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3

Ten people form a line, among which two are Chinese and two are Americans. Find the probability that both Chinese will stand in front of both Americans (not necessarily immediately in the front).


57

In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. Find the probability that Wednesday is the first day Sandy selects matching socks.


68
Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.

69
In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. Find the probability that a randomly chosen Latin student is a sophomore.

84
An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

122
Let $a > b > c$ be three positive integers. If their remainders are $2$, $7$, and $9$ respectively when being divided by $11$. Find the remainder when $(a+b+c)(a-b)(b-c)$ is divided by $11$.

123
Find all positive integer $n$ such that $2^n+1$ is divisible by 3.

155
How many ordered pairs of positive integers $(x, y)$ can satisfy the equation $x^2 + y^2 = x^3$?

159
Solve in integers the equation $2(x+y)=xy+7$.

161
Find the ordered pair of positive integers $(x, y)$ with the largest possible $y$ such that $\frac{1}{x} - \frac{1}{y}=\frac{1}{12}$ holds.

166
Find all ordered pairs of integers $(x, y)$ that satisfy the equation $$\sqrt{y-\frac{1}{5}} + \sqrt{x-\frac{1}{5}} = \sqrt{5}$$

172
Prove the product of $4$ consecutive positive integers is a perfect square minus $1$.

181

Find the number of five-digit positive integers, $n$, that satisfy the following conditions:

  • the number $n$ is divisible by $5$,
  • the first and last digits of $n$ are equal, and
  • the sum of the digits of $n$ is divisible by $5$.

210
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$ and $c \neq 0$ such that both $abc$ and $cba$ are multiples of $4$.

225
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.

227
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.

270
A perfect power is a number that can be written as a positive integer raised to an integer power greater than $1$. For example, $125$ is a perfect power because it is equal $5^3$. The list $2$, $3$, $5$, $6$, $7$, $10$, $11$, $12$, $13$, $14$, $15$, $17$, $\cdots$ contains every positive integer less than $1000$ that is not a perfect power. How many integer are in the list?

272
What is the last digit of $9^{2019}$?

273
What are the last two digits of $8^{88}$?

274
Find the remainder when $3^{2019} + 4^{2019}$ is divided by 5?

277
Find the coefficient of $x^{17}$ in the expansion of $(1+x^5 + x^7)^{20}$.

280
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999}}$ is divided by $1000$.

294
Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$.

309
Prove: randomly select 51 numbers from $1, 2, 3, \cdots, 100$, at least two of them must be relatively prime to each other.

310

Let integer $a$, $b$, and $c$ satisfy $a+b+c=0$, prove $|a^{1999}+b^{1999}+c^{1999}|$ is a composite number.