Combinatorics › Pigeonhole and extremal problems

Coloured beads and balloons

Solve the first problem 18 problems

Red 4-door cars in a parking lot ●○○○○ 10 ptsThere are 20 cars in a parking lot. All of the cars are red or white, and all of the cars are either 2-door or 4-door. Of these cars, 12 are red, 15 are 4-door, and 4 are 2-door and white. How many of the cars are 4-door and red?
Cans of red paint in a mixture ●○○○○ 10 ptsPaint is to be mixed so that the ratio of red paint to white paint is $3$ to $2$ . If Ben wants to make $30$ cans of the mixture and all cans hold the same volume of paint, how many cans of red paint will he need?
Balloon bouquets with equal colors ●○○○○ 10 ptsElizabeth has 84 blue balloons and 96 white balloons. She wants to make identical bouquets: every bouquet must have the same number of blue balloons and the same number of white balloons as every other bouquet. She wants to use all of her balloons. Find the greatest number of bouquets she can make.
Chameleon colour changes in September ●●○○○ 20 ptsOn an island there live red, yellow, green and blue chameleons. On a cloudy day, either one red chameleon changes its color to yellow, or one green chameleon changes its color to blue. On a sunny day, either one red chameleon changes its color to green, or one yellow chameleon changes its color to blue. In September there were 18 sunny days and 12 cloudy days. As a result, the number of yellow chameleons increased by 5. By how much did the number of green chameleons increase?
Blue points on a line ●●○○○ 20 ptsOn a line, one hundred points are marked, each painted green, blue, or red. Between any two red points there is a blue point, and between any two blue points there is a green point. There are no fewer red points than blue points, and no fewer blue points than green points. How many points are painted blue?
Distinct candy flavors by percent red ●●○○○ 20 ptsFour red candies and three green candies can be combined to make many different flavors. Two flavors are different if their percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green, and 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is made using some or all of the seven candies (at least one candy), how many different flavors are possible?
Blue segments in a two-colored grid ●●●○○ 50 ptsA $16\times16$ square grid of dots is colored red and blue. Two dots that are adjacent in the grid (horizontally or vertically) are joined by a line segment. A segment joining two dots of the same color has that color, and a segment joining dots of different colors is yellow. There are 133 red dots in total. Exactly 2 of the 4 corner dots are red, and 32 other red dots lie on the boundary of the grid (not corners). There are 196 yellow segments in total. How many blue segments are there?
Maximum red beads on a string ●●●○○ 50 ptsA string is threaded with 150 beads, each red, blue or green. Among any six consecutive beads there is at least one green bead, and among any eleven consecutive beads there is at least one blue bead. What is the maximum number of red beads on the string?
Blue rectangles on a blackboard ●●●○○ 50 ptsBlue and red rectangles are drawn on a blackboard. Exactly 7 of the rectangles are squares. There are 3 more red rectangles than blue squares. There are 2 more red squares than blue rectangles. How many blue rectangles are there?
Minimum red balloons in a string ●●●●○ 100 ptsDavid is making a string of red, white, and blue balloons by these rules: - no red balloon is adjacent to another red balloon; - white balloons appear in groups of exactly two, and groups of white balloons are separated by at least two non-white balloons; - blue balloons appear in groups of exactly three, and groups of blue balloons are separated by at least three non-blue balloons. If David uses more than 600 balloons, find the smallest number of red balloons he can use.
Minimum red beads in a necklace ●●●●○ 100 ptsA necklace consists of 50 blue beads and a certain number of red beads. In any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? The beads are arranged cyclically, so the last bead is adjacent to the first.

Coloured beads and balloons

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