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?
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.
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 in a $n$-people party, at least two of them have met the same number of other guests before.
Two players, $A$ and $B$, take turns naming positive integers, with $A$ playing first. No player may name an integer that can be expressed as a linear combination, with positive integer coefficients, of previously named integers. The player who names 1 loses. Show that no matter how A and B play, the game will always end.
$\textbf{Cards Game (Sum Fifteen)}$
There are nine cards laid out on a table, numbered $1$ through $9$. Two players, Joe and John, take turns to pick up one card a time (and once a card is picked up, it is out of play). As soon as one of these two players has the sum of three of cards equal $15$ , that player wins. Who will win if both players adopt the best 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?
Find the smallest positive integer $n$ so that $107n$ has the same last two digits as $n$.
Let $A_n$ be the average of all the integers between 1 and 101 which are the multiples of $n$ . Which is the largest among $A_2, A_3, A_4, A_5$ and $A_6$?
$\textbf{Passing the Bridge}$
It is a dark and stormy night. Four people must evacuate from an island to the mainland. The only link is a narrow bridge which allows passage of two people at a time. Moreover, the bridge must be illuminated, and the four people have only one lantern among them. After each passage to the mainland, if there are still people on the island, someone must bring the lantern back. When they cross the bridge individually, the four people take $2$, $4$, $8$ and $16$ minutes, respectively. Crossing the bridge in pairs, the slower speed is used. What is the minimum time for the entire evacuation?
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?
The sum of 2008 consecutive positive integers is a perfect square. What is the minimum value of the largest of these integers?
Find the largest positive integer $n$ such that $(3^{1024} - 1)$ is divisible by $2^n$.
A farmer has four straight fences, with respective lengths 1, 4, 7 and 8 metres. What is the maximum area of the quadrilateral the farmer can enclose?
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?
Let $x$ be a positive number. Denote by $[x]$ the integer part of $x$ and by $\{x\}$ the decimal part of $x$. Find the sum of all positive numbers satisfying $5\{x\} + 0.2[x] = 25$.
A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base $10$ is equal to $n$. For instance, $13$ is good because $7^2 = 49$ and $4 + 9 = 13$. How many good numbers are among $1, 2, 3, \cdots , 2007$?
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?
Determine all positive integers $m$ and $n$ such that $m^2+1$ is a prime number and $10(m^2 + 1) = n^2 + 1$.