Practice With Solutions
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Compute the value of $$\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}$$
Show that $1\cdot 1! + 2\cdot 2! + \cdots + n\cdot n! = (n+1)!-1$
As shown, points $X$ and $Y$ are on the extension of $BC$ in $\triangle{ABC}$ such that the order of these four points are $X$, $B$, $C$, and $Y$. Meanwhile, they satisfy the relation $BX\cdot AC = CY\cdot AB$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle{ACX}$ and $\triangle{ABY}$, respectively. If $O_1O_2$ intersects $AB$ and $AC$ at $U$ and $V$, respectively, show that $\triangle{AUV}$ is isosceles.
Let real numbers $a_1$, $a_2$, $\cdots$, $a_{2016}$ satisfy $9a_i\ge 11a_{i+1}^2$ for $i=1, 2,\cdots, 2015$. Define $a_{2017}=a_1$, find the maximum value of $$P=\displaystyle\prod_{i=1}^{2016}(a_i-a_{i+1}^2)$$
Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$
Let point $P$ inside an equilateral $\triangle{ABC}$ such that $AP=3$, $BP=4$, and $CP=5$. Find the side length of $\triangle{ABC}$.
Let $M$ be a point inside $\triangle{ABC}$. Draw $MA'\perp BC$, $MB'\perp CA$, and $MC'\perp AB$ such that $BA'=BC'$ and $CA'=CB'$. Prove $AB'=AC'$.
Four sides of a concyclic quadrilateral have lengths of 25, 39, 52, and 60, in that order. Find the circumference of its circumcircle.
Let $a$ and $b$ be the two roots of $x^2 - 3x -1=0$. Try to solve the following problems without computing $a$ and $b$:
1) Find a quadratic equation whose roots are $a^2$ and $b^2$
2) Find the value of $\frac{1}{a+1}+\frac{1}{b+1}$
3) Find the recursion relationship of $x_n=a^n + b^n$
Find as many different solutions as possible.
Three of the roots of $x^4 + ax^2 + bx + c = 0$ are $2$, $−3$, and $5$. Find the value of $a + b + c$.
In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that
$$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$
Suppose $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are all positive real numbers. If both $a_1x^2 +b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=$ are solvable in real numbers. Show that their roots must be all negative. Furthermore, prove equation $a_1a_2x^2+b_1b_2x+c_1c_2=0$ has two negative real roots too.
Let $x$, $y$, and $z$ be real numbers satisfying $x=6-y$ and $z^2=xy-9$. Show that $x=y$.
Let $\alpha_n$ and $\beta_n$ be two roots of equation $x^2+(2n+1)x+n^2=0$ where $n$ is a positive integer. Evaluate the following expression $$\frac{1}{(\alpha_3+1)(\beta_3+1)}+\frac{1}{(\alpha_4+1)(\beta_4+1)}+\cdots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}$$
Let real numbers $a$, $b$, and $c$ satisfy
$$
\left\{
\begin{array}{rcl}
a^2 - bc-8a +7&=&0\\
b^2 + c^2 +bc-6a+6&=&0
\end{array}
\right.
$$
Show that $1 \le a \le 9$.
Find one real solution $(a, b, c, d)$ to the following system:
$$
\left\{
\begin{array}{rcl}
a+b+c+d&=&-2\\
ab+ac+ad+bc+bd+cd&=&-3\\
abc+abd+acd+bcd&=&4\\
abcd&=&3
\end{array}
\right.
$$
If $m^2 = m+1, n^2-n=1$ and $m\ne n$, compute $m^7 +n^7$.
Find the range of real number $a$ if the two roots of $x^2+2ax+6-a=0$ satisfy one of the following condition:
- two roots are both greater than 1
- one root is greater than 1 and the other is less than 1
Solve equation $(6x+7)^2(3x+4)(x+1)=6$ in real numbers.
If $x^2 + 11x+16=0, y^2 + 11y+16=0$, and $x\ne y$, what is the value of $$\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}$$
Let $x_1$ and $x_2$ be two real roots of $x^2-x-1=0$. Find the value of $2x_1^5 + 5x_2^3$.
Find integer $m$ such that the equation $x^2+mx-m+1=0$ has two positive integer roots.
Let $\alpha$ and $\beta$ be two real roots of $x^4 +k=3x^2$ and also satisfy $\alpha + \beta = 2$. Find the value of $k$.
While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first
point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For
example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later
in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically
reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a
player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his
or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and
finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How
many possible games could they have played? Two games are considered the same if and only if they
include the exact same sequence of scoring.
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?