Practice using the pigeonhole principle. Record final answers, show reasoning for the extended problems, and compare your method with the worked review.
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The question order, section structure, and point values follow Test 23: The Pigeonhole Principle. Questions 1–10 are short-response items; Questions 11–13 require a written argument. Q2, Q3, and Q7 have explicit wording clarifications so the tasks match the reasoning taught in the lessons.
Questions 1–10 · 6 points each · 60 points
An examination has 10,000 participants. The full score is 150, and every score is an integer. Students scoring at least 60, including 60, make up 4/5 of all participants. Among these students, at least how many must have the same score?
Thirteen points are placed anywhere inside or on a square of side length 1. Divide the square into four equal smaller squares. This partition guarantees that at least how many points share one smaller square, so their convex hull has area no greater than 1/4?
A class has 50 students. All students were born in either 2014 or 2015. At least how many students must have been born in the same year and month?
A storeroom contains unlimited basketballs, volleyballs, soccer balls, and handballs. Each person carries any two balls. Among 101 carriers, at least how many people must be carrying exactly the same unordered pair of ball types?
A school sends 55 students to a mathematics contest. No matter how the participants are divided into four nonempty groups, some group must contain more than 2 girls. Also, every set of 10 participants contains at least one boy. How many boys are participating?
At a New Year party, each student reaches into a bag containing many glass balls and takes two. The bag contains at least two balls of each of five colors: red, yellow, white, blue, and green. Both balls are returned before the next student draws. The balls feel identical; the students cannot see the colors while drawing. What is the least number of participating students that guarantees two students receive the same unordered pair of colors?
A regular hexagon has side length 1. The proof in Lesson 23.4 assigns points in or on it to 24 small triangular drawers, with boundary points assigned so that any two points in one drawer are less than 1/2 apart. What is the least number of distinct points that exceeds these 24 drawers and therefore guarantees two points at distance less than 1/2?
What is the least number of distinct points that must be placed inside a circle of radius 1, or on its boundary, to guarantee that two points are at distance less than 1?
From the natural numbers 1, 2, 3, ..., 30, what is the largest number of distinct values that can be selected so that the sum of any two distinct selected numbers is not a multiple of 9?
There is an ample supply of socks in four colors: red, yellow, blue, and white. Two socks of the same color make one pair. What is the least number of socks that guarantees ten disjoint same-color pairs?
Questions 11–13 · 20 points each · 60 points
Can every cell of a 5 × 5 grid be filled with one of the numbers 1, 2, and 3 so that the sums of the five rows, five columns, and two corner-to-corner diagonals are all pairwise different? Explain your reasoning.
Use only 1, 2, or 3. This grid is optional and is not part of the completion requirement. Invalid entries are marked; they are not silently changed.
Count the lines whose sums must differ. Then find the smallest and largest possible sum of five entries. How many integer sums are available?
For an unaided assessment, leave this hint closed. Completion still records responses only.
The upper-right and lower-left corner squares are removed from an 8 × 8 chessboard. Can 31 rectangular 2 × 1 dominoes cover the remaining board exactly? Explain.
L and D label the alternating light and dark cells; × marks a removed corner. Each domino covers two edge-adjacent remaining cells.
Compare the colors of the two removed corners. How many light and dark cells does each domino cover?
For an unaided assessment, leave this hint closed. Completion still records responses only.
From the natural numbers 1, 2, 3, ..., 80, at least how many must be selected to guarantee that two selected numbers have a divisibility relationship—that is, one number is a multiple of the other?
Write each number as an odd number multiplied by a power of 2. Numbers with the same odd part form one drawer. For the near-miss, consider the upper half of the interval.
For an unaided assessment, leave this hint closed. Completion still records responses only.