Practice (Basic)

1430

For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.

1431

Let $M$ be the product of any four consecutive positive integers. Prove $M+1$ must be a perfect square.

1448

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?

1539

What is the sum of the digits of the square of $111,111,111$?

1540

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?

1570

How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?

1580

What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$?

1606

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?

1610

A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?

1617

A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?

1626

A poll shows that $70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?

1650

Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2019$, inclusive. What is the probability that $(ad-bc)$ is even?

1664

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?

1668

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (asumming that the letter $A$ is one place to the right of the letter $Z$). The second time this same letter appears in the given message, it is replaced by the letter that is $1+2$ places to the right, the third time it is replaced by the letter that is $1+2+3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter $s$ in the message \[\text{"Lee's sis is a Mississippi miss, Chriss!"?}\]

1680

A set of $25$ square blocks is arranged into a $5 \times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?

1689

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?

1694

How many sets of two or more consecutive positive integers have a sum of $15$?

1705

Six distinct positive integers are randomly chosen between $1$ and $2020$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$?

1706

How many four-digit positive integers have at least one digit that is a $2$ or a $3$?

1721

What is the tens digit in the sum $7!+8!+9!+...+2018!$

1774

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?

1777

All of David's telephone numbers have the form $555 - abc - defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have?

1799

How many positive integers less than $1000$ do not have $7$ as any digit?

1807

A bag contains $8$ blue marbles, $4$ red marbles and $3$ green marbles. In a single draw, what is the probability of not drawing a green marble?

1812

The product of the digits of positive integer $n$ is $20$, and the sum of the digits is $13$. What is the smallest possible value of $n$?

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