Algebra, Inequalities

Inequalities are statements comparing two expressions using symbols like <, >, \le, \ge. This topic involves solving inequalities (linear, quadratic, absolute value), proving algebraic or geometric inequalities (e.g., AM-GM), and understanding their properties.

• Question

  Let the sides of a triangle be a, b, c and the lengths of the corresponding medians be `m_a , m_b, m_c`. Show that

  `sum_{cyc} m_a / a >= {3( m_a + m_b + m_c)} /{a + b + c}`

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  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 1

• Question

  Given natural numbers m, n such that `m/n <= sqrt 23`, prove that `m/n+3/{mn} <= sqrt 23`

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Algebra -> Inequalities Proof and Example -> Proof by Contradiction Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 4

• Cyclic Quadrilaterals

  Given two triangles ACE, BDF
  intersecting at 6 points: G,H,I,J,K,L
  as shown in the figure. It is given that in each of the quadrilaterals
  EFGI, DELH, CDKG, BCJL, ABIK a circle can be inscribed.
  Is it possible that a circle can also be inscribed in quadrilateral FAHJ?
  

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  Topics:

  Geometry -> Solid Geometry / Geometry in Space Geometry -> Plane Geometry -> Circles Algebra -> Equations Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 5

• The Number Circle

  The numbers 1 through 6 are written in a circle in order, as shown in the figure.
  In each step, Lior chooses a number a in the circle whose neighbors are b and c, and replaces
  it with the number `(bc)/a`.
  Can Lior reach a state where the product of the numbers in the circle is greater than `10^100`
  (a) in 100 steps.
  (b) in 110 steps

  

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  Topics:

  Number Theory Algebra -> Inequalities Algebra -> Sequences
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 6

• Inscribed Circle in a Triangle

  Inside a triangle there is a point P, whose distances from the lines containing the sides of the triangle are `d_a,d_b,d_c`. Let R denote the radius of the circumscribed circle of the triangle and r the radius of the inscribed circle in the triangle. Show that `sqrt(d_a)+sqrt(d_b)+sqrt(d_3)<= sqrt (2R+5r) `.

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  Topics:

  Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Circles Algebra -> Inequalities
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 7

• Triangle Side Lengths

  Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that

  `(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`

  Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Grossman Math Olympiad, 2006 Question 5

• Polynomial with Integer Coefficients

  Let `p(x)` be a polynomial with integer coefficients such that `p(-2006) < p(2006)=2005`. Prove that `p(-2006)<=-2007`.
   

  Topics:

  Number Theory Algebra -> Algebraic Techniques Algebra -> Inequalities
  Sources:
  • Grossman Math Olympiad, 2006 Question 6

• Question

  Shuki walked for 3.5 hours. In each hour-long period, he traveled 5 km. Does it follow that Shuki's average speed during this entire time was 5 km per hour?

  Topics:

  Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means

• Question

  Can you find `35` integers whose average is equal to `6.35`?

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  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Inequalities -> Averages / Means Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• Question

  The sum of several numbers is equal to `1`. Is it possible that the sum of their squares is less than one-tenth?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences Arithmetic -> Fractions Algebra -> Inequalities -> Averages / Means Minimum and Maximum Problems / Optimization Problems
  Sources:
  • problems.ru web site