Logic
Logic is the study of reasoning and valid inference. It involves analyzing statements, arguments, and deductive processes. Questions may include solving logic puzzles, evaluating the truth of compound statements, using truth tables, and identifying logical fallacies.
Reasoning / Logic Truth-tellers and Liars Problems-
Question
In the following addition problem, different shapes replace different digits, and identical shapes replace identical digits:
`triangle square triangle square triangle+ square triangle square triangle square = o+ triangle triangle triangle triangle o+`
What is the result of the addition?
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Colorful Street
Along the street are 16 houses, in red, blue, and green. There is at least one house of each color. No two adjacent houses are of the same color.
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Between any two blue houses there is a red house. Between any two green houses there is a blue house and a red house.
What is the largest possible number of green houses?
Note: The street is straight, all houses are located on one side of the street. -
The Truth-Teller and Liar Survey
13 truth-tellers and 12 liars participated in a survey. In the survey, each participant was asked about every other participant (including themselves) whether they were a truth-teller. How many "yes" answers were received in the survey in total?
Sources:Topics:Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems -
Pairwise Relatively Prime Composite Numbers
Yossi writes two-digit composite numbers on the board. He wants all the numbers written on the board to be pairwise relatively prime.
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What is the maximum number of integers Yossi can write on the board?
Note: Integers are called relatively prime if they have no common factors other than 1. -
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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Hexagonal Tiling
Given two types of tiles. The shape of each tile of the first type is a regular hexagon with a side of length 1. The shape of each tile of the second type is a regular hexagon with a side of length 2. An unlimited supply of tiles of each type is given. Is it possible to tile the entire plane using these tiles, using both types of tiles?
Sources:Topics:Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation- Grossman Math Olympiad, 2006 Question 4