Practice With Solutions
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Compute the value of $\sin{18^\circ}$ using regular geometry.
Suppose the point $F$ is inside a square $ABCD$ such that $BF=1$, $FA=2$, and $FD=3$, as shown. Find the measurement of $\angle{BFA}$.
Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$
Solve in integers the equation $$(x+y)^2 = x^3 + y^3$$
Let $k$ be a positive integer, show that $(4k+3)$ cannot be a square number.
How many numbers in this series are squares? $$1, 14, 144, 1444, 14444, \cdots$$
Find all positive integer $n$ such that $n$ is a square and its last four digits are the same.
Solve the following equation in positive integers: $15x - 35y + 3 = z^2$
Find a four-digit square number whose first two digits are the same and the last two digits are the same too.
Solve the following equation in positive integers: $3\times (5x + 1)=y^2$
Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.
Let $A$ and $B$ be two positive integers and $A=B^2$. If $A$ satisfies the following conditions, find the value of $B$:
- $A$'s thousands digit is $4$
- $A$'s tens digit is $9$
- The sum of all $A$'s digits is $19$
If the middle term of three consecutive integers is a perfect square, then the product of these three numbers is called a $\textit{beautiful}$ number. What is the greatest common divisor of all the $\textit{beautiful}$ numbers?
Find the smallest square whose last three digits are the same but not equal $0$.
Let both $A$ and $B$ be two-digit numbers, and their difference is $14$. If the last two digits of $A^2$ and $B^2$ are the same, what are all the possible values of $A$ and $B$.
Find such a positive integer $n$ such that both $(n-100)$ and $(n-63)$ are square numbers.
Find such a positive integer $n$ such that both $(n+23)$ and $(n-30)$ are square numbers.
Find the smallest positive integer $n$ such that $\frac{12!}{n}$ is a square.
Find all the integer solutions to the equation $xy - 10(x+ y)= 1$.
Solve in integers the equation $x^2 - xy +2x -3 y = 0$
Solve the following system in integers:
$$
\left\{
\begin{array}{ll}
x_1 + x_2 + \cdots + x_n &= n \\
x_1^2 + x_2^2 + \cdots + x_n^2 &= n \\
\cdots\\
x_1^n + x_2^n + \cdots + x_n^n &= n
\end{array}
\right.
$$
Show that for any positive integer $n$, the following relationship holds: $$2^n+2 > n^2$$
Let $\triangle{ABC}$ be an acute triangle. If the distance between the vertex $A$ and the orthocenter $H$ is equal to the radius of its circumcircle, find the measurement of $\angle{A}$.
Let $AD$ be the altitude in $\triangle{ABC}$ from the vertex $A$. If $\angle{A}=45^\circ$, $BD=3$, $DC=2$, find the area of $\triangle{ABC}$.
Let $O$ be the incenter of $\triangle{ABC}$. Connect $AO$, $BO$, and $CO$ and extends so that they intersect with $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Let $DE$ intersect $AC$ at $G$, and $DF$ intersects $AB$ at $H$. Show that $G$, $H$ and $O$ are collinear.