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• Question

  I went to the market to buy vegetables. I paid for tomatoes half of the money I had plus one shekel. After that, I paid for cucumbers half of the remaining money plus one shekel, and then I paid for onions half of the remaining money plus one shekel, and that used up all the money I had.

  What is the amount of money I came to the market with?

  Topics:

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

• Question - The Sophisticated Task

  Hannah has a basket with `13` apples. Hannah wants to know the total weight of all these apples. Rachel has a digital scale, and she is willing to help Hannah, but only under the following conditions: In each weighing, Hannah can weigh exactly `2` apples, and the number of weighings cannot exceed `8`.

  Explain how, under these conditions, Hannah can know the total weight of the apples.

  Topics:

  Combinatorics -> Double Counting Algebra -> Equations Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Algorithm Theory -> Weighing

• Question

  The number `100` is written on the board. Find a digit that satisfies the following condition:

  If we add it to the notation of the number written on the board once to the left and once to the right, we get a number that is divisible by `12`.

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  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Prime Numbers -> Prime Factorization

• Question

  A group of children went to an amusement park. It is known that the number of girls wearing hats is equal to the number of boys not wearing hats.

  Who are there more of in this group: children not wearing hats or girls?

  Topics:

  Logic -> Reasoning / Logic Set Theory

• Question

  On a circle, `2016` blue points and one red point are marked. Consider all possible polygons whose vertices are at these points. Which polygons are more numerous – those that contain the red point or those whose vertices are all blue?

  Topics:

  Combinatorics -> Matchings Geometry -> Plane Geometry -> Circles

• Question

  The sum of several numbers is equal to `1`. Is it possible that the sum of their squares is less than one-tenth?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences Arithmetic -> Fractions Algebra -> Inequalities -> Averages / Means Minimum and Maximum Problems / Optimization Problems

• Question

  Prove that there exist two powers of `2` such that their difference is divisible by `2017`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Combinatorics -> Pigeonhole Principle

• Question

  The following numbers are written on the board:

  `1,2^1,2^2,2^3,2^4,2^5`

  In one operation, you are allowed to erase two numbers written on the board and write their (non-negative) difference in their place.

  Is it possible to reach a state, through such operations, where only the number `15` is written on the board?

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  Topics:

  Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences

• Question

  In a class of `38` students, prove that there are four students who celebrate their birthday in the same month.

  Topics:

  Combinatorics -> Pigeonhole Principle

• Question

  It is known that every prime number has two divisors – `1` and the number itself. What numbers have exactly three divisors?

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization