Beno Arbel Olympiad, 2017, Grade 8

• Question 1

  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

• Question 2

  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

• Question 3 - Paths in a Triangular Park

  In a park, there are 3 straight paths that form a triangle (there are no additional paths). The entrances to the park are at the midpoints of the paths, and a lamp hangs at each vertex of the triangle. From each entrance, the shortest walking distance along the park's paths to the lamp at the opposite vertex was measured. It turned out that 2 out of the 3 distances are equal to each other. Is the triangle necessarily isosceles?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Triangle Inequality

• Question 4

  In this letter exercise, identical letters represent identical digits, different letters represent different digits, and asterisks represent any digit. Find all the digits.
  `(ABCD)^2 = A B ** ** ** C D `

  Topics:

  Number Theory Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Question 5

  Given several heavy boxes that can be transported using 7 trucks with a capacity of 6 tons each, but cannot be transported using 6 trucks of the same type.
  a. Is it possible to transport them with 3 trucks with a capacity of 7 tons each?
  b. Is it possible to transport them with 3 trucks with a capacity of 10 tons each?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Minimum and Maximum Problems / Optimization Problems

• Question 6

  Inside a square ABCD with side length 1, a point E is marked, and outside the square, a point F is marked, such that triangles ABE and DAF are equilateral. Calculate the area of the pentagon CBEFD.
  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation

• Question 7

  Find the largest natural number in which all digits are distinct, and if you look at every 3 consecutive digits, you get a number divisible by 13.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 8

  There are 5778 extinguished lamps arranged at equal distances on a circle. Below each lamp is a button. Pressing a button changes the state of 4 lamps: the lamp next to the button, the next two lamps in the circle clockwise, and the lamp opposite the button (an extinguished lamp lights up when its state is changed, and a lit lamp is extinguished). What is the maximum number of lamps that can be lit simultaneously?
  

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings