Practice (90)
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Let integer $a$, $b$, and $c$ satisfy $a+b+c=0$, prove $|a^{1999}+b^{1999}+c^{1999}|$ is a composite number.
The number $2^{29}$ is a nine-digit number whose digits are all distinct. Which digit of $0$ to $9$ does not appear?
True or false: Let $\tau({n})$ denotes the number of positive divisors that $n$ has. Then $$\tau({1}) + \tau({2}) +\tau({3}) +\cdots + \tau({2015})$$ is an odd number.
Is it possible to equally divide the set {$1, 2, 3, \cdots, 972$} into 12 non-intersect subsets so that each subset has exactly 81 elements, and the sums of those subsets are all equal?
Joe wrote $3$ positive integers on the whiteboard, then erased one number and replaced with the sum of the other two numbers minus $1$. He continued doing this and stopped when there're $17$, $1967$ and $1983$ on the whiteboard. Is it possible that the initial three numbers Joe wrote are $2$, $2$ and $2$?
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker \$ $\underline{1}\underline{A}\underline{2}$. What is the missing digit A of this 3-digit number?
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal?
Back in 1930, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?
For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50) = 12$ because the prime factorization of $50$ is $2 \times 5^{2}$, and $2 + 5 + 5 = 12$. What is the range of the function $r$, $\{r(n): n \text{ is a composite positive integer}\}$ ?
What is the units digit of the sum of the squares of the integers from $1$ to $2015$, inclusive?
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
A restaurant sells three sizes of drinks: small for \$1.20, medium for \$1.30 and large for \$1.80. Each person at a table of ten ordered one drink, for a total cost of \$14.90, before sales tax. How many people ordered a large drink?
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?
Let $S$ be the set of positive integers $n$ for which $\tfrac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. What is the sum of the elements of $S$?
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^\text{th}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive?
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$ ?
The number $2013$ is expressed in the form
$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?
Julia's age is a two-digit multiple of $5$, and when Julia's age is divided by $2$, $3$, $4$, $6$ or $8$, the remainder is always $1$. If Julia is five times as old as Bart, how old is Bart?
If $p$ is the greatest prime whose digits are distinct prime numbers, what is the units digit of $p^2$?
When the integers 1 through 7 are written in base two, what fraction of the digits are 1s? Express your answer as a common fraction.
Six different prime numbers are placed in the six different circles shown. The three circles on each side of the triangle have the same sum. What is the least possible value of the side sum?
Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?
Two integers have a sum of $26$. when two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?