Find the value of \(m\) such that \(2 x^2+3 x+m\) has a minimum value of \(9\) .
(A) \(\frac{9}{8}\)
(B) \(-\frac{9}{8}\)
(C) \(\frac{81}{8}\)
(D) \(-\frac{81}{8}\)
(E) \(\frac{63}{8}\)
If \(\log {\sqrt{2}} x=10-3 \log {\sqrt{2}} 10\), find \(x\).
(A) 0.32
(B) 0.032
(C) 0.0032
(D) 0.64
(E) 0.064
If \((x-5)^2+(y-5)^2=5^2\), find the smallest value of \((x+5)^2+(y+5)^2\).
(A) \(225-100 \sqrt{2}\)
(B) \(225+100 \sqrt{2}\)
(C) \(225 \sqrt{2}\)
(D) \(100 \sqrt{2}\)
(E) None of the above
Suppose the roots of \(x^2+11 x+3=0\) are \(p\) and \(q\), and the roots of \(x^2+B x-C=0\) are \(p+1\) and \(q+1\). Find \(C\).
If the smallest possible value of \((A-x)(23-x)(A+x)(23+x)\) is \(-(48)^2\), find the value of \(A\) if \(A>0\).
Consider the following simultaneous equations:
\( x y^2+x y z=91\)
\( x y z-y^2 z=72\),
where \(x, y\), and \(z\) are positive integers. Find the maximum value of \(x z\).
A sequence \(a_1, a_2, \ldots\), is defined by
\(a_1=5, a_2=7, a_{n+1}=\frac{a_n+1}{a_{n-1}} \text { for } n \geq 2\) .
Find the value of \(100 \times a_{2023}\)
Let \(C\) be a constant such that the equation \(5 \cos x+6 \sin x-C=0\) have two distint roots \(a\) and \(b\), where \(0<b<a<\pi\). Find the value of \(61 \times \sin (a+b)\).
Let \(f(x)=\cos ^2\left(\frac{\pi x}{2}\right)\). Find the value of
\(f \left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\cdots+f\left(\frac{2021}{2023}\right)+f\left(\frac{2022}{2023}\right)\) .
Suppose that there exist numbers \(a, b, c\) and a function \(f\) such that for any real numbers \(x\) and \(y\),
\(f(x+y)+f(x-y)=2 f(x)+2 f(y)+a x+b y+c\) .
It is given that
\(f(2)=3, \quad f(3)=-5, \quad\)and \(\quad f(5)=7\) . Find the value of \(f(123)\).
Let \(f\) be a function such that for any nonzero number \(x\),
\(6 x f(x)+5 x^2 f(1 / x)+10=0\) .
Find the value of \(f(10)\).
Suppose the roots of \(\frac{x^2}{2}+m x+n=0\) are \(\frac{m}{2}\) and \(\frac{n}{3}\). Find the smallest value of \(mn\).
(A) -1080
(B) -90
(C) 0
(D) 90
(E) 1080
Suppose \(x^{20}+\frac{x^{10}}{2}-\frac{3^{2 x}}{9}+\frac{1}{16}=0\) for some positive real number \(x\). Find the value of
\(4 \cdot 3^x-12 x^{10}\).
How many distinct terms are there if \(\left(x^2+y^2\right)^{11}\left(x^{11}+y^{11}\right)^9\) is algebraically expanded and simplified?
If \(\sqrt{x^2+7 x-4}+\sqrt{x^2-x+4}=x-1\), find the value of \(3 x^2+14 x\).
If \(\frac{12}{x}+\frac{48}{y}=1\), where \(x\) and \(y\) are positive real numbers, find the smallest possible value of \(x+y\).
Find the largest value of \(40 x+60 y\) if \(x-y \leq 2,5 x+y \geq 5\) and \(5 x+3 y \leq 15\).
Find the largest positive integer \(n\) for which \(\frac{20 n+2020}{3 n-6}\) is a positive integer.
Let \(f(x)\) be a function such that \(3 f\left(x^2\right)+f(13-4 x)=3 x^2-4 x+293\)
for all real number \(x\). Find the value of \(f(1)\).
Let \(p\) be a real number such that the equation \(x^2-10 x=p\) has no real solution. of the following is true?
(A) \(0<p<25\)
(B) \(p = 25\)
(C) \(p>25\)
(D) \(p<-25\)
(E) \(-25<p<0\)
If \(\cos A-\cos B=\frac{1}{2}\) and \(\sin A-\sin B=-\frac{1}{4}\), find the value of \(100 \sin (A+B)\).
Find the sum of all the solutions to the equation \(\sqrt[3]{x-110}-\sqrt[3]{x-381}=1\) .
If \(f(x)=\left(2 x+4+\frac{x-2}{x+3}\right)^2\), where \(-2 \leq x \leq 2\), find the maximum value of \(f(x)\).
Given that \(D=\sqrt{\sqrt{x^2+(y-1)^2}+\sqrt{(x-1)^2+y^2}}\) for real values of (x) and (y), find the minimum value of \(D^8\).
A function \(f\) satisfies \(f(x) f(x+1)=x^2+3 x\) for all real numbers \(x\). If \(f(1)+f(2)=\frac{25}{6}) and (0<f(1)<2\), determine the value of \(11 \times f(10)\).
