Found problems: 236
1981 AMC 12/AHSME, 30
If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are
\[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2}
\]is
$ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$
$ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$
2012 National Olympiad First Round, 23
$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is
$ \textbf{(A)}\ 156 \qquad \textbf{(B)}\ 171 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 186 \qquad \textbf{(E)}\ 201$
2013 Canada National Olympiad, 1
Determine all polynomials $P(x)$ with real coefficients such that
\[(x+1)P(x-1)-(x-1)P(x)\]
is a constant polynomial.
1958 AMC 12/AHSME, 33
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$
2009 AMC 12/AHSME, 19
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?
$ \textbf{(A)}\ 794\qquad
\textbf{(B)}\ 796\qquad
\textbf{(C)}\ 798\qquad
\textbf{(D)}\ 800\qquad
\textbf{(E)}\ 802$
2003 AMC 10, 18
What is the sum of the reciprocals of the roots of the equation
\[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0?
\]
$ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$
2008 Harvard-MIT Mathematics Tournament, 5
Let $ f(x) \equal{} x^3 \plus{} x \plus{} 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) \equal{} \minus{} 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.
2014 AIME Problems, 6
The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
2011 Mediterranean Mathematics Olympiad, 1
A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2010 Stanford Mathematics Tournament, 10
Find the sum of all solutions of the equation
$\frac{1}{x^2-1}+\frac{2}{x^2-2}+\frac{3}{x^2-3}+\frac{4}{x^2-4}=2010x-4$
2000 Turkey Team Selection Test, 1
$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$
$(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$
2022 Bulgarian Autumn Math Competition, Problem 9.1
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
2013 IFYM, Sozopol, 2
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]
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2010 Contests, 1
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
PEN A Problems, 4
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
1988 IMO Longlists, 42
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.
2017 CMIMC Team, 10
The polynomial $P(x) = x^3 - 6x - 2$ has three real roots, $\alpha$, $\beta$, and $\gamma$. Depending on the assignment of the roots, there exist two different quadratics $Q$ such that the graph of $y=Q(x)$ pass through the points $(\alpha,\beta)$, $(\beta,\gamma)$, and $(\gamma,\alpha)$. What is the larger of the two values of $Q(1)$?
2014 Paenza, 3
Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.
1998 All-Russian Olympiad, 1
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.
PEN A Problems, 3
Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.
2013 Korea National Olympiad, 7
For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows:
\[ b_1 = c_1 = 1 \]
\[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \]
\[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \]
Let $a_k = b_{2014} $. Find the value of
\[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]
1958 AMC 12/AHSME, 41
The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of
\[ x^2 \plus{} px \plus{} q \equal{} 0
\]
to be $ r^2$ and $ s^2$, $ p$ must equal:
$ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad
\textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad
\textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\
\textbf{(D)}\ B^2 \minus{} 2C\qquad
\textbf{(E)}\ 2C \minus{} B^2$
2002 AMC 10, 11
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$
1983 AIME Problems, 3
What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]