Practice (TheColoringMethod)
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Point $M$ of rectangle $ABCD$ is the midpoint of side $BC$ and point $N$ lies on $CD$ such that $DN:NC = 1:4$. Segment $BN$ intersects $AM$ and $AC$ at points $R$ and $S$. If $NS:SR:RB$ = $x:y:z$, what is the minimum possible value of $x + y + z$?
Dr. Gru took 30 seconds to pump $9.00 of gasoline. It took a total of 105 seconds to pump 15 gallons of gas. We must find the cost for one gallon of gas at the station.
If $kx + 12 = 3k$, for how many integer values of $k$ is $x$ a positive integer?
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.
The sum of the first $n$ terms of a sequence, $a_1 + a_2 + \cdots + a_n$, is given by the formula $S_n = n^2 + 4n + 8$ What is the value of $a_6$?
The first and last initials of the 348 students form a unique ordered letter pair. We must find how many more students are required to guarantee that there are two students whose initials form the same ordered letter pair.
A semicircle and circle are placed inside a square with sides of length 4. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a side of the square (or 4). We must find the radius of the circle
The diameter of a spherical balloon is increased by 150%. We must find by what percent the volume increases.
In one roll of four standard six-sided dice, what is the probability of rolling exactly three different numbers?
Adult tickets are \$5 and student tickets are \$2. Five times as many student tickets were sold as adult tickets for a total of \$1125. We must find the number of tickets sold.
CDs sell for $3$ different amounts. Three customers bought $3$ CDs each but none bought three of the same price. The first customer spent $\$4$, the second spent $\$9$ and the third spent $\$12$. We must find the price of the most expensive CD.
When one integer is removed from a list of 5 integers, the mean of the remaining four integers is 3 less than the mean of the original 5 integers. So what is the positive difference between the mean of the original five integers and the integer that was removed?
The legs of a right triangle are in the ratio 3:4. One of the altitudes is 30 ft. What is the greatest possible area of this triangle?
The positive difference of the cubes of two consecutive positive integers is 111 less than 5 times the product of the two consecutive integers. We must find the sum of the two consecutive integers.
In how many ways can $18$ be written as the sum of four distinct positive integers?
Four towns are located at $A(0,0)$, $B(2,12)$, $C(12,8)$, and $D(7,2)$. A warehouse is built at point $P$ so the sum of the distances $PA + PB + PC + PD$ is minimized. We must find the coordinates of point $P$.
A hot-air balloon descends at a constant rate of 15 ft per minute starting from 1200 ft above the ground. A helium-filled balloon is released at a height of 10 ft above the ground. It goes up at a rate of 5 ft per second. We must find how many minutes expire before the two balloons are at the same height.
There is more than one four-digit positive integer in which the thousands digit is the number of 0s in the four-digit number, the hundreds digit is the number of 1s, the tens digit is the number of 2s and the units digit is the number of 3s. What is the sum of all such integers?
Segments AD and BC are the radii of the top and bottom bases of the frustum. AD = 8, BC = 12 and AC = 15, what is the volume of the frustum?
Mrs. Smith teaches for $5\frac{1}{2}$ hours each day so how many hours does she teach in a 6 day period?
What percent of the positive integers $\le$ 36 are factors of 36?
Three questions are asked in one round of a game show. The second question is worth twice as much as the first. The third is worth three times as much as the second. The third question is worth $12,000. So what is the value of the first question?
We are asked to find the area of the shaded (grey) region where the distance between each dot is 1 cm.
Seven consecutive positive integers have a sum of 91. So what is the largest of these integers?
We are asked to find the largest sum of calendar dates for seven consecutive Fridays in any given year.