Pigeonhole principle
61 problems · Pigeonhole principle and generalized pigeonhole applications.
Ex.259The pigeonhole principle on birthdays
Pigeonhole principle
High school
Ex.260Socks in the dark
Pigeonhole principle
High school
Ex.331Pigeonhole modulo 4
Pigeonhole principle
L1
Ex.525Among $n+1$ from $2n$, two consecutive
Pigeonhole principle
High school
Ex.526Consecutive integers
Pigeonhole principle
L1
Ex.527Difference divisible by $n$
Pigeonhole principle
L1
Ex.528Pigeonhole with numbers
Pigeonhole principle
L1
Ex.529More socks
Pigeonhole principle
High school
Ex.530Summary of pigeonhole techniques
Pigeonhole principle
High school
Ex.531Generalized pigeonhole with $100$ people
Pigeonhole principle
High school
Ex.532Balls in boxes
Pigeonhole principle
High school
Ex.533Proof of the generalized pigeonhole principle
Pigeonhole principle
L1
Ex.534A pair with difference of $n$
Pigeonhole principle
L1
Ex.535Pairs summing to $2n+1$
Pigeonhole principle
L1
Ex.601A block of consecutive terms divisible by n
Pigeonhole principle
M1
Ex.606Two people with the same number of friends
Pigeonhole principle
M1
Ex.637Six people: three friends or three strangers
Pigeonhole principle
M1
Ex.1088$\{1,2,3,\ldots\}$ and pairwise differences
Pigeonhole principle
Olympiad
Ex.1089Eratosthenes via pigeonhole
Pigeonhole principle
L1
Ex.1090Sum of digits
Pigeonhole principle
Olympiad
Ex.1091Binomial coefficient bound
Pigeonhole principle
Olympiad
Ex.1092Consecutive integers whose sum is divisible
Pigeonhole principle
L1
Ex.1093Divisibility via pigeonhole
Pigeonhole principle
L1
Ex.1094Three same-color integers
Pigeonhole principle
Olympiad
Ex.1095Points in an equilateral triangle
Pigeonhole principle
Olympiad
Ex.1096An even number of odd vertices
Pigeonhole principle
Olympiad
Ex.1097The handshake lemma by pigeonhole
Pigeonhole principle
L1
Ex.1098Integer grid points
Pigeonhole principle
Olympiad
Ex.1099Pigeonhole in a Latin square
Pigeonhole principle
Olympiad
Ex.1100Many odd numbers
Pigeonhole principle
Olympiad
Ex.1101Modular pigeonhole
Pigeonhole principle
L1
Ex.1102A multiple of $n$ from $0$s and $1$s
Pigeonhole principle
Olympiad
Ex.1103An odd number in a triple
Pigeonhole principle
Olympiad
Ex.1104Points in a unit square
Pigeonhole principle
L1
Ex.1105Subset sums
Pigeonhole principle
L1
Ex.1106Remainders of consecutive integers
Pigeonhole principle
Olympiad
Ex.1107Pigeonhole in a table
Pigeonhole principle
L1
Ex.1108A sequence of $0$s and $1$s
Pigeonhole principle
Olympiad
Ex.1240Monotone subsequences (Erdős–Szekeres)
Pigeonhole principle
Olympiad
Ex.1242Choosing n + 1 numbers among 1, …, 2n
Pigeonhole principle
Olympiad
Ex.1251The fewest kings controlling the board
Pigeonhole principle
M1
Ex.1252Knights and liars around a table
Pigeonhole principle
M1
Ex.1311Five points contain a convex quadrilateral
Pigeonhole principle
Olympiad
Ex.1803Difference exactly $10$
Pigeonhole principle
Olympiad
Ex.1804Difference of at most $10$
Pigeonhole principle
Olympiad
Ex.1805Circles and pigeonhole
Pigeonhole principle
Olympiad
Ex.1806Coloring integers
Pigeonhole principle
Olympiad
Ex.1807Friends and enemies
Pigeonhole principle
Olympiad
Ex.1808Graph theory via pigeonhole
Pigeonhole principle
Olympiad
Ex.1809Increasing or decreasing subsequence
Pigeonhole principle
Olympiad
Ex.1810Monochromatic triangle
Pigeonhole principle
Olympiad
Ex.1811Rectangle with lattice points
Pigeonhole principle
Olympiad
Ex.1812Separated integers
Pigeonhole principle
Olympiad
Ex.1813Subsets of fixed size
Pigeonhole principle
Olympiad
Ex.1814A subset with sum divisible by $n$
Pigeonhole principle
Olympiad
Ex.1815Sum of two distinct elements
Pigeonhole principle
Olympiad
Ex.1816A triple summing to a multiple
Pigeonhole principle
Olympiad
Ex.1817Triangle area via pigeonhole
Pigeonhole principle
Olympiad
Ex.1977The Sylvester–Gallai theorem
Pigeonhole principle
Olympiad
Ex.2187Ramsey-type pigeonhole
Pigeonhole principle
Olympiad
Ex.2227The Sylvester-Gallai theorem
Pigeonhole principle
Olympiad
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