There is a set of $2024$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card lies beautifully if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $150$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be? K.A.Sukhov
Problem
Source: Assara 2024 S1 (seniors)
Tags: combinatorics
CrazyInMath
24.11.2024 12:25
We just say a card is "beautiful" if it's lying beautifully, and "ugly" otherwise.
Claim: if a beautiful card is flipped, it become ugly.
Proof: if it's upper side is red, then when it's flipped it's underside would be red (and not blue), also it's upperside would be not red (as the two sides are different colors). if it's under side is blue, then when it's flipped it's upper side would be blue (and not red), also it's under side would not be blue (as the two sides are different colors).
Claim: if an ugly card is flipped, it become beautiful
Proof: Let a card be ugly. if the card's under side is red, then it would become beautiful when flipped, if the card's under side is white, then it would depend on the upper side. Let the card's under side be white, then the upper side would not be white (as the two sides) and not be red (as it would be already beautiful), so the only choice is that the upper side is blue, which when flipped would become beautiful.
So the answer would be the number of cards that are originally ugly. Which is $2024-150=1874$