Algebra

Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.

• The Confrontation

  Ehud and Benjamin are participating in a public confrontation. Each presents, in turn, a question to their opponent. Ehud is chosen to be the first to present a question. A "tricky question" is a question for which the opponent has no answer. A contestant who manages to ask a tricky question immediately wins the confrontation. The probability of each of the two contestants finding (in turn) a tricky question is exactly 1/2. It is also known that there is no dependence between the questions. What is the probability of Ehud winning the confrontation?

  Topics:

  Algebra -> Sequences Probability Theory

• The Confrontation

  Ehud and Benjamin are participating in a public debate. Each one presents a question to his opponent in turn. Ehud is chosen to be the first to present a question. A "tricky question" is a question that the opponent has no answer to. A contestant who manages to ask a tricky question immediately wins the debate. The probability of each of the two contestants finding (in turn) a tricky question is exactly 1/2. Also, it is known that there is no dependence between the questions. What is the probability of Ehud winning the debate?

  Topics:

  Algebra -> Sequences Probability Theory
  Sources:
  • Grossman Math Olympiad, 2006 Question 1

• Divisible by 13

  All natural numbers from 1 to 2006 are written on a sheet of paper, and a series of operations is performed as described below. At each step, any number of numbers are deleted from the list and their sum is denoted by S. Instead of the deleted numbers, a single number is added, which is the remainder obtained from the division of S by 13. After some number of such steps, only two numbers remain on the paper. One of them is 100. Find the other number.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence
  Sources:
  • Grossman Math Olympiad, 2006 Question 2

• The 1224th Digit

  We write the natural numbers in order, one after the other from left to right:

  1234567891011...

  Note, for example, that the digit in the 10th place is 1 and the digit in the 11th place is 0.

  Continuing with this writing as much as needed...

  Which digit will be in the 1224th place in the sequence?

  Topics:

  Number Theory Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 1

• Maximum Length Game

  The game begins with a row of 42 coins, all showing 'Heads' up.

  Each player, in their turn, chooses one of the coins showing 'Heads' up - flips the chosen coin, and also the coin immediately to its right (if one exists).

  If you manage to make all the coins show 'Tails' up on your turn, you win the game!

  At most, after how many turns will the game end?

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  Topics:

  Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 2

• Triangle Side Lengths

  Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that

  `(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`

  Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Grossman Math Olympiad, 2006 Question 5

• Polynomial with Integer Coefficients

  Let `p(x)` be a polynomial with integer coefficients such that `p(-2006) < p(2006)=2005`. Prove that `p(-2006)<=-2007`.
   

  Topics:

  Number Theory Algebra -> Algebraic Techniques Algebra -> Inequalities
  Sources:
  • Grossman Math Olympiad, 2006 Question 6

• How a Wheel Turns

  All 6 wheels in the diagram rotate as they touch each other without slipping. The diameter of the leftmost wheel is 15.7 cm and it makes 12 revolutions per minute.

  It is known that the smallest wheel makes one revolution per second.

  What is the diameter of the smallest wheel?

  

  Topics:

  Arithmetic Geometry -> Plane Geometry -> Circles Algebra -> Word Problems
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 7

• 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?

  Topics:

  Number Theory Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 1

• Question

  The numbers from 1 to `10^9` (inclusive) are written on the board. The numbers divisible by 3 are written in red, and the rest of the numbers are written in blue. The sum of all the red numbers is equal to `X`, and the sum of all the blue numbers is equal to `Y`. Which number is larger, `2X` or `Y`, and by how much?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Number Theory -> Division
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 2