)
Processes and operations
(13 problems)
Proof by exhaustion
(20 problems)
Symmetry and involutions
(2 problems)
Working backwards
(1 problems)
Colouring
(17 problems)
Chessboard colouring
(10 problems)
Colouring (other)
(7 problems)
Examples and counterexamples. Constructive proofs
(69 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
(32 problems)
Invariants
(38 problems)
Invariants and semi-invariants (other)
(1 problems)
Mathematical induction
(32 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
(395 problems)
Pigeonhole principle (angles and lengths)
(62 problems)
Pigeonhole principle (area and volume)
(13 problems)
Pigeonhole principle (finite number of poits, lines etc.)
(46 problems)
Pigeonhole principle (other)
(249 problems)
Proof by contradiction
(58 problems)
Trial and error
(2 problems)
Paintings
(1 problems)
Probability and statistics
(143 problems)
Probability theory
(133 problems)
Conditional probability
(21 problems)
Continuous distribution
(12 problems)
Discrete distribution
(101 problems)
Probability theory (other)
(15 problems)
Statistics
(10 problems)
Set theory and logic
(541 problems)
Joke problem
(22 problems)
Linguistics
(3 problems)
Logic and set theory
(35 problems)
Mathematical logic
(326 problems)
Algorithm theory
(1 problems)
Liars and Knights
(1 problems)
Mathematical logic (other)
(136 problems)
Paradoxes
(14 problems)
Puzzles
(54 problems)
Weightings
(2 problems)
Order relations
(10 problems)
Set theory
(17 problems)
Inclusion-exclusion principle
(6 problems)
Set cardinality. One-to-one correspondence
(2 problems)
Set theory (other)
(2 problems)
Union, intersection, and set difference
(4 problems)
Unusual constructions
(3 problems)
Constructing with right angle
(1 problems)
Theory of algotithms
(200 problems)
Balance puzzles
(19 problems)
Cooperative algorithms
(2 problems)
Game theory
(84 problems)
Fun games
(5 problems)
Game theory (other)
(34 problems)
Symmetric strategies
(22 problems)
Winning and loosing positions
(25 problems)
Theory of algorithms (other)
(92 problems)
Tilings
(1 problems)
Rotations of homothetic transformations
(0 problems)
A homothetic transformation helps solve the problem
(1 problems)
Homothetic circles
(2 problems)
Uncategorized
(17 problems)