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).

• The broken chain

  There are five segments of a broken chain, each segment having three links. Moses wants to repair the chain. What is the minimum number of links he needs to open and close again to join all these segments together?

  Note: The chain is not circular!

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems

• Question

  The "Sweet Math" candies are sold in boxes of `12` units, and the "Geometry with Nuts" candies – in boxes of `15` units.

  What is the minimum number of boxes that must be purchased so that there are equal quantities of candies of both types?

  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Minimum and Maximum Problems / Optimization Problems
  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?

  Show hint

  Log in here to receive the hint
  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

• Question

  The game takes place on an infinite plane. One player moves the wolf, and another player moves K sheep. After the wolf's move, one of the sheep makes a move, then the wolf again, and so on. In one move, the wolf or a sheep cannot move more than one meter in any direction. Can the wolf always catch at least one sheep, regardless of the initial positions?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Geometry -> Plane Geometry Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 2

• Question

  K friends simultaneously learn K pieces of news (one piece of news per friend). They begin to phone each other and exchange news. Each call lasts one hour. How long will it take for all friends to know all the news? Consider the cases:
  (a) (5 points) K=64
  (b) (10 points) K=55
  (c) (12 points) K=100
  (a) Answer

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 4

• Question

  Given a line `l` and two points `A, B` at different distances from the line. Find the point `C` on the line such that the difference between the lengths of the segments `AC`, `AB` is maximal.

  Topics:

  Geometry -> Plane Geometry -> Triangle Inequality Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 3

• Question

  A `29×29` table contains all integers from `1` to `29`, each appearing exactly `29` times. The sum of all numbers above the main diagonal is exactly three times greater than the sum of all numbers below the main diagonal. What number is written in the central cell of the table?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Combinatorics -> Number Tables Minimum and Maximum Problems / Optimization Problems

• Question

  A total of `21` children have `200` nuts. Prove that there exist two children who have the same number of nuts.

  Topics:

  Combinatorics -> Pigeonhole Principle Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems

• Question

  There are `30` students in a class. During a test, Pinchas made `13` mistakes, and the rest made fewer mistakes. Prove that there are three students who made the same number of mistakes.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems

• Question

  A square is divided into several convex polygons (more than `1`), each of which has a different number of sides. Prove that among these polygons there is a triangle.

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Combinatorics -> Graph Theory Geometry -> Plane Geometry -> Triangles Proof and Example -> Proof by Contradiction Geometry -> Solid Geometry / Geometry in Space -> Polyhedra Minimum and Maximum Problems / Optimization Problems