Compute $50^{250} \pmod{83}$ .
$\textbf{Cover the Board}$
Joe cuts off the top left corner and the bottom right corner of an $8\times 8$ board, and then tries to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.
$\textbf{Cover the Board (II)}$
Joe cuts off a $2\times 2$ corner from an $8\times 8$ board, and then tries to cover the remaining part using $15$ L-shaped grids made of $4$ grids as shown. Is it possible?
Show that among any $6$ people in the world, there must exist $3$ people who either know each other or do not know each other.
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.
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!$?
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the five coefficients ($a$, $b$, $c$, $d$, $d$, and $e$) cannot be zero?
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.
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 in a $n$-people party, at least two of them have met the same number of other guests before.
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?
Find the smallest positive integer $n$ so that $107n$ has the same last two digits as $n$.
A regiment had 48 soldiers but only half of them had uniforms. During inspection, they form a 6 × 8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?
In the diagram , $PA = QB = PC = QC = PD = QD = 1, CE = CF = EF$ and $EA = BF = 2AB$. Determine $BD$.
Each of the numbers 2, 3, 4, 5, 6, 7, 8 and 9 is used once to fill in one of the boxes in the equation below to make it correct. Of the three fractions being added, what is the value of the largest one?
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.
Mary found a $3$-digit number that, when multiplied by itself, produced a number which ended in her original $3$-digit number. What is the sum of all the numbers which have this property?
Find $8$ prime numbers, not necessarily distinct such that the sum of the squares of these numbers is $992$ less than $4$ times of the product of these numbers.
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$?
Let $p(x)$ be a polynomial with integer coefficients. Assume that $p(a) = p(b) = p(c) = -1$, where $a, b, c$ are three different integers. Prove that $p(x)$ has no integral zeros.
The product of two of the four zeros of the quartic equation $$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$ is $-32$. Find $k$.
Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.
Find the remainder when you divide $(x^{81} + x^{49} + x^{25} + x^9 + x)$ by $(x^3 - x)$.
Does there exist a polynomial $f(x)$ for which $xf(x - 1) = (x + 1)f(x)$
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?