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

  The numbers `1,2,3,4,5` are written at the vertices of a regular pentagon, with each number at exactly one vertex. A trio of vertices is called successful if it forms an isosceles triangle, where the number at its apex is greater than the numbers at the other two vertices, or where the number at its apex is smaller than the numbers at the other two vertices.

  Find the maximum number of successful trios that can exist.

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Symmetry Minimum and Maximum Problems / Optimization Problems

• More Game Cubes

  Aviv has game cubes, where two opposite faces of each are painted red and the rest are blue.
  Aviv glued together a `3 xx 3 xx 3` cube from the game cubes. Then his friend Kfir came and calculated the total red area on the surface of the large cube.
  What is the largest result Kfir can get?

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Combinatorics -> Combinatorial Geometry Minimum and Maximum Problems / Optimization Problems Combinatorics -> Colorings
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 6

• Two Hashes

  What is the maximum number of "domino" shapes (rectangles `1 times 2` or `2 times 1`) that can be placed inside the orange shape,
  such that they do not overlap and do not extend beyond the boundaries of the shape?

  

  Topics:

  Combinatorics -> Invariants Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems Combinatorics -> Colorings -> Chessboard Coloring Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 1

• Largest Perimeter

  A polygon with an area of 12 is drawn on grid paper, with all its sides passing through the grid lines. What is the largest possible perimeter of this polygon?

  Topics:

  Geometry -> Area Calculation Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 3-4 Question 5

• Weighing Coins

  Given are seven outwardly identical coins; four are genuine and three are counterfeit. The three counterfeit coins are of identical weight, as are the four genuine coins.
  It is known that a counterfeit coin is lighter than a genuine coin. In one weighing, you can select two groups of coins and determine which is lighter, or if they have the same weight.
  How many weighings are needed to locate at least one counterfeit coin?

  Topics:

  Logic -> Reasoning / Logic Algorithm Theory -> Weighing Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 1

• Drawing of a Relation

  Given a 5x5 grid divided into 1x1 squares. Two squares are considered related if they are in the same row or column, and the distance between their centers is 2 or 3.
  For example, in the drawing, all the squares related to the red square are marked in gray. Sammy receives a blank grid and wants to mark as many squares as possible such that no two of them are related. What is the maximum number of squares he can mark?
  

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Gillis Mathematical Olympiad, 2018-2019 Question 2

• Sets in the Plane

  A. Does there exist a set A in the plane such that its intersection with every circle contains exactly two points?

  B. Does there exist a set B in the plane such that its intersection with every circle of radius 1 contains exactly two points?

  Topics:

  Geometry -> Plane Geometry -> Circles Proof and Example -> Constructing an Example / Counterexample Set Theory Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Grossman Math Olympiad, 2006 Question 3

• Question

  Given several heavy boxes that can be transported using 7 trucks with a capacity of 6 tons each, but cannot be transported using 6 trucks of the same type.
  a. Is it possible to transport them with 3 trucks with a capacity of 7 tons each?
  b. Is it possible to transport them with 3 trucks with a capacity of 10 tons each?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 5

• THE MARKET WOMEN

  A number of market women sold their various products at a certain price per pound (different in every case), and each received the same amount—`2`s. `2`½d. What is the greatest number of women there could have been? The price per pound in every case must be such as could be paid in current money.
  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 18

• A SHOPPING PERPLEXITY

  Two ladies went into a shop where, through some curious eccentricity, no change was given, and made purchases amounting together to less than five shillings. "Do you know," said one lady, "I find I shall require no fewer than six current coins of the realm to pay for what I have bought." The other lady considered a moment, and then exclaimed: "By a peculiar coincidence, I am exactly in the same dilemma." "Then we will pay the two bills together." But, to their astonishment, they still required six coins. What is the smallest possible amount of their purchases—both different?
  Topics:

  Arithmetic Algebra -> Word Problems Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 24