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• Question - 8-5=7?

  Remove two matchsticks to make the following equation true:

  

  Topics:

  Puzzles and Rebuses -> Matchstick Puzzles

• Question

  Solve the equation:

  `(x + 2010)(x + 2011)(x + 2012) = (x + 2011)(x + 2012)(x + 2013) `

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Given the polynomial `P(n)=n^2+n+41`. Is it true that this polynomial yields prime numbers for all natural numbers `n`?

  Topics:

  Number Theory -> Prime Numbers Algebra Proof and Example -> Constructing an Example / Counterexample

• Question

  Find a two-digit number that is twice as large as the product of its digits.

  Topics:

  Number Theory Arithmetic Algebra -> Algebraic Techniques Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations

• 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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  Topics:

  Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems

• Question - 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?

  Topics:

  Combinatorics -> Invariants Combinatorics -> Game Theory Logic -> Reasoning / Logic

• Question

  Take any three-digit number and arbitrarily rearrange its digits. Prove that the difference between the original number and the new number is divisible by `3`. Is this also true for four-digit numbers?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9

• Question

  Is the number  `10^2016+8` divisible by `9`? Justify your answer!

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9

• Question - The Cookies

  Avi, Benny, and Danny's mother left a plate of cookies on the table in the morning. Avi woke up, ate one-third of the total number of cookies, and went to school. After that, Benny woke up, unaware that Avi had already eaten some, so he ate one-third of the remaining cookies and also went to school. After that, Danny woke up and ate one-third of the cookies that were left. When their mother returned home, she saw that there were `8` cookies left. How many cookies were there at the beginning?

  Topics:

  Arithmetic -> Fractions Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards

• 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?

  Topics:

  Logic -> Reasoning / Logic