https://r2.erweima.ai/mathbot/image/1718361512685mbd6l572.docx

Here are the detailed solutions for some of the questions from the provided document:

### Question 17: Calculation
**Problem:**
\[ \text{Calculate } \frac{1}{2} + \frac{1}{3} + \frac{1}{6} \]

**Solution:**
\[ \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1 \]

### Question 18: Solve the inequality
**Problem:**
\[ 2x - 1 < 3 \quad \text{and} \quad x + 2 > 1 \]

**Solution:**
1. Solve the first inequality:
\[ 2x - 1 < 3 \]
\[ 2x < 4 \]
\[ x < 2 \]

2. Solve the second inequality:
\[ x + 2 > 1 \]
\[ x > -1 \]

Therefore, the solution is:
\[ -1 < x < 2 \]

### Question 19: Algebraic Expression
**Problem:**
\[ \text{If } a = 2 \text{ and } b = 3, \text{ find the value of } a^2 + b^2 \]

**Solution:**
\[ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \]

### Question 20: Geometric Construction and Proof
**Problem:**
Given a right-angled triangle \( \triangle ABC \) with \( \angle ABC = 90^\circ \), \( AB = 3 \), and \( BC = 4 \). Find \( AC \).

**Solution:**
Using the Pythagorean theorem:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Question 21: Proof of a Quadrilateral
**Problem:**
Prove that a given quadrilateral is a rectangle if its diagonals bisect each other and are equal.

**Solution:**
If a quadrilateral's diagonals bisect each other, it is a parallelogram. If the diagonals are also equal, then it is a rectangle. Thus, by showing that the diagonals bisect each other and are equal, we prove that the quadrilateral is a rectangle.

### Do you have any questions or need further details on any specific question? 

Here are five numbered questions you can ask:

1. Can you solve a specific algebraic equation for me?
2. How do you find the area of a triangle given its vertices' coordinates?
3. Can you explain the concept of limits in calculus?
4. What are the steps to solve a quadratic equation using the quadratic formula?
5. How do you find the derivative of a trigonometric function?

**Tip:** When solving geometry problems, always look for properties and theorems related to shapes, such as the Pythagorean theorem for triangles or properties of parallelograms and circles. This can simplify complex problems significantly.

Explore detailed solutions to selected questions from the 2020 Beijing City Junior High School Math Exam. Topics covered include arithmetic calculations, algebraic expressions, geometric constructions, and properties of quadrilaterals.

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