Practice (TheColoringMethod)
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Evaluate $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{1368}+\sqrt{1369}}$.
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n)) + f(n) = 2n + 3$ for all nonnegative integers $n$. Find $f(2014)$.
Given that $a_n a_{n-2} - a_{n-1}^2 +a_n-na_{n-2}=-n^2+3n-1$ and $a_0=1$, $a_1=3$, find $a_{20}$.
A girl and a guy are going to arrive at a train station. If they arrive within $10$ minutes of each other, they will instantly fall in love and live happily ever after. But after $10$ minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between $8$ AM and $9$ AM with equal probability. The guy will arrive between $7$ AM and $8:30$ AM, also with equal probability. Find the probability that the probability that they fall in love.
Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\cdots$, $8$, $1$, $2$, $3$, $\cdots$, $8$, $\cdots$ in order. Line segments may only be drawn to connect points labeled with the same number. What is the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).
Consider an orange and black coloring of a $20\times 14$ square grid. Let $n$ be the number of coloring such that every row and column has an even number of orange square. Evaluate $\log_2 n$.
Find the number of fractions in the following list that is in its lowest form (i.e. the denominator and the numerator are co-prime). $$\frac{1}{2014}, \frac{2}{2013}, \frac{3}{2012}, \cdots, \frac{1007}{1008}$$
For all positive integer $n$, show that $$\sum_{k=1}^n\frac{k\cdot k! \cdot\binom{n}{k}}{n^k}=n$$
Solve the equation $$\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=(a+1)\sqrt{\frac{x}{x+\sqrt{x}}}$$
There are $n$ circles inside a square $ABC$ whose side's length is $a$. If the area of any circle is no more than 1, and every line that is parallel to one side of $ABCD$ intersects at most one such circle, show that the sum of the area of all these $n$ circles is less than $a$.
Show that:
(a) It is possible to divide all positive integers into three groups $A_1$, $A_2$, and $A_3$ such that for every integer $n\ge 15$, it is possible to find two distinct elements whose sum equals $n$ from all of $A_1$, $A_2$, and $A_3$.
(b) Dividing all positive integers into four groups, then regardless of the partition, there must exists an integer $n\ge 15$ such that it is impossible to find two distinct numbers whose sum is $n$ in at least one of these four groups.
How many square yards of carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.)
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?
Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before Jack does Jill arrive?
The Centerville Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
Billy's basketball team scored the following points over the course of the first 11 games of the season:
\[42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73\]If his team scores 40 in the 12th game, which of the following statistics will show an increase?
In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$?
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$?
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days?
How many integers between $1000$ and $9999$ have four distinct digits?
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6?
Which of the following integers cannot be written as the sum of four consecutive odd integers?