Math Problem Statement

The image contains a math exam paper with four problems. The problems involve algebraic expressions, solving equations, and geometric concepts. The tasks range from calculating expressions, expanding polynomials, proving algebraic identities, finding integer solutions for equations, to solving geometric problems in triangles and squares.

Solution

This image appears to be a math exam paper written in Vietnamese. It consists of four problems. Here's a quick overview of the content:

Problem 1 (4 points):

  1. Calculate the value of the expression: A=63247221521052A = \frac{63^2 - 47^2}{215^2 - 105^2}
  2. Calculate the value of the expression: B=x650x5+50x450x3+50x250x+50B = x^6 - 50x^5 + 50x^4 - 50x^3 + 50x^2 - 50x + 50 at x=49x = 49

Problem 2 (4 points):

  1. Expand the expression: x32x2+xxy2x^3 - 2x^2 + x - xy^2
  2. Find nn such that n2+2n+20n^2 + 2n + 20 is a perfect square.

Problem 3 (5 points):

  1. Show that for three real numbers aa, bb, and cc such that abc=1abc = 1, the following holds: aab+1+bbc+1+cca+1=1\frac{a}{ab+1} + \frac{b}{bc+1} + \frac{c}{ca+1} = 1
  2. Show that a+b+ca + b + c is divisible by 6 when a3+b3+c3=a+b+2024ca^3 + b^3 + c^3 = a + b + 2024c.
  3. Find all integer pairs (x,y)(x, y) that satisfy: x2xy2022x+2023y2024=0x^2 - xy - 2022x + 2023y - 2024 = 0

Problem 4 (6 points):

  1. In the square ABCDABCD, the sides ABAB, BCBC, CDCD, and DADA are extended to points MM, NN, PP, and QQ such that AM=BN=CP=DQAM = BN = CP = DQ. Prove that AM=BN=CP=DQAM = BN = CP = DQ.
  2. Calculate the area of triangle MNPQMNPQ.
  3. Prove the following geometric relations in triangle ABCABC (with AB<ACAB < AC):
    • II is the incenter of triangle ABCABC.
    • A line passing through II and perpendicular to BCBC intersects ABAB at points EE and FF.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometric Proofs
Polynomials
Systems of Equations

Formulas

A = (63^2 - 47^2) / (215^2 - 105^2)
B = x^6 - 50x^5 + 50x^4 - 50x^3 + 50x^2 - 50x + 50
x^3 - 2x^2 + x - xy^2
a / (ab+1) + b / (bc+1) + c / (ca+1) = 1
x^2 - xy - 2022x + 2023y - 2024 = 0

Theorems

Properties of Polynomials
Perfect Square Theorem
Geometric Relations in Triangles and Squares

Suitable Grade Level

Grades 8-10