Beno Arbel Olympiad, 2013, Grade 7

• Question 1

  `a,b` are two distinct natural numbers. The sum of the divisors of each is equal to the same natural number `n`. What is the smallest possible value of `n`?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization

• Question 2 - Coins and Cakes

  Miriam has coins of two shekels and of five shekels.

  If she pays only with two-shekel coins, she will be 60 shekels short of buying 4 cakes.

  If she pays only with five-shekel coins, she will be 60 shekels short of buying 5 cakes.

  In total, she is 60 shekels short of buying 6 cakes. How much does a cake cost?

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  Topics:

  Algebra -> Equations Algebra -> Word Problems

• Question 3

  Given a line `l` and two points `A, B` at different distances from the line. Find the point `C` on the line such that the difference between the lengths of the segments `AC`, `AB` is maximal.

  Topics:

  Geometry -> Plane Geometry -> Triangle Inequality Minimum and Maximum Problems / Optimization Problems

• Question 4 - A mistake in the exercise

  Prove that there is an error in the following multiplication problem:

  \(\begin{array}& & & * & * & * & 2 & 7 \\ \times & & & & & * & * \\ \hline & * & * & * & * & * & 6 \\ + & * & * & * & * & * & \\ \hline & * & * & * & * & 4 & 6 \end{array}\)

  Topics:

  Arithmetic Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Question 5 - 5 Degrees on a Clock

  At what time is there a 5-degree angle between the clock hands?

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation

• Question 6

  How many solutions in natural numbers are there to the equation `(2013 - x)(2013-y)=2013^2`?

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Algebra -> Equations -> Diophantine Equations