Practice (TheColoringMethod)
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A pyramid has 6 vertices and 6 faces. How many edges does it have?
The product of the integers from 1 through 7 is equal to $2^j\cdot 3^k\cdot 5 \cdot 7$ What is the value of $j - k$?
A six-sided die was rolled repeatedly and 1 was rolled 4 times, 2 - 4 times, 3 -3 times, 4 - 4 times, 5 - 2 times and 6 - 3 times. We need to find the mean of the 20 numbers that were rolled.
Let's name the coordinates of the vertices of a trapezoid are A(1, 7), B(1, 11), C(8, 4) and D(4, 4). What is the area of the trapezoid?
Malika ran 3 miles. She ran the first mile in 6 minutes and 45 seconds. Each of the remaining two miles after that took 1/9 longer than the previous mile. We must find, in seconds, how long it took Malika to run all 3 miles.
Four consecutive integers are substituted in every possible way for distinct values $a$, $b$, $c$ and $d$. What is the positive difference between the smallest and largest possible values of $(ab + cd)$?
Triangle $\triangle{MNO}$ is an isosceles trianglewith MN = NO = 25. A line segment drawn from the midpoint of MO perpendicular to MN, intersects MN at point P with NP:PM = 4:1. We must find the length of the altitude drawn from point N to side MO.
{$a,b,c,d$} is a set of numbers chosen from the first nine positive integers. If you add every possible pair of these four numbers you get 7, 9, 10, 12, 13 and 15. We must find the smallest possible product of these four numbers.
In a sequence of positive integers, every term after the first two terms is the sum of the previous two terms of the sequence. The fifth term is 2012 so what is the maximum possible value of the first term?
The figure shows the first three stages of a fractal, respectively. We must find how many circles in Stage 5 of the fractal.
We have a set of numbers {1, 2, 3, 4, 5} and we take products of three different numbers. We must find now many pairs of relatively prime numbers there are.
In rectangle $ABCD$, $AB = 6$ units. m$\angle{DBC} = 30^{\circ}$, $M$ is the midpoint of segment $AD$, and segments $CM$ and $BD$ intersect at point $K$. We must find the length of segment $MK$.
Jack and Jill drove the same distance. Jill drove 20% faster than Jack and she arrived half an hour earlier. We must find how many hours Jack drove.
What is the largest five-digit integer such that the product of the digits is $2520$?
A rectangular prism is composed of unit cubes. The outside faces of the prism are painted blue and the seven unit cubes in the interior are unpainted. We must find how many unit cubes have exactly one painted face.
Let $f(x) = x^2 + 5$, and $g(x) = 2(f(x))$. What is the greatest possible value of $f(x + 1)$ when $g(x)$ = 108?
A right triangle has sides with lengths 8, 15 and 17. A circle is inscribed in the triangle, as shown, and we must find the radius of the circle.
Given that $x \ne 0$, what quantity can be added to $\frac{x +1}{x}$ or multiplied by $\frac{x +1}{x}$ to give the same result?
In trapezoid ABCD segments AB and CD are parallel. Point P is the intersection of diagonals AC and BD. The area of $\triangle{PAB}$ is 16 and $\triangle{PCD}$ is 25. We must find the area of the trapezoid.
The sum of the squares of two positive numbers is 20 and the sum of their reciprocals is 2. We must find their product.
A triangle has angles measuring $15^{\circ}$, $45^{\circ}$ and $120^{\circ}$. The side opposite the $45^{\circ}$ angle is 20 units. The area of the triangle can be expressed as $m -n\sqrt{q}$ and we must find the sum $m + n + q$.
A semicircle is positioned above a square. The diameter of the semicircle is 2 units. We must find the radius, r, of the smallest circle that contains this figure.
In how many ways can 6 different gifts be given to five different children with each child receiving at least one gift and each gift being given to exactly one child?
If the cost of a dozen eggs is reduced by $x$ cents, a buyer will pay one cent less for $x$ + 1 eggs than if the cost of a dozen eggs is increased by $x$ cents. What is the value of $x$?
For how many two-element subsets {$a,b$} of the set {$1, 2, 3, \cdots, 36$} is the product of $ab$ a perfect square?