Combinatorics, Case Analysis / Checking Cases, Processes / Procedures
Case analysis is a problem-solving technique where a problem is divided into several distinct, exhaustive cases. Each case is then analyzed separately to arrive at a solution or proof. Questions suitable for this involve conditions that naturally split the problem into different scenarios.
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The Units Digit
Miriam has eight cards with consecutive three-digit numbers. The units digit of the smallest number is 1,
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the units digit of the largest number is 8. Miriam arranged the cards in a row such that the first number is divisible by 2,
the second number is divisible by 3, the third number is divisible by 4, and so on until the eighth number which is divisible by 9.
What is the units digit of the number divisible by 7? -
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?Sources:Topics:Logic -> Reasoning / Logic Algorithm Theory -> Weighing Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems- Gillis Mathematical Olympiad, 2019-2020 Question 1
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Children's Clubs
In a kindergarten, there are three clubs: Judo, Agriculture, and Mathematics. Each child participates in exactly one club, and each club has at least one participant. The total number of children in the kindergarten is 32. On Friday, the kindergarten teacher gathered 6 children to tidy up the classroom. The teacher counted and found that exactly half of the Judo club members, a quarter of the Agriculture club members, and an eighth of the Mathematics club members volunteered for the task. How many students are in each club?
Sources:Topics:Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division- Gillis Mathematical Olympiad, 2018-2019 Question 1
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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?
Sources: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- Gillis Mathematical Olympiad, 2018-2019 Question 2
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Equality in Stages
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written on the board, and David is supposed to change them in stages. At each stage, David is allowed to choose two numbers and change them by 1, that is, to add 1 to both, subtract 1 from both, or add 1 to one and subtract 1 from the other.
Can David, after a number of stages, reach a situation where all the numbers on the board are equal? If so, show an example, and if not, explain your answer in detail.
Sources:Topics:Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures- Grossman Math Olympiad, 2017, Juniors Question 3
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The Hidden Number
Bar and Ilan each chose an integer between 1 and 30.
Ilan: Is your number double mine?
Bar: I don't know. Is your number double mine?
Ilan: I don't know. Is your number half of mine?
Bar: I don't know. Is your number half of mine?
Ilan: I don't know.
Bar: I know what your number is.
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Two Argue, the Third Takes
The entire class is in dispute!
42 think yes, 43 think maybe, and 36 think no.
When two people who think differently from each other meet, they both change their position to the third.
What is the minimum number of meetings that must take place until everyone agrees on the same position?
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Question
Eighth-grade students threw rubber balls into a box and then tried to guess how many balls had accumulated there. Five students tried to guess: 45, 41, 55, 50, 43, but no one guessed the exact amount. The guesses differed from the truth by 3, 7, 5, 7, and 2 balls (not necessarily in the same order as the guesses). How many balls were in the box?
Sources:Topics:Number Theory Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures- Beno Arbel Olympiad, 2017, Grade 8 Question 1
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Question
In this letter exercise, identical letters represent identical digits, different letters represent different digits, and asterisks represent any digit. Find all the digits.
`(ABCD)^2 = A B ** ** ** C D `Sources:Topics:Number Theory Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic- Beno Arbel Olympiad, 2017, Grade 8 Question 4
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Question
Find the largest natural number in which all digits are distinct, and if you look at every 3 consecutive digits, you get a number divisible by 13.
Sources:Topics:Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures- Beno Arbel Olympiad, 2017, Grade 8 Question 7