In Quadrilateral $ABCD, AB=3, BC=4, CD=5, DA=6$ and $\angle ABC=90$. Find the area of $ABCD$.
Line segment $AB$ is $7$ units long. The distance of point $P$ and $AB$ is $3$ units. Find the minimum value of $AP \times BP$.
Four distinct one-digit numbers have lowest common multiple, $L$. Find the maximum value of $L$.
Square $ABCD$ has sides of length $1$. Circle $O$ is drawn with diameter $AD$. Point $E$ is on side $AB$ such that $CE$ is tangent to circle $O$. Find the area of $\triangle CBE$.
Cards numbered $2, 3, 4, 5$ and $6$ are shuffled and randomly arranged in a row. Find the probably that
*Give your answer as a fraction in simplest form.
In how many different ways can you express $100$ as the sum of positive powers of $3$? For example, $3^4+3^2+3^2+3^0=100$ or $3^3+3^3+3^3+3^2+3^1+3^1+3^1+3^0=100$. Sums with the same terms in different order, such as $3^4+3^2+3^2+3^0$ and $3^0+3^2+3^2+3^4$, are considered the same.
Japan’s currency has 1-yen, 5-yen, 10-yen, 50-yen, 100-yen and 500-yen coins.
Taro has one 1-yen, 10-yen and 100-yen coin each and a 1000-yen bill. He buys an item and uses all of his money to pay for it. How many different prices for the item are possible?
Note: Taro wants as few coins in change as possible. For example, if an item costs 900 yen, he would pay with his 1000-yen bill and not use his coins. However, if an item costs 911, he would pay with his 1000-yen bill, 10-yen coin and 1-yen coin as this would minimize the number of coins in change.
Cards numbered $1, 2, 3, \ldots ,15$ are in a pile. When $n$ cards are taken, where $n \ge 1$, the number(s) on the card(s) are all greater than or equal to $n$. In how many total ways can this be done?
$n$ is a positive integer such that the sum of the factors of $n$ that do not have a remainder of $2$ when divided by $4$ equals $1000$. Find all possible values of $n$.
*Separate your answers with commas
At a competition with $n$ participants, $\frac{n}{a}$ rounded down to the nearest integer gold medals are awarded, $\frac{n}{b}$ rounded down to the nearest integer silver medals are awarded and $\frac{n}{c}$ rounded down to the nearest integer bronze medals are awarded, where $a,b$ and $c$ are integers and $a \ge b \ge c$. No participant earns more than one medal.
For $k \ge 3$, when $k$ participants do not earn medals, there are exactly two values of $n$. For example, if $\left(a,b,c\right)=\left(6,6,6\right)$, when $k=3, n=3 \; and \; 6$, when $k=4, n=4 \; and \; 7$, when $k=5, n=5 \; and \; 8$ and so on.
Other than $\left(6,6,6\right)$, find the other possible sets of $\left(a,b,c\right)$.
*Separate your answers with commas