Problem solving strategies 01

Problem solving strategies 01

By Ngo Van Minh

​P1. Given X={√1, √2, √3,...,√2022}. Prove that out of 90 numbers in set X we can always choose 2 numbers a and b such that |a-b| ≤1/2

P2. Suppose the positive integer n is odd. First Giang writes the numbers 1, 2,..., 2n on the blackboard. Then he picks any two numbers a, b, erases them, and writes, instead, |a − b|. Prove that an odd number will remain at the end

P3. A circle is divided into six sectors. Then the numbers 1, 0, 1, 0, 0, 0 are written into the sectors counterclockwise, say). You may increase two neighboring numbers by 1. Is it possible to equalize all numbers by a sequence of such steps?

P3. Solution:

​Suppose a1,...,a6 are the numbers currently on the sectors.

Set: I=​a1 −a2 +a3 −a4 +a5 −a6

Then I is an invariant. Initially I = 2.

The goal I = 0 cannot be reached.

P3. Solution:

​Suppose a1,...,a6 are the numbers currently on the sectors.

Set: I=​a1 −a2 +a3 −a4 +a5 −a6

Then I is an invariant. Initially I = 2.

The goal I = 0 cannot be reached.

P4. Let d(n) be the digital sum of n ∈ N. Solve:

a) n + d(n) + d(d(n)) = 1997.

b) n + d(n) + d(d(n)) = 2022.

P5. A rectangular floor is covered by 2×2 and 1×4 tiles. One tile got smashed. There is a tile of the other kind available. Show that the floor cannot be covered by rearranging the tiles.

P4. Let d(n) be the digital sum of n ∈ N. Solve:

a) n + d(n) + d(d(n)) = 1997.

b) n + d(n) + d(d(n)) = 2022.

P5. Is it possible to form a rectangle with the five tetrominoes in the figure?

P6. Show that a 10 × 10 board cannot be covered by 25 straight tetrominoes 1x4

P7. Alice, Bob and Charlie solved 100 math problems together. Each of them solved 60 problems. Let’s call a problem difficult if only one person solved it, and easy if all three people solved it. How different is the number of difficult problems from the number of easy ones?

P8. There are 27 students in the class. Each of the students in the class is engaged in no more than two extracurricular activity clubs, and for every two students there is a club in which they joint together. Prove that there is a club with at least 18 students.

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