Colourful Mathematics 10

Contents
Mathematical reasoning	10
1. What does it imply?	10
2. The pigeonhole principle	21
3. Arrangement (ordering) problems	29
4. Picking problems	32
Computing the root	36
1. Rational numbers, irrational numbers	36
2. The identities (laws) of the square root	40
3. Applying the identities (laws) of the square root	44
4. The nth root of numbers	50
5. The identities (laws) of the nth root .	53
The quadratic equation	60
1. The quadratic equation and function	60
2. The quadratic formula	64
3. The zero product form. The relation between the roots and the coefficients	69
4. Equations of higher degree which can be reduced to quadratic equations	74
5. Quadratic inequalities	80
6. Parametric quadratic equations (higher level courseware)	84
7. Equations involving square roots	90
8. Quadratic simultaneous equations	96
9. Arithmetic and geometric mean	101
10. Extreme value exercises (higher level courseware)	106
11. Problems leading to quadratic equations	110
Geometry	116
Widening the knowledge about circles	116
1. Reminder	116
2. The theorem of the central and inscribed/tangent-chord angles	117
3. The theorem of inscribed angles; the arc of viewing angles	121
4. The theorem of inscribed quadrilaterals (higher level courseware)	125
The similarity transformation and its applications	129
1. Parallel intercepting lines, parallel intercepting line segments (higher level courseware)	129
2. The angle bisector theorem (higher level courseware)	135
3. The transformation of central dilation (or homothety)	137
4. The similarity transformation	141
5. Similarity of figures; the simple cases of similar triangles	143
6. A few applications of similarity	147
7. The ratio of the area of similar planar figures	154
8. The ratio of the volume of similar solids	158
Trigonometric functions of acute angles	161
1. Determining distances with the help of similarity	161
2. Trigonometric functions of acute angles	164
3. Relations between the trigonometric functions of acute angles	168
4. Trigonometric functions of special angles	172
5. Determining several data of a triangle with the help of trigonometric functions	175
6. Calculations in the plane and in space with the help of trigonometric functions	180
Vectors	184
1. The concept of a vector; the sum and the difference of vectors; scalar multiplication of vectors (reminder)	184
2. Expressing vectors as the sum of components in different directions	188
3. Applying vectors in the plane and in space	194
4. Vectors in the coordinate system, the coordinates of a vector, operations with vectors given with coordinates	199
Trigonometric functions	204
1. The definition and the simple properties of the sine and the cosine function	204
2. The graph of the sine function	209
3. The graph of the cosine function, equations, inequalities	214
4. The tangent and the cotangent function	221
5. Compound exercises and applications .	228
6. Geometric applications	232
Calculation of probability	238
1. Events	238
2. Operations with events	243
3. Experiments, frequency, relative frequency, probability	248
4. The classical model of probability ..	251