Practice With Solutions
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Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?
Let $n$ be a positive integer greater than $4$ such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?
If integer $a$, $b$, $c$, and $d$ satisfy $ad-bc=1$. Prove $a+b$ and $c+d$ are relatively prime.
Prove for any positive integer $n$, the fraction $\frac{21n+4}{14n+3}$ cannot be further simplified.
Prove: there exists a rational number $\frac{c}{d}$, where $d<1000$, such that $$\Big[k\cdot\frac{c}{d}\Big]=\Big[k\cdot\frac{73}{100}\Big]$$ holds for every positive integer $k$ that is less than 1000. Here $\Big[x\Big]$ denotes the largest integer that is not exceeding $x$.
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?
Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?
In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?
Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate--in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Find $p$.
For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.
Let $M$ be the product of any four consecutive positive integers. Prove $M+1$ must be a perfect square.
Prove: if $(2^n+1)$ is a prime number, then $n$ must be some power of 2.
If $2^n-1$ is a prime number, prove $n$ must be a prime number too.
A pair of standard $6$-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
Seven students count from $1$ to $1000$ as follows:
- Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1$, $3$, $4$, $6$, $7$, $9$, . . ., $997$, $999$, $1000$.
- Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
- Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
- Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
- Finally, George says the only number that no one else says.
What number does George say?
What is the hundreds digit of $2011^{2011}?$
The number obtained from the last two non-zero digits of $90!$ is equal to $n$. What is $n$?
What is the sum of the digits of the square of $111,111,111$?
A circle of radius $2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$?
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be $6$. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts $1$. If it comes up tails, he takes half of the previous term and subtracts $1$. What is the probability that the fourth term in Jacob's sequence is an integer?
Two subsets of the set $S=\{ a,b,c,d,e\}$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?