Minimum and Maximum Problems / Optimization Problems

These problems, also known as optimization problems, involve finding the smallest (minimum) or largest (maximum) value of a quantity or function under given constraints. Techniques can range from algebraic inequalities, geometric reasoning, to calculus (if applicable).

• Question

  Given a sheet of paper of size `10×10` cm. Can you cut out a number of circles from this sheet such that the sum of their diameters is greater than `5` meters?

  Topics:

  Arithmetic Geometry -> Area Calculation Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• 9 Coins

  Given `9` coins that look identical. One of the coins is counterfeit, and its weight is less than the weight of a regular coin. How to find the counterfeit coin using two weighings on a balance scale without weights?

  Topics:

  Logic -> Reasoning / Logic Algorithm Theory -> Weighing Minimum and Maximum Problems / Optimization Problems

• 26 Coins

  Given `26` coins that look identical. One of the coins is counterfeit, and it weighs less than a regular coin. How can you find the counterfeit coin using three weighings on a balance scale without weights?

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  Topics:

  Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Algorithm Theory -> Weighing Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems

• 80 Coins

  Given `80` coins that look identical. One of the coins is counterfeit and weighs less than a regular coin. How can you find the counterfeit coin using four weighings on a balance scale without weights?

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  Topics:

  Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Algorithm Theory -> Weighing Minimum and Maximum Problems / Optimization Problems

• The King and the Corrupt Ministers

  The king of the magical land has `100` ministers. It is known that among any `10` ministers we choose, there is at least one corrupt minister. What is the minimum possible number of corrupt ministers in the magical land?

  Topics:

  Combinatorics -> Pigeonhole Principle Minimum and Maximum Problems / Optimization Problems

• Question

  What is the maximum number of chess kings that can be placed on an `8xx8` board such that they do not threaten each other?

  Topics:

  Combinatorics -> Pigeonhole Principle Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry

• Question

  What is the maximum number of rooks that can be placed on an `8xx8` board so that they do not threaten each other?

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems

• Question

  In a quadrilateral, the lengths of all diagonals and all sides are less than 1. Prove that the quadrilateral can be covered by a circle with a radius of 0.9.

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  Topics:

  Geometry -> Plane Geometry -> Circles Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1981-1982, Spring

• Squares

  You have many cardboard squares of sizes `1 times 1`, `2 times 2`, and `3 times 3`, and you must assemble them into a square of size `7 times 7`.
  What is the smallest possible number of squares you will need?

  Topics:

  Geometry Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 5

• Parentheses

  Add parentheses to make the result as large as possible:

  `10000-1000-100-10-1`

  Topics:

  Arithmetic Algebra -> Equations Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 1