What is the greatest possible area of a triangle with vertices on or above the $x$-axis and on or below the parabola $y = -(x\u200a - \frac{1}{2})^2+ 3$? Express your answer in simplest radical form.
This figure consists of eight squares labeled A through H. The area of square F is16 units$^2$. The area of square B is 25 units$^2$. The area of square H is 25 units$^2$. In square units, what is the area of square D?
Nine consecutive positive even integers are entered into the 3 $\times$ 3 grid shown so that the sums of the three numbers in each row, each column and each diagonal are the same. What is the average value of the five numbers that are missing? Express your answer as a decimal to the nearest tenth.
Quatro Airlines flies between four major cities. To provide direct flights from each city to the other three cities requires a total of six different direct routes, as shown. How many routes are needed to connect 15 cities, with exactly one route directly connecting each pair?
Abigail, Bartholomew and Cromwell play a game in which they take turns adding 1, 2, 3 or 4 to a sum in order to create an increasing sequence of primes. For example, Abigail must start with either 2 or 3. If she chooses 2, then Bartholomew can add 1 to make 3, or he can add 3 to make 5. If Bartholomew makes 3, then Cromwell can add 2 to make 5, or he can add 4 to make 7. Abigail, Bartholomew and Cromwell take turns, in that order, until no more primes can be made, and the game ends. The player who makes the last prime wins. If Bartholomew wins, how many primes were made?
How many positive integers less than $1000$ do not have $7$ as any digit?
Using each of the digits 1 to 6, inclusive, exactly once, how many six-digit integers can be formed that are divisible by 6?
Points D, E and F lie along the perimeter of $\triangle ABC$ such that $\overline{AD}$ , $\overline{BE}$ and $\overline{CF}$ intersect at point G. If AF = 3, BF = BD = CD = 2 and AE = 5, then what is $\frac{BG}{EG}$ ? Express your answer as a common fraction.
If $0 \le a_1 \le a_2 \le a_3 \le \cdots \le a_n \le 1$, find the maximum value of $$\sum_{1 \le i < j \le n}(a_j-a_i+1)^2+4 \sum_{i=i}^n a_i^2$$
Let the sequence {$a_n$} satisfy $a_0=0$, $a_1=1$, $a_{n+2} = (n+3)a_{n+1} -(n+2)a_n$. Find whether the following equation is solvable in rational numbers:$$\sum_{i=1}^n\frac{x^i}{a_i-a_{i-1}}=-1\qquad\qquad(n \ge 2)$$
If the sum of two numbers is 4 and their difference is 2, what is their product?
The arithmetic mean of 11 numbers is 78. If 1 is subtracted from the first, 2 is subtracted from the second, 3 is subtracted from the third, and so forth, until 11 is subtracted from the eleventh, what is the arithmetic mean of the 11 resulting numbers?
An optometrist has this logo on his storefront. The center circle has area 36\u03c0 $in^2$, and it is tangent to each crescent at its widest point (A and B). The shortest distance from A to the outer circle is $\frac{1}{3}$ the diameter of the smaller circle. What is the area of the larger circle? Express your answer in terms of \u03c0.
What is the value of $\frac{444^2-111^2}{444-111}$ ?
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 $n$?
Quadrilateral ABCD is a square with BC = 12 cm. $\overset{\frown} {BOC}$ and $\overset{\frown} {DOC}$ are semicircles. what is the area of the shaded region?
Real numbers a and b satisfy the equation $\frac{2a-4}{5}+\frac{3a+1}{5}=b$. What is the value of $a - b$? Express your answer as a common fraction.
If the point $(x, x)$ is equidistant from (-2, 5) and (3, -2), what is the value of $x$?
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 $b$?
What is the units digit of the product $7^{23} \times 8^{105} \times 3^{18}$?
If $4(a - 3) - 2(b + 5) = 14$ and $5b -a = 0$, what is the value of $a + b$?
The two cones shown have parallel bases and common apex $T$. $TW = 32$ m, $WV = 8$ m and $ZY = 5$ m. What is the volume of the frustum with circle $W$ and circle $Z$ as its bases? Express your answer in terms of $\pi$.
A coin is flipped until it has either landed heads two times or tails two times, not necessarily in a row. If the first flip lands heads, what is the probability that a second head occurs before two tails? Express your answer as a common fraction.
The product of two consecutive integers is five more than their sum. What is the smallest possible sum of two such consecutive integers?
Four nickels, one penny and one dime were divided among three piggy banks so that each bank received two coins. Labels indicating the amount in each bank were made (6 cents, 10 cents and 15 cents), but when the labels were put on the banks, no bank had the correct label attached. Soraya shook the piggy bank labeled as 15 cents, and out fell a penny. What was the actual combined value of the two coins contained in the piggy bank that was labeled 6 cents?