Following are a few more problems. (If the solution is available the exclamation mark is
clickable - guess why did I say that.) A good half of the problems (of which some are original) were kindly suggested by William A McWorter Jr.
17 rooks are placed on an 8×8 chessboard. Prove that there are at least 3 rooks that do not threaten each other.
Chinese Remainder Theorem.
Let's mark the centers of all squares of an 8x8 chess board. Is it possible to cut the board with 13 straight lines (none passing through a single midpoint) so that every piece had at most 1 marked point?
Each of the given 9 lines cuts a given square into two quadrilaterals whose areas are in proportion 2:3. Prove that at least three of these lines pass through the same point.
Suppose each point of the plane is colored red or blue. Show that some rectangle has its vertices all the same color.
Suppose each point on a circumference of a circle is colored either red or blue. Prove that, no matter how colors may be distributed, there exist 3 equally spaced points of the same color.
Suppose f(x) is a polynomial with integral coefficients. If f(x) = 2 for three different integers a, b, and c, prove that, for no integer, f(x) can be equal to 3.
Prove that there exists a power of three that ends with 001.
Show that if more than half of the subsets of an n-element set are selected, then some two of the selected subsets have the property that one is a subset of the other.
Let a and b be postive integers, with a < b < 2a. Then, given more than half of the integers in the set {1, 2, ..., a + b}, some two of the given integers differ by a or by b.
Given any 6 points inside a circle of radius 1, some two of the 6 points are within 1 of each other.
Let n be a positive integer greater than 3. Let m be the largest integer in (n+2)/2. Then, given more than m of the integers in the set {1, 2, ..., n}, some three of the integers in the given set have the property that one of the three is the sum of the other two.
If more than half of the integers from {1, 2, ..., 2n} are selected, then some two of the selected integers have the property that one divides the other.
If more than half of the integers from {1, 2, ..., 2n} are selected, then some two of the selected integers are mutually prime.
Given any sequence of mn+1 real numbers, some subsequence of (m + 1) numbers is increasing or some subsequence of (n+1) numbers is decreasing.
Given any 1000 integers, some two of them differ by, or sum to, a multiple of 1997.
Given any 10 4-element subsets of an 11-set, some two of the subsets intersect in at least two elements.
A person takes at least one aspirin a day for 30 days. If he takes 45 aspirin altogether, in some sequence of consecutive days he takes exactly 14 aspirin.
A theater club gives 7 plays one season. Five women in the club are each cast in 3 of the plays. Then some play has at least 3 women in its cast.
At a party of n people, some pair of people are friends with the same number of people at the party.
Given any 6 integers from 1 to 10, some two of them have an odd sum.
Given a set A 
{1, 2, ..., 100} of ten integers. Prove that it is possible to select two disjoint non-empty subsets of A, S and T, whose members have the same sum.
If a 14×14 0-1 matrix A has 58 1's, then some 2×2 submatrix of A consists of all 1's.
Given 14 or more integers from {1, 2, ... , 28} there exist four of the given integers which can be split into two groups of two with the same sum.
Arrange words HEN, HUT, WIT, SAW, CAR, CUB, MOB, DIM, RED and SON in a circuit so that any two adjacent words share a letter.
The "Math Lotto" betting card is a 6×6 table. The gambler is to mark 6 of the 36 slots in the card. The official result is published with 6 slots chosen as the "LOSING SLOTS". The gambler wins if he doesn't pick any losing slot.
Proizvolov's Identity [Savchev]. In any partition of the first 2N natural numbers into decreasing and increasing sequences of N members each, the sum of absolute values of the differences of the corresponding members of the two sequences is always N2.
Given any nm+1 intervals, there exist n+1 pairwise disjoint intervals or m+1 intervals with a nonempty intersection.
All antichains in Nk with the lexicographic order are finite.
Euclid's theorem: For any two coprime intergers a, b there are two other integers x, y such that ax - by = 1.
In a circle are light bulbs numbered 1 through N, all initially on. At time t, you examine bulb number t, and if it's on, you change the state of bulb t+1 (mod N); i.e., you turn it off if it's on, and on if it's off. If bulb t is off you do nothing. Prove that if you continue around and around the circle in this manner, eventually all the bulbs will again be on.
Twenty five boys and twenty five girls sit around a table. Prove that it is always possible to find a person both of whose neighbors are girls.
A chess master who has 11 weeks to prepare for a tournament decides to play at least one game every day but, in order not to tire himself, he decides not to play more than 12 games during any calendar week. Show that there exists a succession of consecutive days during which the chess master will have played exactly 21 games.
Given two disks, one smaller than the other. Each disk is divided into 200 congruent sectors. In the
larger disk 100 sectors are chosen arbitrarily and painted red; the other 100 sectors are painted blue. In the smaller
disk each sector is painted either red or blue with no stipulation on the number of red and blue sectors. The smaller
disk is placed on the larger disk so that the centers and sectors coincide. Show that it is possible to align the two
disks so that the number of sectors of the smaller disk whose color matches the corresponding sector of the larger
disk is at least 100.
My wife and I recently attended a party at which there were four other married couples. Various handshakes took place. No one shook hands with oneself, nor with one's spouse, and no one shook hands with the same person more than once. After all the handshakes were over, I asked each person, including my wife, how many hands he (or she) had shaken. To my surprise each gave a different answer. How many hands did my wife shake?
Is it possible to move the counters from the left half of the board to the right half? The only allowed move is to jump over an adjacent (vertically, horizontally, or diagonally) counter. (The jumped over counters are not removed.)
Assume in a class of students each of the number of committees contains more than half of all the students. Prove that there is a student who is a member in more than half of the committees.
Any number S is written down, and beneath it are written its multiples 2S, 3S,...up to 10S. Consider the ten digits in any column. There does not appear to be any general law; yet every complete column must contain either a 9 or a 0. Prove this. (In the first column the gaps may be filled by 0's, and the same result holds for all columns.)
21 girls and 21 boys participate in a math competition. The results show that: a) Each contestant solved at most six problems, and b) For each pair of a girl and a boy, there was at least one problem that they both solved. Prove that there is a problem that was solved by at least three girls and at least three boys.
In a tournament where each team meets every other team once, there are always two teams that played the same number of games.
There is a repunit divisible by 2009.
There are 2009 clubs, each with 45 members, any two of which have exactly one person in common. Show that there is one person who is a member of all the clubs.
[s/Q, (s+1)/Q) and {q2a}
[s/Q, (s+1)/Q)
