Practice With Solutions

1608

Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?

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?

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?

1685

How many pairs of positive integers (a,b) are there such that $a$ and $b$ have no common factors greater than 1 and: $\frac{a}{b} + \frac{14b}{9a}$ is an integer?

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?

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?

1818

How many ways can all six numbers in the set $\{4, 3, 2, 12, 1, 6\}$ be ordered so that $a$ comes before $b$ whenever $a$ is a divisor of $b$?

1819

What is the units digit of the product $7^{23} \times 8^{105} \times 3^{18}$?

1850

How many collections of six positive, odd integers have a sum of $18$? Note that $1 + 1 + 1 + 3 + 3 + 9$ and $9 + 1 + 3 + 1 + 3 + 1$ are considered to be the same collection.

1852

Call a positive integer squarish if it contains the digits of the squares of its digits in order but not necessarily contiguous. For example, $14263$ contains $1^2 = 1$, $4^2 = 16$ and $2^2 = 4$. However, it is not squarish because it does not contain $3^2 = 9$, and $6^2 = 36$ is not in order. What is the smallest squarish number that includes at least one digit greater than $1$?

1855

Place 9 points in a unit square. Prove it is possible to select 3 points from them to create a triangle whose area is no more than $\frac{1}{8}$.

1861

When $(37 \times 45) - 15$ is simplified, what is the units digit?

1869

Farmer Hank has fewer than $100$ pigs on his farm. If he groups the pigs five to a pen, there are always three pigs left over. If he groups the pigs seven to a pen, there is always one pig left over. However, if he groups the pigs three to a pen, there are no pigs left over. What is the greatest number of pigs that Farmer Hank could have on his farm?

1905

How many diagonals does a convex octagon have?

1935

In $\triangle{ABC}$, segments AB and AC have each been divided into four congruent segments. We must find the fraction of the triangle that is shaded.

1978

How many positive integers not exceeding $2000$ have an odd number of factors?

2003

Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?

2009

For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?

2016

Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.)

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