Found problems: 236
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?
$ \textbf {(A) } 15 \qquad \textbf {(B) } 15.5 \qquad \textbf {(C) } 16 \qquad \textbf {(D) } 16.5 \qquad \textbf {(E) } 17 $
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is:
$ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$
$\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is:
$ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$
Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]
For how many positive real numbers $a$ has the equation $a^2x^2 + ax+1-7a^2 = 0$ two distinct integer roots?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of above}
$
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of
$$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$
Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is
$ \textbf{(A)}\ (\minus{}2,1) \qquad
\textbf{(B)}\ (\minus{}1,2) \qquad
\textbf{(C)}\ (1,\minus{}2) \qquad
\textbf{(D)}\ (2,\minus{}1) \qquad
\textbf{(E)}\ (4,4)$
Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows:
$ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad
\textbf{(B)}\ 2b^2 \equal{} 9ac\qquad
\textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\
\textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad
\textbf{(E)}\ 9b^2 \equal{} 2ac$
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
$\textbf{(A) }6\qquad
\textbf{(B) }12\qquad
\textbf{(C) }24\qquad
\textbf{(D) }48\qquad
\textbf{(E) }78\qquad$
Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be distinct integers such that
\[ (6 \minus{} a)(6 \minus{} b)(6 \minus{} c)(6 \minus{} d)(6 \minus{} e) \equal{} 45.
\]What is $ a \plus{} b \plus{} c \plus{} d \plus{} e?$
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$
Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.
Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is
$\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$
Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that:
\[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3}
\]
[19 points out of 100 for the 6 problems]
Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$.
[i]Aaron Pixton.[/i]
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]
Given is the equation:
\[x^2+mx+2022=0\]
a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers
b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers
The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible?
$ \textbf{(A)}\ 250,\!000 \qquad
\textbf{(B)}\ 250,\!250 \qquad
\textbf{(C)}\ 250,\!500 \qquad
\textbf{(D)}\ 250,\!750 \qquad
\textbf{(E)}\ 251,\!000$