The product of the ages of three adults is 26390 . Find the sum of their ages. (A person is an adult if he or she is at least 21 years old.)
The product of the two-digit number \(\overline{x 4}\) and the three-digit number \(\overline{3 y z}\) is 7656 . Find the value of (x+y+z).
If (x) and (y) are real numbers such that (x+y=12) and (x y=10), find the value of (x^4+y^4).
Find the value of the integer (n) such that the following equation holds:
\([
\frac{\sqrt{5}+n \sqrt{3}-2 \sqrt{2}}{(\sqrt{5}+\sqrt{3})(\sqrt{3}-\sqrt{2})}=\sqrt{5}+\sqrt{2} .
]\)
Let (x) be a real number. What is the minimum value of the following expression?
\([
\frac{20 x^2+10 x+3}{2 x^2+x+1}
]\)
If \(\sqrt{19-8 \sqrt{3}}\) is a root of the equation (x^2-a x+b=0) where (a) and (b) are rational numbers, find the value of (a+b).
Four positive integers (x, y, z) and (w) satisfy the following equations:
\([
\begin{aligned}
& x y+x+y=104 \
& y z+y+z=146 \
& z w+z+w=524
\end{aligned}
]\)
Consider the following product of two mixed fractions
\(m\frac{6}{7} \times n \frac{1}{3}=23\),
where (m) and (n) are positive integers. What is the value of (m+n) ?
If (a) and (b) are distinct solutions to the equation
\(x^2+10 x+20=0\),
what is the value of \(a^4+b^4\) ?
If we have
\(\frac{\sqrt{15}+\sqrt{35}+\sqrt{21}+5}{\sqrt{3}+2 \sqrt{5}+\sqrt{7}}=\frac{a \sqrt{7}+b \sqrt{5}+c \sqrt{3}}{2}\)
for some integers (a, b, c). What is the value of (a+b+c) ?
How many integers (k) are there such that the quadratic equation \(k x^2+20 x+20-k=0\) has only integer solutions?
Let (x) be the positive real number that satisfies \(\sqrt{x^2-4 x+5}+\sqrt{x^2+4 x+5}=3 x \).
What is the value of \(\left\lfloor 10^4 x^2\right\rfloor\) ?
What is the number of positive integers (c) such that the equation \(x^2-2021 x+100 c=0\) has real roots?
Let \(A=\frac{1}{7} \times 3.14 \mathrm{i} \dot{5}), where (3.14 \mathrm{i} \dot{5}\) is the rational number with recurring digits 15 . In other words,
\([
3.14 \dot{1} \dot{5}=3.14+0.0015+0.000015+0.00000015+\cdots .
]\) Suppose that \(A=\frac{m}{n}\), where (m) and (n) are positive integers with no common factors larger than 1 . What is the value of (m+n) ?
If we have
\([
(1-3 x)+(1-3 x)^2+\cdots+(1-3 x)^{100}=a_0+a_1 x+a_2 x^2+\cdots+a_{100} x^{100},
]\)
for some integers (a_0, a_1, \ldots, a_{100}), what is the value of
\([
\left|\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{100}}{3^{100}}\right| ?
]\)
What is the value of
\([
\begin{aligned}
\left(\frac{1}{2}+\frac{1}{3}\right. & \left.+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{37}\right)+\left(\frac{2}{3}+\frac{2}{4}+\frac{2}{5}+\cdots+\frac{2}{37}\right) \
& +\left(\frac{3}{4}+\frac{3}{5}+\frac{3}{6}+\cdots+\frac{3}{37}\right)+\cdots+\left(\frac{35}{36}+\frac{35}{37}\right)+\frac{36}{37} ?
\end{aligned}
]\)
Let \(x, y\) and \(z\) be positive integers satisfying
\(x^2+y^2+z^2=2(x y+1) \quad \text { and } \quad x+y+z=2022\) .
If \(x_1\) and \(x_2\) are two distinct solutions for \(x\), what is the value of \(x_1+x_2\) ?
Suppose that \(m\) and \(n\) are positive integers where \(\frac{100 m}{n}\) is a perfect cube greater than 1 . What is the minimum value of \(m+n\) ?
If the equation \(\frac{x-1}{x-5}=\frac{m}{10-2 x}\) has no solutions in \(x\), what is the value of \(|m| ?\)
If (x) is a nonnegative real number, find the minimum value of
\[
\sqrt{x^2+4}+\sqrt{x^2-24 x+153} .
\]
