Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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This table shows every category of Martin's monthly expenses. Once completed, it will depict how 100% of Martin's monthly income is budgeted. Based on the information provided, what is Martin's monthly income?
The points $A(-1, 2)$ and $B(3, 2)$ are graphed on a coordinate plane. Point $C$ is the reflection of point $A$ over the $x$-axis. What is the area of triangle $ABC$?
What is the greatest prime factor of the product $6 \times 14 \times 22 $?
If A+B = 5 and A-B = 3, what is the value of A+A\u200a?
Ms. Spice tells her class that she is thinking of a positive common fraction. The product of the numerator and denominator is 60. When she adds 1 to the numerator and divides the denominator by 2, the resulting fraction is equal to 1. What common fraction is Ms. Spice thinking of?
Working alone, a professor grades a paper every 10 minutes. The professor spends 20 minutes training an assistant. Then, working together, they grade 2 papers every 15 minutes. For how many graded papers is the amount of time it would take the professor working alone the same as the amount of time it would take the professor and her assistant working together, including the time required for training?
If $b = a^2$ and $c = 3b - 2$, what is the product of all values of $a$ for which $b = c$?
A set of six distinct positive integers has a mean of 8, a median of 8 and no term greater than 13. What is the least possible value of any term in the set?
If $x \uparrow y = (x + y)^2$ for positive integers x and y, what is the value of $(1 \uparrow 2) \uparrow 3$?
Segment XY is drawn parallel to the base of triangle $ABC$. If the area of trapezoid $BCYX$ is 10 units$^2$ and the area of triangle AXY is 8 units$^2$, what is the ratio of $XY$ to $BC$? Express your answer as a common fraction.
What is the greatest possible absolute difference between the mean and the median of five single-digit positive integers? Express your answer as a common fraction.
If $f$ is a function such that $f(f(x)) = x^2 - 1$, what is $f(f(f(f(3))))$?
If the sum of an arithmetic progression of six positive integer terms is 78, what is the greatest possible difference between consecutive terms?
Points A, B and C have coordinates (-4, 2), (1, 2) and (-1, 5), respectively. If triangle ABC is reflected across the y-axis, what is the area of the region that is the intersection of triangle ABC and its reflection? Express your answer as a decimal to the nearest tenth.
On a standard die with six faces, each face contains a different number from 1 through 6. Jake has a non-standard die with six faces, and each face on Jake's die contains an expression with a different value from 1 through 6. In no particular order, the six expressions are $a + 1$, $2a - 5$, $3a - 10$, $b + 8$, $2b + 5$ and $3b + 10$. If $a$ and $b$ are integers, what is the value of the product $a \times b$?
For a particular sequence $a_1 = 3$, $a_2 = 5$ and $a_n = a_{n -1} - a_{n -2}$, for $n \ge 3$. What is the $2015^{th}$ term in this sequence?
If Desi flips a fair coin eight times, what is the probability that he will get the same number of heads and tails? Express your answer as a common fraction.
How many ordered pairs of integers (x, y) satisfy the equation x + |\u200ay| = y + |\u200ax| if \u221210 \u2264 x \u2264 10 and \u221210 \u2264 y \u2264 10?
When $\frac{1}{98}$ is expressed as a decimal, what is the $10^{th}$ digit to the right of the decimal point?
For positive integers $n$ and $m$, each exterior angle of a regular $n$-sided polygon is 45 degrees larger than each exterior angle of a regular $m$-sided polygon. One example is $n = 4$ and $m = 8$ because the measures of each exterior angle of a square and a regular octagon are 90 degrees and 45 degrees, respectively. What is the greatest of all possible values of $m$?
What are the last two digits in the sum of the factorials of the first $100$ positive integers?
Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.
Prove: randomly select $51$ numbers from $1$, $2$, $3$, $\dots$, $100$, there must exist two numbers for which one is a multiple of the other.
Let four positive integers $a$, $b$, $c$, and $d$ satisfy $a+b+c+d=2019$. Prove $\left(a^3+b^3+c^3+d^3\right)$ cannot be an even number.
Prove: it is impossible to have two positive integers such that the product of their sum and their difference equals 1990.