In trapezoid $ABCD$ we have $\overline{AB}$ parallel to $\overline{DC}$, $E$ as the midpoint of $\overline{BC}$, and $F$ as the midpoint of $\overline{DA}$. The area of $ABEF$ is twice the area of $FECD$. What is $\frac{AB}{DC}$?
Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^2 - y^2 = m^2$ for some positive integer $m$. What is $x + y + m$?
A subset $B$ of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of $B$ sum to $125$. What is the maximum possible number of elements in $B$?
Box A contains 142 marbles, box B contains 152 marbles and box C contains 136 marbles. Marbles are transferred only from box B to box C. What is the least number of marbles that must be transferred so that box C contains more marbles than each of the other two boxes?
What is the largest prime that divides both $20! + 14!$ and $20!-14!$?
In how many distinguishable ways can the four letters in the word NINE be arranged?
A particular online vendor offers discounts for orders of 11 or more shirts, as the table shows. For how many different quantities of shirts would the cost exceed the cost of buying the least number of shirts at the next discount level?
Each term in the sequence that begins 13, 9, 18, $\cdots$ is the sum of three times the tens digit and two times the units digit of the previous term. What is the greatest value of any term in this sequence?
In square ABCD, shown here, sector BCD was drawn with a center C and BC = 24 cm. A semicircle with diameter AE is drawn tangent to the sector BCD. If points A, E and D are collinear, what is AE?
How many distinct unit cubes are there with two faces painted red, two faces painted green and two faces painted blue? Two unit cubes are considered distinct if one unit cube cannot be obtained by rotating the other.
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?
Xiang needs to print T-shirts for a class project. For what number of shirts will the cost under Plan A and Plan B be the same?
Jackie invested \$1000 into an account that earns 3% interest, compounded annually. If she has \$1125.51 now, for how many years did she have her money in her account? Express your answer as a whole number.
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?
In a particular word game, there are two types of letters: vowels and consonants. Vowels are worth 1 point each and consonants are worth 2 points each. (The letter Y is always considered a consonant.) When more than one letter of the same type appears consecutively, each letter is worth twice as much as the one before. For example, CUP is worth 2 + 1 + 2 = 5 points and SLY is worth 2 + 4 + 8 = 14 points. What is the absolute difference between the values of QUEUEING and SYZYGY?
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?
Mary and Ann live at opposite ends of the same road. They plan to leave home at the same time and ride their bikes to meet somewhere between the two houses. At 11:00 a.m. Mary has traveled half of the distance between their houses. Ann is riding more slowly and has covered only 38 of the distance between the houses. They are still one mile apart. How many miles apart are their houses?