A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What probability that a reader guesses at random will match three
- all correctly?
- all incorrectly?
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
Determine all roots, real or complex, of the following system
\begin{align}
x+y+z &= 3\\
x^2+y^2+z^2 &= 3\\
x^3+y^3+z^3 &= 3
\end{align}
Given that $x^2+5x+6=20$, find the value of $3x^2 + 15x+17$.
Express $\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}$ in the simplest possible form.
Let $r_1, \cdots, r_5$ be the roots of the polynomial $x^5 + 5x^4 - 79x^3 +64x^2 + 60x+144$. What is $r_1^2 +\cdots + r_5^2$?
Find all pairs of real numbers $(a, b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Find the greatest integer less than $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$$
Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.
Solve this equation $(x-2)(x+1)(x+4)(x+7)=19$.
Let real numbers $x, y,$ and $z$ satisfy $$x+\frac{1}{y}=4\quad\text{,}\quad y+\frac{1}{z}=1\quad\text{,}\quad z +\frac{1}{x}=\frac{7}{3}$$ Find the value of $xyz$.
Find the range of real number $a$ if equation $\mid\frac{x^2}{x-1}\mid=a$ has exactly two distinct real roots.
Solve this equation $$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$$
Let non-zero real numbers $a, b, c$ satisfy $a+b+c\ne 0$. If the following relations hold $$\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}$$
Find the value of $$\frac{(a+b)(b+c)(c+a)}{abc}$$
Solve this equation $2x^4 + 3x^3 -16x^2+3x + 2 =0$.
Solve this equation: $(x^2-x-1)^{x+2}=1$.
The sum of two positive integers is $2310$. Show that their product is not divisible by $2310$.
Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.
Suppose $a, b, c$ are all real numbers. If the quadratic polynomial $P(x)=ax^2 + bx + c$ satisfies the condition that $\mid P(x)\mid \le 1$ for all $-1 \le x \le 1$, find the maximum possible value of $b$.
Let $a_1=a_2=1$ and $a_{n}=(a_{n-1}^2+2)/a_{n-2}$ for $n=3, 4, \cdots$. Show that $a_n$ is an integer for $n=3, 4, \cdots$.
Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.
Let $a, b, c, p$ be real numbers, with $a, b, c$ not all equal, such that $$a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=p$$ Determine all possible values of $p$ and prove $abc+p=0$.
Show that if a polynomial $P(x)$ satisfies $P(2x^2-1)=(P(x))^2/2$, then it must be a constant.
Compute $$S_n=\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\cdots+\frac{n+1}{2^n}$$
Suppose sequence $\{a_n\}$ satisfies $a_1=0$, $a_2=1$, $a_3=9$, and $S_n^2S_{n-2}=10S_{n-1}^3$ for $n > 3$ where $S_n$ is the sum of the first $n$ terms of this sequence. Find $a_n$ when $n\ge 3$.