Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.
Show that it is impossible to cover an $8\times 8$ square using fifteen $4\times 1$ rectangles and one $2\times 2$ square.
Is it possible to arrange these numbers, $1, 1, 2, 2, 3, 3, \cdots, 1986, 1986$ to form a sequence for such there is $1$ number between two $1$'s, $2$ numbers between two $2$'s, $\cdots$, $1986$ numbers between two 1986's?
It is possible to cover a $6\times 6$ grid using one L-shaped piece made of 3 grids and eleven $3\times 1$ smaller grid ?
There are $100$ guests attending a party. If everyone knows at least $67$ other guests, show that there must exist $4$ guests who know each other.
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?
Show that among any four randomly selected integers, there must exist two whose difference is a multiple of 3.
Show that it is possible to find an integer whose digits are all 4 and it is a multiple of 2016.
There are 13 randomly selected points inside a square of side length 2. Show that there must exist quadrilateral whose vertice are among these 13 points and area is no more than 1.
Joe randomly selects 50 different numbers from 1, 2, $\cdots$, 97, 98, and finds there are always two of them whose difference is a multiple of 7. Can you explain why?
Among nine randomly selected even numbers from $2$, $4$, $6$, $\cdots$, $28$, $30$, show that at least two of them whose sum is $34$.
Show that, among randomly selected $11$ numbers from $1$, $2$, $3$, $\cdots$, $19$, $20$, one of them must be a multiple of another.
Show that among any 5 integers, three of them must satisfy the condition that their sum is a multiple of 3.
Show that in a $n$-people party, at least two of them have met the same number of other guests before.
There are only two problems in a math test. Ten points will be awarded for every correct answer. Two point will be given for any skipped problem. No point will be given for wrong answer. The teacher claims regardlessly there must be at least 3 students will end up with the same score. Can you figure out the minimal number of students who will take this test?
Alan and Barbara play a game in which they take turns filling entries of an initially empty $1024$ by $1024$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
$\textbf{Heaps of Beans}$
A game starts with four heaps of beans, containing $3$, $4$, $5$ and $6$ beans, respectively. The two players move alternately. A move consists of taking either one bean from a heap, provided at least two beans are left behind in that heap, or a complete heap of two or three beans. The player who takes the last bean wins. Does the first or second player have a winning strategy?
A prime number is called an absolute prime if every permutation of its digits in base 10 is also a prime number. For example: 2, 3, 5, 7, 11, 13 (31), 17 (71), 37 (73) 79 (97), 113 (131, 311), 199 (919, 991) and 337 (373, 733) are absolute primes. Prove that no absolute prime contains all of the digits 1, 3, 7 and 9 in base 10.
Let $n$ be an even positive integer, and let $p(x)$ be an $n$-degree polynomial such that $p(-k) = p(k)$ for $k = 1, 2, \dots , n$. Prove that there is a polynomial $q(x)$ such that $p(x) = q(x^2)$.
Let $a, b, c$ be distinct integers. Can the polynomial $(x - a)(x - b)(x - c) - 1$ be factored into the product of two polynomials with integer coefficients?
Let $p_1, p_2, \cdots, p_n$ be distinct integers and let $f(x)$ be the polynomial of degree $n$ given by $$f(x) = (x - p_1)(x - p_2)\cdots (x -p_n)$$ Prove that the polynomial $g(x) = (f(x))^2 + 1$ cannot be expressed as the product of two non-constant polynomials with integral coefficients.
Find the remainder when you divide $(x^{81} + x^{49} + x^{25} + x^9 + x)$ by $(x^3 - x)$.
Is it possible to write the polynomial $f(x) = x^{105}-9$ as the product of two polynomials of degree less than 105 with integer coefficients?