Let \(a_1, a_2, a_3\) be three distinct integers where \(1000>a_1>a_2>a_3>0\). Suppose there exist real numbers \(x, y, z\) such that
\(\left(a_1-a_2\right) y+\left(a_1-a_3\right) z=a_1+a_2+a_3 \)
\( \left(a_1-a_2\right) x+\left(a_2-a_3\right) z=a_1+a_2+a_3 \)
\(\left(a_1-a_3\right) x+\left(a_2-a_3\right) y=a_1+a_2+a_3\) .
Find the largest possible value of \(x+y+z\).
Find the number of real solutions \(x, y\) of the system of equations
\(x^3+y^3+y^2 =0, \)
\(x^2+x^2 y+x y^2 =0\) .
Let \(n\) be a positive integer such that \(\frac{2021 n}{2021+n}\) is also a positive integer. Determine the smallest possible value of (n).
Let \(b\) be a positive integer. If the minimum possible value of the quadratic function \(5 x^2+b x+506\) is \(6\) , find the value of \(b\).
(A) 90
(B) 100
(C) 110
(D) 120
(E) 130
Let \(p=\log _{10}(\sin x), q=(\sin x)^{10}, r=10^{\sin x}\), where \(0<x<\frac{\pi}{2}\). Which of the following is true? following is true?
(A) \(p<q<r\)
(B) \(p<r<q\)
(C) \(q<r<p\)
(D) \(q<p<r\)
(E) \(r<p<q\)
Find the number of ordered pairs \(x, y\), where \(x\) and \(y\) are integers, such that
\(x^2+y^2-20 x-14 y+140<0\) .
If \(\frac{x^2}{5}+\frac{y^2}{7}=1\), find the largest possible value of \((x+y)^2\).
Find the coefficient of \(x^6\) in the expansion of \(\left(1+x+2 x^2\right)^7\).
Suppose \(3 x-y)^2+\sqrt{x+38+14 \sqrt{x-11}}+|z+x-y|=7\). Find the value of \(|x+y+z|\).
Suppose there are real numbers (x, y, z) satisfying the following equations: \(x+y+z=60, x y-z^2=900\) Find the maximum possible value of \(|z|\).
Find the sum \(\sum_{k=1}^{16} \log _2\left(\sqrt{\sin ^2 \frac{k \pi}{8}+1}-\sin \frac{k \pi}{8}\right)\)
The roots of the quadratic equation \(x^2-7 m x+5 n=0\) are \(m\) and \(n\), where \(m \neq 0\) and \(n \neq 0\). Find a quadratic equation whose roots are \(\frac{m}{n}\) and \(\frac{n}{m}\).
(A) \(6 x^2-37 x+1=0\)
(B) \(6 x^2-50 x-7=0\)
(C) \(6 x^2-50 x+7=0\)
(D) \(6 x^2-37 x+6=0\)
(E) \(x^2-37 x+1=0\)
Suppose (x) and (y) are real numbers such that
\[
|x-y|+3 x-y=70, \text { and } \
|y-x|+3 y+x=50 .
\]
Let \(P(x)\) be the polynomial that results from the expansion of the following expression:
\[
\left(2 x^3+3 x^2+x\right)^5\left(\frac{x}{6}+\frac{1}{2}\right)^5 .
\]
Find the sum of the coefficients of \(x^{2 k+1}\), where \(k=0,1,2,3, \ldots, 9\).
Let \(M\) be the maximum possible value of \(\frac{15 x^2-x^3-39 x-54}{x+1}\), where \(x\) is a positive integer. Find the value of \(9 M\).
Find the maximum possible value of \(x+y+z\) where \(x, y, z\) are integers satisfying the following system of equations:
\[
x^2 z+y^2 z+8 x y=200 \
2 x^2+2 y^2+x y z=50 .
\]
Suppose \(\left(\log _2 x\right)^2\)+\(4\left(\log _2\left(\log _3 y\right)\right)^2\) = \(4\left(\log _2 x\right)\left(\log _2\left(\log _3 y\right)\right)\). If \(x = 49\) and \(y\) is a positive integer, find \(y\).
A sequence \(x_0, x_1, x_2, x_3, \ldots\) of integers satisfies the following conditions: \(x_0=1\), and for any positive integer \(n \geq 1,\left|x_n-1\right|=\left|x_{n-1}+2\right|\). Find the maximum possible value of \(2019-\left(x_1+x_2+\cdots+x_{2018}\right)\).
Let \(k\) be a positive integer and let the function \(f\) be defined as follows:
\[
f(x)=\frac{\pi^x}{\pi^x+\pi^{2 k-x}} .
\]
Suppose the function \(g(k)\) is defined as follows:
\[
g(k)=f(0)+f\left(\frac{k}{2019}\right)+f\left(\frac{2 k}{2019}\right)+f\left(\frac{3 k}{2019}\right)+\ldots+f\left(\frac{4037 k}{2019}\right)+f(2 k) .
\]
Find the greatest positive integer \(n\) such that \(g(k) \geq n\) for all \(k \geq 1\).
Suppose (a) and (b) are positive integers satisfying
\(a^2-2 b^2=1\) .
If \(500<a+b<1000\), find \(a+b\).
