Carl decided to in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?
Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$
Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is \[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?\]
For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?
Given $\triangle{ABC}$, let $m_a, m_b,$ and $m_c$ be the lengths of three medians. Find its area $S_{\triangle{ABC}}$ with respect to $m_a, m_b,$ and $m_c$.
Let $b$ and $c$ be two given real numbers. What is the maximum number of distinct integers $x$ are there such that $$\mid 101x^2 + bx + c\mid \le 50$$?
A highly valued secretive recipe is kept in a safe with multiple locks. Keys to these locks are distributed among 9 members of the managing committee. The established rule requires that at least 6 members must present in order to open the safe. What is the minimal number of locks required? Correspondingly, how many keys are required?
There exist some integers, $a$, such that the equation $(a+1)x^2 -(a^2+1)x+2a^2-6=0$ is solvable in integers. Find the sum of all such $a$.
Let $a_n=\binom{200}{n}(\sqrt[3]{6})^{200-n}\left(\frac{1}{\sqrt{2}}\right)^n$, where $n=1$, $2$, $\cdots$, $95$. Find the number of integer terms in $\{a_n\}$.
Let sequence $\{a_n\}$ satisfy the condition: $a_1=\frac{\pi}{6}$ and $a_{n+1}=\arctan(\sec a_n)$, where $n\in Z^+$. There exists a positive integer $m$ such that $\sin{a_1}\cdot\sin{a_2}\cdots\sin{a_m}=\frac{1}{100}$. Find $m$.
As shown in diagram below, find the degree measure of $\angle{ADB}$.
How many ordered integeral triples $(x, y, z)$ have the property that each number is the product of the other two?
Randomly choosing two numbers from the set $\{1, 3, 5, 7, 9\}$ with replacement, what is the probability that the product is greater than 40?
A book contains $250$ pages. How many times is the digit used in numbering the pages?
Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.
Let even function $f(x)$ and odd function $g(x)$ satisfy the relationship of $f(x)+g(x)=\sqrt{1+x+x^2}$. Find $f(3)$.
Let $f\Big(\dfrac{1}{x}\Big)=\dfrac{1}{x^2+1}$. Compute $$f\Big(\dfrac{1}{2013}\Big)+f\Big(\dfrac{1}{2012}\Big)+f\Big(\dfrac{1}{2011}\Big)+\cdots +f\Big(\dfrac{1}{2}\Big)+f(1)+f(2)+\cdots +f(2011)+f(2012)+f(2013)$$
Let $f(x)=x^{-\frac{k^2}{2}+\frac{3}{2}k+1}$ be an odd function where $k$ is an integer. If $f(x)$ is monotonically increasing when $x\in(0,+\infty)$, find all the possible values of $k$.