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
Yossi's mother told him: "All champions excel in their studies." Yossi replied: "I excel in my studies, therefore I am a champion!" Is Yossi's reasoning correct?
Sources:Topics:Logic -> Reasoning / Logic -
The Restless Fly
Cities A and B are 300 kilometers apart. Two cyclists start simultaneously from A and B, heading towards each other. Their speeds are constant and equal to `30` km/h and `20` km/h, respectively. At the same moment, a fly departs from city A, flying at a speed of `100` km/h. The fly overtakes the first cyclist and flies until it meets the cyclist who left from city B. The moment the fly meets the cyclist, it immediately turns around and flies back until it meets the first cyclist again, and then it turns around again, and so on, until the cyclists meet. How many kilometers did the fly travel?
Topics:Arithmetic Algebra -> Equations Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems -
Question
At `12:00`, the two hands of a clock coincide. In exactly how much time will they coincide again? How many times within a 24-hour period will they coincide?
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Question
How long will it take a train traveling at a speed of `60` km/h and with a length of `50` meters to pass through a tunnel with a length of `50` meters?
Topics:Arithmetic Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems Units of Measurement -
Playing with Chocolate
Yossi and Danny are playing the following game. They have a chocolate bar of size `5xx7` squares, which is placed on a table. Each player, in turn, breaks one of the pieces of chocolate on the table along a straight line and returns the resulting pieces to the table. That is, in the first turn, the original chocolate bar is broken, and in the following turns, one piece is chosen from the pieces created up to that point and broken. It is only possible to break along the lines of the squares, and each break is from edge to edge. The player who cannot make a move loses. The winner eats all the chocolate.
Which of the two can guarantee a victory for themselves, if Yossi makes the first move?
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Game with Piles of Stones
Two players are playing the following game. On the table are three piles of stones. The first pile has `10` stones, the second – `15`, and the third – `20`. Each player, in their turn, chooses one of the piles currently on the table and divides it into two smaller piles. The player who cannot make a move loses.
Which of the two players has a winning strategy, and what is it?
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Question
On an island live liars and truth-tellers (truth-tellers always tell the truth, and liars always lie). What question should you ask a random person from the island's inhabitants to find out if they keep a crocodile as a pet?
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Question
A knight exited the square `a1` and, after several moves, returned to the same square.
Is it possible that the knight made an odd number of moves?
Topics:Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Colorings -> Chessboard Coloring -
Question
A knight moves from square `a1` to square `h8`. Is it possible that along the way it visited every square on the board exactly once?
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Question
Given an `M times N` matrix, where each cell contains a real number. It is known that the sum of the numbers in each row and each column is equal to `1`.
Prove that `M = N`.
Topics:Combinatorics -> Double Counting Logic -> Reasoning / Logic Algebra -> Inequalities -> Averages / Means